Chapter 3

Differential Calculus

[668/69%]
Show
LaTeX
Section 3.1Theory of Limits[116/100%]
Problem 717

Find the limit lim⁡x→0x4+3x2x5+x3+2x2\lim_{x \to 0} \frac{x^{4}+3 x^{2}}{x^{5}+x^{3}+2 x^{2}}.

?
Problem 718

Find the limit lim⁡x→1x4+2x2−3x2−3x+2\lim_{x \to 1} \frac{x^{4}+2 x^{2}-3}{x^{2}-3 x+2}.

?
Problem 719

Find the limit lim⁡x→13x4−4x3+1(x−1)2\lim_{x \to 1} \frac{3 x^{4}-4 x^{3}+1}{(x-1)^{2}}.

?
Problem 720

Find the limit lim⁡x→4x2−6x+8x2−5x+4\lim_{x \to 4} \frac{x^{2}-6 x+8}{x^{2}-5 x+4}.

?
Problem 721

Find the limit lim⁡x→1xn−1x−1\lim_{x \to 1} \frac{x^{n}-1}{x-1}; nn is an integer.

?
Problem 722

Find the limit lim⁡x→1xm−1xn−1\lim_{x \to 1} \frac{x^{m}-1}{x^{n}-1}; mm and nn are integers.

?
Problem 723

Find the limit lim⁡x→1(11−x−31−x3)\lim_{x \to 1} \left(\frac{1}{1-x} - \frac{3}{1-x^{3}}\right).

?
Problem 724

Find the limit lim⁡x→1(a1−xa−b1−xb)\lim_{x \to 1} \left(\frac{a}{1-x^{a}} - \frac{b}{1-x^{b}}\right); aa and bb are integers.

?
Problem 725

Find the limit lim⁡x→1(xn−1)(xn−1−1)…(xn−k+1−1)(x−1)(x2−1)…(xk−1)\lim_{x \to 1} \frac{(x^{n}-1)(x^{n-1}-1) \ldots (x^{n-k+1}-1)}{(x-1)(x^{2}-1) \ldots (x^{k}-1)}.

?
Problem 726

Find the limit lim⁡x→1xp/q−1xr/s−1\lim_{x \to 1} \frac{x^{p/q}-1}{x^{r/s}-1}.

?
Problem 727

Find the limit lim⁡x→01+x3−1x\lim_{x \to 0} \frac{\sqrt[3]{1+x}-1}{x}.

?
Problem 728

Find the limit lim⁡x→−11+x31+x5\lim_{x \to -1} \frac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}.

?
Problem 729

Find the limit lim⁡x→01+xn−1x\lim_{x \to 0} \frac{\sqrt[n]{1+x}-1}{x}.

?
Problem 730

Find the limit lim⁡n→∞n2+n+1(n−1)2\lim_{n \to \infty } \frac{n^{2}+n+1}{(n-1)^{2}}.

?
Problem 731

Find the limit lim⁡n→∞(n+1)(n+2)(n+3)n4+n2+1\lim_{n \to \infty } \frac{(n+1)(n+2)(n+3)}{n^{4}+n^{2}+1}.

?
Problem 732

Find the limit lim⁡n→∞n2+n3n+2\lim_{n \to \infty } \frac{\sqrt[3]{n^{2}+n}}{n+2}.

?
Problem 733

Find the limit lim⁡n→∞n3+1+n4n2+n−n\lim_{n \to \infty } \frac{\sqrt{n^{3}+1}+\sqrt{n}}{4 \sqrt{n^{2}+n}-n}.

?
Problem 734

Find the limit lim⁡x→±∞x2+1x+1\lim_{x \to \pm \infty } \frac{\sqrt{x^{2}+1}}{x+1}.

?
Problem 735

Find the limit lim⁡x→±∞x2+3xx3−2x23\lim_{x \to \pm \infty } \frac{\sqrt{x^{2}+3 x}}{\sqrt[3]{x^{3}-2 x^{2}}}.

?
Problem 736

Find the limit lim⁡n→∞(1+2+3+…+nn+2−n2)\lim_{n \to \infty } \left(\frac{1+2+3+\ldots +n}{n+2} - \frac{n}{2}\right).

?
Problem 737

Find the limit lim⁡n→∞1−2+3−…−2nn2+1\lim_{n \to \infty } \frac{1-2+3-\ldots -2 n}{\sqrt{n^{2}+1}}.

?
Problem 738

Find the limit lim⁡n→∞1n((a+1n)2+(a+2n)2+…+(a+n−1n)2)\lim_{n \to \infty } \frac{1}{n} \left(\left(a+\frac{1}{n}\right)^{2} + \left(a+\frac{2}{n}\right)^{2} + \ldots + \left(a+\frac{n-1}{n}\right)^{2}\right).

?
Remark.
?

In this problem, as well as in the following ones (739--741), it is convenient to use the formula for the sum of squares of the natural numbers:

12+22+32+…+(n−1)2=n(n−1)(2n−1)6, 1^{2}+2^{2}+3^{2}+\ldots +(n-1)^{2} = \frac{n(n-1)(2n-1)}{6},

which is easily proved by mathematical induction.

Problem 739

Find the limit lim⁡n→∞1+22+32+…+n2n3\lim_{n \to \infty } \frac{1+2^{2}+3^{2}+\ldots +n^{2}}{n^{3}}.

?
Problem 740

Find the limit lim⁡n→∞(1+22+…+n2n2−n3)\lim_{n \to \infty } \left(\frac{1+2^{2}+\ldots +n^{2}}{n^{2}} - \frac{n}{3}\right).

?
Problem 741

Find the limit lim⁡n→∞12+32+52+…+(2n−1)2n3\lim_{n \to \infty } \frac{1^{2}+3^{2}+5^{2}+\ldots +(2n-1)^{2}}{n^{3}}.

?
Problem 742

Find the limit lim⁡n→∞1⋅2+2⋅3+…+n(n+1)n3\lim_{n \to \infty } \frac{1 \cdot 2+2 \cdot 3+\ldots +n(n+1)}{n^{3}}.

?
Problem 743

Find the limit lim⁡x→01+x+x2−1x\lim_{x \to 0} \frac{\sqrt{1+x+x^{2}}-1}{x}.

?
Problem 744

Find the limit lim⁡x→01+x−1+x21+x−1\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1+x^{2}}}{\sqrt{1+x}-1}.

?
Problem 745

Find the limit lim⁡x→03+x+x2−9−2x+x2x2−3x+2\lim_{x \to 0} \frac{\sqrt{3+x+x^{2}}-\sqrt{9-2 x+x^{2}}}{x^{2}-3 x+2}.

?
Problem 746

Find the limit lim⁡x→05x1+x3−1−x3\lim_{x \to 0} \frac{5 x}{\sqrt[3]{1+x}-\sqrt[3]{1-x}}.

?
Problem 747

Find the limit lim⁡x→01+3x23−1x2+x3\lim_{x \to 0} \frac{\sqrt[3]{1+3 x^{2}}-1}{x^{2}+x^{3}}.

?
Problem 748

Find the limit lim⁡x→−11+2x3+12+x3+x\lim_{x \to -1} \frac{\sqrt[3]{1+2 x}+1}{\sqrt[3]{2+x}+x}.

?
Problem 749

Find the limit lim⁡x→01+3x3−1−2xx+x2\lim_{x \to 0} \frac{\sqrt[3]{1+3 x}-\sqrt{1-2 x}}{x+x^{2}}.

?
Problem 750

Find the limit lim⁡x→±∞(x2+1−1)\lim_{x \to \pm \infty } \left(\sqrt{x^{2}+1}-1\right).

?
Problem 751

Find the limit lim⁡x→∞(x2+1−x2−1)\lim_{x \to \infty } \left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right).

?
Problem 752

Find the limit lim⁡x→∞(x2+x+1−x2−x+1)\lim_{x \to \infty } \left(\sqrt{x^{2}+x+1}-\sqrt{x^{2}-x+1}\right).

?
Problem 753

Find the limit lim⁡x→∞(1−x33+x)\lim_{x \to \infty } \left(\sqrt[3]{1-x^{3}}+x\right).

?
Problem 754

Find the limit lim⁡x→+∞x(x2+1−x)\lim_{x \to +\infty } x \left(\sqrt{x^{2}+1}-x\right).

?
Problem 755

Find the limit lim⁡x→∞((x+1)2/3−(x−1)2/3)\lim_{x \to \infty } \left((x+1)^{2/3}-(x-1)^{2/3}\right).

?
Problem 756

Find the limit lim⁡x→∞(x4/3−(x2−1)2/3)\lim_{x \to \infty } \left(x^{4/3}-(x^{2}-1)^{2/3}\right).

?
Problem 757

Find the limit lim⁡x→+∞x3/2(x+1+x−1−2x)\lim_{x \to +\infty } x^{3/2} \left(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\right).

?
Problem 758

Find the limit lim⁡x→+∞x3(x2+x4+1−x2)\lim_{x \to +\infty } x^{3} \left(\sqrt{x^{2}+\sqrt{x^{4}+1}}-x\sqrt{2}\right).

?
Problem 759

Find the limit lim⁡x→+∞((x+a1)(x+a2)…(x+an)n−x)\lim_{x \to +\infty } \left(\sqrt[n]{(x+a_{1})(x+a_{2}) \ldots (x+a_{n})}-x\right).

?
Problem 760

Find the limit lim⁡x→∞x4/3(x2+13−x2−13)\lim_{x \to \infty } x^{4/3} \left(\sqrt[3]{x^{2}+1}-\sqrt[3]{x^{2}-1}\right).

?
Problem 761

Determine λ\lambda and μ\mu from the condition

lim⁡x→∞(1−x33−λx−μ)=0. \lim _{x \to \infty } \left(\sqrt[3]{1-x^{3}}-\lambda x-\mu \right) = 0.
?
Problem 762

Determine λ\lambda and μ\mu from the condition

lim⁡x→+∞(∑ν=1naνx2+bνx+cν−λx−μ)=0,aν>0. \lim _{x \to +\infty } \left(\sum _{\nu =1}^{n} \sqrt{a_{\nu } x^{2}+b_{\nu } x+c_{\nu }}-\lambda x-\mu \right) = 0, \quad a_{\nu } > 0.
?
Problem 763

Find the limit lim⁡x→0a+xn−a−xnx\lim_{x \to 0} \frac{\sqrt[n]{a+x}-\sqrt[n]{a-x}}{x} (nn is a positive integer).

?
Problem 764

Find the limit lim⁡x→01+3x+1+x3−1+x5−1+x71+2x4+x−1+x6\lim_{x \to 0} \frac{\sqrt{1+3x}+\sqrt[3]{1+x}-\sqrt[5]{1+x}-\sqrt[7]{1+x}}{\sqrt[4]{1+2x}+x-\sqrt[6]{1+x}}.

?
Problem 765

Find the limit lim⁡x→0a2+ax+x23−a2−ax+x23a+x−a−x\lim_{x \to 0} \frac{\sqrt[3]{a^{2}+a x+x^{2}}-\sqrt[3]{a^{2}-a x+x^{2}}}{\sqrt{a+x}-\sqrt{a-x}}.

?
Problem 766

Find the limit lim⁡x→0(1+x2+x)n−(1+x2−x)nx\lim_{x \to 0} \frac{\left(\sqrt{1+x^{2}}+x\right)^{n}-\left(\sqrt{1+x^{2}}-x\right)^{n}}{x}.

?
Problem 767

Find the limit lim⁡n→∞an1+an\lim_{n \to \infty } \frac{a^{n}}{1+a^{n}}; a>0a > 0.

?
Problem 768

Find the limit lim⁡n→∞an1+a2n\lim_{n \to \infty } \frac{a^{n}}{1+a^{2n}}; a>0a > 0.

?
Problem 769

Find the limit lim⁡n→∞an−a−nan+a−n\lim_{n \to \infty } \frac{a^{n}-a^{-n}}{a^{n}+a^{-n}}; a>0a > 0.

?
Problem 770

Find the limit lim⁡n→∞an(1+a)(1+a2)⋯(1+an)\lim_{n \to \infty } \frac{a^{n}}{(1+a)(1+a^{2}) \cdots (1+a^{n})}; a>0a > 0.

?
Problem 771

Find the limit lim⁡x→0sin⁡4xx\lim_{x \to 0} \frac{\sin 4x}{x}.

?
Problem 772

Find the limit lim⁡x→0sin⁡2xsin⁡3x\lim_{x \to 0} \frac{\sin 2x}{\sin 3x}.

?
Problem 773

Find the limit lim⁡x→0sin⁡5xtg⁡x\lim_{x \to 0} \frac{\sin 5x}{\operatorname {tg} x}.

?
Problem 774

Find the limit lim⁡x→0sin⁡mxsin⁡nx\lim_{x \to 0} \frac{\sin mx}{\sin nx}.

?
Problem 775

Find the limit lim⁡x→πsin⁡mxsin⁡nx\lim_{x \to \pi } \frac{\sin mx}{\sin nx}, where mm and nn are integers.

?
Problem 776

Find the limit lim⁡x→0xctg⁡x\lim_{x \to 0} x \operatorname {ctg} x.

?
Problem 777

Find the limit lim⁡n→∞2nsin⁡x2n\lim_{n \to \infty } 2^{n} \sin \frac{x}{2^{n}}.

?
Problem 778

Find the limit lim⁡x→01−cos⁡xx2\lim_{x \to 0} \frac{1-\cos x}{x^{2}}.

?
Problem 779

Find the limit lim⁡x→0cos⁡mx−cos⁡nxx2\lim_{x \to 0} \frac{\cos mx-\cos nx}{x^{2}}.

?
Problem 780

Find the limit lim⁡x→1(1−x)tg⁡πx2\lim_{x \to 1} (1-x) \operatorname {tg} \frac{\pi x}{2}.

?
Problem 781

Find the limit lim⁡x→0tg⁡x−sin⁡xx3\lim_{x \to 0} \frac{\operatorname {tg} x-\sin x}{x^{3}}.

?
Problem 782

Find the limit lim⁡x→0sin⁡(a+x)−sin⁡(a−x)x\lim_{x \to 0} \frac{\sin (a+x)-\sin (a-x)}{x}.

?
Problem 783

Find the limit lim⁡x→0sin⁡(a+x)+sin⁡(a−x)−2sin⁡ax2\lim_{x \to 0} \frac{\sin (a+x)+\sin (a-x)-2 \sin a}{x^{2}}.

?
Problem 784

Find the limit lim⁡x→0cos⁡(a+x)+cos⁡(a−x)−2cos⁡a1−cos⁡x\lim_{x \to 0} \frac{\cos (a+x)+\cos (a-x)-2 \cos a}{1-\cos x}.

?
Problem 785

Find the limit lim⁡x→∞(sin⁡x+1−sin⁡x)\lim_{x \to \infty } \left(\sin \sqrt{x+1}-\sin \sqrt{x}\right).

?
Problem 786

Find the limit lim⁡x→0cos⁡x−1x2\lim_{x \to 0} \frac{\sqrt{\cos x}-1}{x^{2}}.

?
Problem 787

Find the limit lim⁡x→π/2cos⁡x2−sin⁡x2cos⁡x\lim_{x \to \pi /2} \frac{\cos \frac{x}{2}-\sin \frac{x}{2}}{\cos x}.

?
Problem 788

Find the limit lim⁡x→π/3sin⁡(x−π3)1−2cos⁡x\lim_{x \to \pi /3} \frac{\sin \left(x-\frac{\pi }{3}\right)}{1-2\cos x}.

?
Problem 789

Find the limit lim⁡x→π/42cos⁡x−11−tg⁡2x\lim_{x \to \pi /4} \frac{\sqrt{2}\cos x-1}{1-\operatorname {tg}^{2} x}.

?
Problem 790

Find the limit lim⁡x→01+tg⁡x−1−tg⁡xsin⁡x\lim_{x \to 0} \frac{\sqrt{1+\operatorname {tg} x}-\sqrt{1-\operatorname {tg} x}}{\sin x}.

?
Problem 791

Find the limit lim⁡x→0cos⁡αxm−cos⁡βxmx2\lim_{x \to 0} \frac{\sqrt[m]{\cos \alpha x}-\sqrt[m]{\cos \beta x}}{x^{2}}.

?
Problem 792

Find the limit lim⁡x→0cos⁡x−cos⁡x3sin⁡2x\lim_{x \to 0} \frac{\cos x-\sqrt[3]{\cos x}}{\sin^{2} x}.

?
Problem 793

Find the limit lim⁡x→01−cos⁡xcos⁡2xx2\lim_{x \to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^{2}}.

?
Problem 794

Find the limit lim⁡x→01+xsin⁡x−cos⁡xsin⁡2x2\lim_{x \to 0} \frac{\sqrt{1+x \sin x}-\cos x}{\sin^{2} \frac{x}{2}}.

?
Problem 795

Find the limit lim⁡x→∞(x+1x−1)x\lim_{x \to \infty } \left(\frac{x+1}{x-1}\right)^{x}.

?
Problem 796

Find the limit lim⁡x→∞(2x+32x+1)x+1\lim_{x \to \infty } \left(\frac{2x+3}{2x+1}\right)^{x+1}.

?
Problem 797

Find the limit lim⁡x→∞(x2−1x2)x4\lim_{x \to \infty } \left(\frac{x^{2}-1}{x^{2}}\right)^{x^{4}}.

?
Problem 798

Find the limit lim⁡x→0(1+tg⁡x)ctg⁡x\lim_{x \to 0} (1+\operatorname {tg} x)^{\operatorname {ctg} x}.

?
Problem 799

Find the limit lim⁡x→0(1+3tg⁡2x)ctg⁡2x\lim_{x \to 0} (1+3 \operatorname {tg}^{2} x)^{\operatorname {ctg}^{2} x}.

?
Problem 800

Find the limit lim⁡x→0(cos⁡xcos⁡2x)1/x2\lim_{x \to 0} \left(\frac{\cos x}{\cos 2x}\right)^{1/x^{2}}.

?
Problem 801

Find the limit lim⁡m→∞(cos⁡xm)m\lim_{m \to \infty } \left(\cos \frac{x}{m}\right)^{m}.

?
Problem 802

Find the limit lim⁡m→∞(cos⁡xm)m\lim_{m \to \infty } \left(\cos \frac{x}{\sqrt{m}}\right)^{m}.

?
Problem 803

Find the limit lim⁡m→∞(cos⁡xm+λsin⁡xm)m\lim_{m \to \infty } \left(\cos \frac{x}{m}+\lambda \sin \frac{x}{m}\right)^{m}.

?
Problem 804

Find the limit lim⁡x→a(sin⁡xsin⁡a)1/(x−a)\lim_{x \to a} \left(\frac{\sin x}{\sin a}\right)^{1/(x-a)}.

?
Problem 805

Find the limit lim⁡m→∞(sin⁡(a+bm)sin⁡a)m\lim_{m \to \infty } \left(\frac{\sin \left(a+\frac{b}{m}\right)}{\sin a}\right)^{m}.

?
Problem 806

Find the limit lim⁡x→π/4tg⁡xtg⁡2x\lim_{x \to \pi /4} \operatorname {tg} x^{\operatorname {tg} 2x}.

?
Problem 807

Find the limit lim⁡n→∞(sin⁡n+1n−1π2)tg⁡n+1n−1π2\lim_{n \to \infty } \left(\sin \frac{n+1}{n-1} \frac{\pi }{2}\right)^{\operatorname {tg} \frac{n+1}{n-1} \frac{\pi }{2}}.

?
Problem 808

Find the limit lim⁡x→0(tg⁡(π4+x))ctg⁡2x\lim_{x \to 0} \left(\operatorname {tg}\left(\frac{\pi }{4}+x\right)\right)^{\operatorname {ctg} 2x}.

?
Problem 809

Find the limit lim⁡x→∞(x2+x+12x2−x+1)x2\lim_{x \to \infty } \left(\frac{x^{2}+x+1}{2x^{2}-x+1}\right)^{x^{2}}.

?
Problem 810

Find the limit lim⁡x→π/4(tg⁡(π8+x))tg⁡2x\lim_{x \to \pi /4} \left(\operatorname {tg}\left(\frac{\pi }{8}+x\right)\right)^{\operatorname {tg} 2x}.

?
Problem 811

Find the limit lim⁡n→∞(sin⁡(2n+12n−1⋅π4))tg⁡n+1n+3⋅π2\lim_{n \to \infty } \left(\sin \left(\frac{2n+1}{2n-1} \cdot \frac{\pi }{4}\right)\right)^{\operatorname {tg} \frac{n+1}{n+3} \cdot \frac{\pi }{2}}; nn is a positive integer.

?
Problem 812

Find the limit lim⁡x→eln⁡x−1x−e\lim_{x \to e} \frac{\ln x-1}{x-e}.

?
Problem 813

Find the limit lim⁡x→10lg⁡x−1x−10\lim_{x \to 10} \frac{\lg x-1}{x-10}.

?
Problem 814

Find the limit lim⁡x→π/4ln⁡tg⁡xcos⁡2x\lim_{x \to \pi /4} \frac{\ln \operatorname {tg} x}{\cos 2x}.

?
Problem 815

Find the limit lim⁡n→∞n2ln⁡cos⁡πn\lim_{n \to \infty } n^{2} \ln \cos \frac{\pi }{n}.

?
Problem 816

Find the limit lim⁡x→0ln⁡cos⁡αxln⁡cos⁡βx\lim_{x \to 0} \frac{\ln \cos \alpha x}{\ln \cos \beta x}.

?
Problem 817

Find the limit lim⁡n→∞nln⁡tg⁡(π4+πn)\lim_{n \to \infty } n \ln \operatorname {tg}\left(\frac{\pi }{4}+\frac{\pi }{n}\right).

?
Problem 818

Find the limit lim⁡x→π/4ln⁡tg⁡x1−ctg⁡x\lim_{x \to \pi /4} \frac{\ln \operatorname {tg} x}{1-\operatorname {ctg} x}.

?
Problem 819

Find the limit lim⁡x→+∞ln⁡(ex+1)x\lim_{x \to +\infty } \frac{\ln (e^{x}+1)}{x}.

?
Problem 820

Find the limit lim⁡x→+∞ln⁡(1+eαx)ln⁡(1+eβx)\lim_{x \to +\infty } \frac{\ln (1+e^{\alpha x})}{\ln (1+e^{\beta x})}; α>0\alpha > 0, β>0\beta > 0.

?
Problem 821

Find the limit lim⁡n→∞(a1/n−1)n\lim_{n \to \infty } \left(a^{1/n}-1\right) n.

?
Problem 822

Find the limit lim⁡n→∞n2(an−an+1)\lim_{n \to \infty } n^{2}\left(\sqrt[n]{a}-\sqrt[n+1]{a}\right); a>0a > 0.

?
Problem 823

Find the limit lim⁡n→∞n2(a1/n+a−1/n−2)\lim_{n \to \infty } n^{2} \left(a^{1/n}+a^{-1/n}-2\right).

?
Problem 824

Find the limit lim⁡x→0ax−bxx\lim_{x \to 0} \frac{a^{x}-b^{x}}{x}; a>0a > 0, b>0b > 0.

?
Problem 825

Find the limit lim⁡x→0eαx−eβxx\lim_{x \to 0} \frac{e^{\alpha x}-e^{\beta x}}{x}.

?
Problem 826

Find the limit lim⁡x→0eαx−eβxsin⁡αx−sin⁡βx\lim_{x \to 0} \frac{e^{\alpha x}-e^{\beta x}}{\sin \alpha x-\sin \beta x}.

?
Problem 827

Find the limit lim⁡x→cax−acx−c\lim_{x \to c} \frac{a^{x}-a^{c}}{x-c}; a>0a > 0.

?
Problem 828

Find the limit lim⁡n→∞(1+an)1/n\lim_{n \to \infty } (1+a^{n})^{1/n}; a>0a > 0.

?
Problem 829

Find the limit lim⁡n→∞(an+bn2)n\lim_{n \to \infty } \left(\frac{\sqrt[n]{a}+\sqrt[n]{b}}{2}\right)^{n}; a>0a > 0, b>0b > 0.

?
Problem 830

Find the limit lim⁡n→∞(a1n+a2n+…+amnm)n\lim_{n \to \infty } \left(\frac{\sqrt[n]{a_{1}}+\sqrt[n]{a_{2}}+\ldots +\sqrt[n]{a_{m}}}{m}\right)^{n}; all ai>0a_{i} > 0.

?
Problem 831

Find the limit lim⁡x→0ln⁡(1+x+x2)+ln⁡(1−x+x2)x2\lim_{x \to 0} \frac{\ln (1+x+x^{2})+\ln (1-x+x^{2})}{x^{2}}.

?
Problem 832

Find the limit lim⁡x→∞xln⁡2a+xa+x\lim_{x \to \infty } x \ln \frac{2a+x}{a+x}.

?
Section 3.2Miscellaneous Problems[37/14%]
Problem 833

Prove Bernoulli's inequality:

(1+α)(1+β)(1+γ)…(1+λ)>1+α+β+γ+…+λ, (1+\alpha )(1+\beta )(1+\gamma ) \ldots (1+\lambda ) > 1+\alpha +\beta +\gamma +\ldots +\lambda ,

where the numbers α,β,γ,…,λ\alpha , \beta , \gamma , \ldots , \lambda have the same sign and are greater than −1-1.

?
Problem 834

Using Bernoulli's inequality, prove that for integer nn, (1−1n2)n>1−1n\left(1-\frac{1}{n^{2}}\right)^{n} > 1-\frac{1}{n}, if n>1n > 1.

?
Problem 835

Prove that (1+1n)n>(1+1n−1)n−1\left(1+\frac{1}{n}\right)^{n} > \left(1+\frac{1}{n-1}\right)^{n-1}, if n>1n > 1.

?
Problem 836

Prove that (1+1n)n+1<(1+1n−1)n\left(1+\frac{1}{n}\right)^{n+1} < \left(1+\frac{1}{n-1}\right)^{n}.

?
Problem 837

The quantity (1+1n)n\left(1+\frac{1}{n}\right)^{n} increases, while the quantity (1+1n)n+1\left(1+\frac{1}{n}\right)^{n+1} decreases, as the integer nn increases. From this and from the obvious inequality (1+1n)n<(1+1n)n+1\left(1+\frac{1}{n}\right)^{n} < \left(1+\frac{1}{n}\right)^{n+1}, deduce that both of these quantities tend to a common limit as n→∞n \to \infty. This limit is denoted by ee:

lim⁡n→∞(1+1n)n=e. \lim _{n \to \infty } \left(1+\frac{1}{n}\right)^{n} = e.
?
Problem 838

Prove the equality:

lim⁡n→∞u1+u2+u3+…+unn=lim⁡n→∞un, \lim _{n \to \infty } \frac{u_{1}+u_{2}+u_{3}+\ldots +u_{n}}{n} = \lim _{n \to \infty } u_{n},

assuming that the limit on the right-hand side exists.

?
Problem 839

Prove the equality:

lim⁡n→∞u1+u2+u3+…+unv1+v2+v3+…+vn=lim⁡n→∞unvn, \lim _{n \to \infty } \frac{u_{1}+u_{2}+u_{3}+\ldots +u_{n}}{v_{1}+v_{2}+v_{3}+\ldots +v_{n}} = \lim _{n \to \infty } \frac{u_{n}}{v_{n}},

assuming that lim⁡n→∞unvn\lim_{n \to \infty } \frac{u_{n}}{v_{n}} exists and that v1+v2+v3+…+vn→∞v_{1}+v_{2}+v_{3}+\ldots +v_{n} \to \infty as n→∞n \to \infty, with the numbers vnv_{n} positive.

?
Problem 840

Prove the equality:

lim⁡n→∞unvn=lim⁡n→∞un−un−1vn−vn−1, \lim _{n \to \infty } \frac{u_{n}}{v_{n}} = \lim _{n \to \infty } \frac{u_{n}-u_{n-1}}{v_{n}-v_{n-1}},

assuming that the limit on the right-hand side exists and vn→∞v_{n} \to \infty as n→∞n \to \infty, with vn>vn−1v_{n} > v_{n-1}.

?
Problem 841

For positive integer mm, prove the equality:

lim⁡n→∞1m+2m+3m+…+nmnm+1=1m+1. \lim _{n \to \infty } \frac{1^{m}+2^{m}+3^{m}+\ldots +n^{m}}{n^{m+1}} = \frac{1}{m+1}.
?
Problem 842

Under the same condition, prove the equality:

lim⁡n→∞(1m+2m+3m+…+nmnm−nm+1)=12. \lim _{n \to \infty } \left(\frac{1^{m}+2^{m}+3^{m}+\ldots +n^{m}}{n^{m}} - \frac{n}{m+1}\right) = \frac{1}{2}.
?
Problem 843

Prove the equality:

lim⁡n→∞(1m+3m+5m+…+(2n−1)mnm−2mnm+1)=0; \lim _{n \to \infty } \left(\frac{1^{m}+3^{m}+5^{m}+\ldots +(2n-1)^{m}}{n^{m}} - \frac{2^{m} n}{m+1}\right) = 0;

mm is an integer ≥0\geq 0.

?
Problem 844

Prove the equality:

lim⁡n→∞1m+3m+5m+…+(2n−1)mnm+1=2mnm+1; \lim _{n \to \infty } \frac{1^{m}+3^{m}+5^{m}+\ldots +(2n-1)^{m}}{n^{m+1}} = \frac{2^{m}n}{m+1};

mm is an integer ≥0\geq 0.

?
Problem 845

Find lim⁡n→∞∑k=1n(1+kn2−1)\lim_{n \to \infty } \sum_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^{2}}}-1\right).

?
Problem 846

Find lim⁡n→∞∑k=1n(1+k2n33−1)\lim_{n \to \infty } \sum_{k=1}^{n} \left(\sqrt[3]{1+\frac{k^{2}}{n^{3}}}-1\right).

?
Problem 847

From the inequalities (1+1n)n<e<(1+1n)n+1\left(1+\frac{1}{n}\right)^{n} < e < \left(1+\frac{1}{n}\right)^{n+1} it follows that

1n+1<ln⁡(1+1n)<1n. \frac{1}{n+1} < \ln \left(1+\frac{1}{n}\right) < \frac{1}{n}.

From this, prove the equality:

1+12+13+…+1n=ln⁡n+C+ε(n). 1+\frac{1}{2}+\frac{1}{3}+\ldots +\frac{1}{n} = \ln n+C+\varepsilon (n).

Here CC is a constant called Euler's constant, and ε(n)→0\varepsilon (n) \to 0 as n→∞n \to \infty.

?
Problem 848

Using the previous result, prove the equality:

lim⁡n→∞(1n+1n+1+1n+2+…+12n)=ln⁡2. \lim _{n \to \infty } \left(\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots +\frac{1}{2n}\right) = \ln 2.
?
Problem 849

Starting from the equality lim⁡x→0ln⁡(1+x)x=1\lim_{x \to 0} \frac{\ln (1+x)}{x} = 1, find

lim⁡n→∞(1+1n2)(1+2n2)…(1+nn2). \lim _{n \to \infty } \left(1+\frac{1}{n^{2}}\right)\left(1+\frac{2}{n^{2}}\right) \ldots \left(1+\frac{n}{n^{2}}\right).
?
Problem 850

Using the formula lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1, prove that

lim⁡n→∞(sin⁡an2+sin⁡3an2+…+sin⁡(2n−1)an2)=a. \lim _{n \to \infty } \left(\sin \frac{a}{n^{2}}+\sin \frac{3a}{n^{2}}+\ldots +\sin \frac{(2n-1)a}{n^{2}}\right) = a.
?
Problem 851

Prove that as n→∞n \to \infty

cos⁡ann⋅cos⁡2ann⋯cos⁡nann→e−a2/6. \cos \frac{a}{n\sqrt{n}} \cdot \cos \frac{2a}{n\sqrt{n}} \cdots \cos \frac{na}{n\sqrt{n}} \to e^{-a^{2}/6}.
?
Problem 852

Prove that

lim⁡n→∞1n(cos⁡π4n+cos⁡3π4n+…+cos⁡(2n+1)π4n)=2π. \lim _{n \to \infty } \frac{1}{n} \left(\cos \frac{\pi }{4n}+\cos \frac{3\pi }{4n}+\ldots +\cos \frac{(2n+1)\pi }{4n}\right) = \frac{2}{\pi }.
?
Problem 853

Prove the equality

lim⁡n→∞(1−122)(1−132)…(1−1n2)=12. \lim _{n \to \infty } \left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right) \ldots \left(1-\frac{1}{n^{2}}\right) = \frac{1}{2}.
?
Problem 854

Prove the equality lim⁡n→∞(1−13)(1−16)…(1−1n(n+1)/2)=13\lim_{n \to \infty } \left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right) \ldots \left(1-\frac{1}{n(n+1)/2}\right) = \frac{1}{3}.

?
Problem 855

Prove the equality lim⁡n→∞23−123+1⋅33−133+1⋯n3−1n3+1=23\lim_{n \to \infty } \frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdots \frac{n^{3}-1}{n^{3}+1} = \frac{2}{3}.

?
Problem 856

Prove the equality lim⁡n→∞(1+x)(1+x2)(1+x4)⋯(1+x2n)=11−x\lim_{n \to \infty } (1+x)(1+x^{2})(1+x^{4}) \cdots (1+x^{2^{n}}) = \frac{1}{1-x}; ∣x∣<1\left|x\right| < 1.

?
Problem 857

Prove the equality lim⁡n→∞cos⁡x2⋅cos⁡x4⋯cos⁡x2n=sin⁡xx\lim_{n \to \infty } \cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdots \cos \frac{x}{2^{n}} = \frac{\sin x}{x} (Euler).

?
Problem 858

Prove the equality

lim⁡n→∞22⋅22+2⋯22+2+…+2⏟n roots=π2 \lim _{n \to \infty } \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{2+\sqrt{2}}} \cdots \frac{2}{\underbrace{\sqrt{2+\sqrt{2+\sqrt{\ldots +\sqrt{2}}}}}_{n \text{ roots}}} = \frac{\pi }{2}

(Viete).

?
Problem 859

A sequence of values unu_{n} is given by the formulas: u1=2u_{1} = \sqrt{2}, u2=2+2u_{2} = \sqrt{2+\sqrt{2}}, u3=2+2+2u_{3} = \sqrt{2+\sqrt{2+\sqrt{2}}} and in general un=2+un−1u_{n} = \sqrt{2+u_{n-1}}. Replacing the last 2 under the radical by 4, verify that unu_{n} is bounded. Having proved that lim⁡n→∞un\lim_{n \to \infty } u_{n} exists, find it from the equality un2=2+un−1u_{n}^{2} = 2+u_{n-1}.

?
Remark.
?

This limit is found even more easily from the equality un=2cos⁡π2n+1u_{n} = 2\cos \frac{\pi }{2^{n+1}}.

Problem 860

A sequence vnv_{n} is given by the equalities:

v1=a,v2=a+a,v3=a+a+a,…;a>0. v_{1} = \sqrt{a}, \quad v_{2} = \sqrt{a+\sqrt{a}}, \quad v_{3} = \sqrt{a+\sqrt{a+\sqrt{a}}}, \ldots ; \quad a > 0.

Replacing the number aa under the last radical by the quantity a+xa+x, where xx is the positive root of the equation x2=a+xx^{2} = a+x, verify that vnv_{n} is bounded, and prove that vn→xv_{n} \to x as n→∞n \to \infty.

?
Problem 861

A sequence of numbers is given by the equalities

x0=a;x1=b;x2=x0+x12,…,xn=xn−1+xn−22. x_{0} = a; \quad x_{1} = b; \quad x_{2} = \frac{x_{0}+x_{1}}{2}, \ldots , x_{n} = \frac{x_{n-1}+x_{n-2}}{2}.

Find an expression for xnx_{n} in terms of aa and bb, and find lim⁡n→∞xn\lim_{n \to \infty } x_{n}.

?
Problem 862

Numbers aa and bb are given, with a>b>0a > b > 0. New pairs of numbers are then formed by the formulas: an+1=an+bn2a_{n+1} = \frac{a_{n}+b_{n}}{2}, bn+1=2anbnan+bnb_{n+1} = \frac{2 a_{n} b_{n}}{a_{n}+b_{n}}. Here a0=aa_{0} = a and b0=bb_{0} = b. Prove that anbn=aba_{n} b_{n} = ab, that

an−anbnan+anbn=(a−aba+ab)2n \frac{a_{n}-\sqrt{a_{n} b_{n}}}{a_{n}+\sqrt{a_{n} b_{n}}} = \left(\frac{a-\sqrt{ab}}{a+\sqrt{ab}}\right)^{2^{n}}

and, finally, that lim⁡n→∞an=lim⁡n→∞bn=ab\lim_{n \to \infty } a_{n} = \lim_{n \to \infty } b_{n} = \sqrt{ab}.

?
Problem 863

From two given positive numbers aa and bb one can construct a series of new numbers using the equalities: a0=aa_{0} = a, b0=bb_{0} = b, a1=a0+b02a_{1} = \frac{a_{0}+b_{0}}{2}, b1=a0b0b_{1} = \sqrt{a_{0} b_{0}}, and in general, an+1=an+bn2a_{n+1} = \frac{a_{n}+b_{n}}{2}, bn+1=anbnb_{n+1} = \sqrt{a_{n} b_{n}}. Prove that as n→∞n \to \infty the numbers ana_{n} and bnb_{n} tend to a common limit. (Gauss's arithmetic-geometric mean.)

?
Problem 864

Two sequences of numbers are formed by the formulas:

an=an−1+bn−12,bn=anbn−1. a_{n} = \frac{a_{n-1}+b_{n-1}}{2}, \qquad b_{n} = \sqrt{a_{n} b_{n-1}}.

Here a0=aa_{0} = a, b0=bb_{0} = b are given positive numbers. Find lim⁡n→∞an\lim_{n \to \infty } a_{n}, which equals lim⁡n→∞bn\lim_{n \to \infty } b_{n}, setting a=bcos⁡φa = b \cos \varphi for a<ba < b and a=bch⁡φa = b \operatorname {ch} \varphi for a>ba > b.

?
Problem 865

A sequence of numbers is formed by the formulas: xn=12(xn−1+axn−1)x_{n} = \frac{1}{2}\left(x_{n-1}+\frac{a}{x_{n-1}}\right), where a>0a > 0 is a given number and x0x_{0} is an arbitrary positive number. Prove that xn→ax_{n} \to \sqrt{a} as n→∞n \to \infty.

?
Problem 866

The numbers of a sequence are formed according to the following law: x0x_{0} is an arbitrary positive number, and the subsequent ones are found by the formula

xn+1=xn(xn2+3a)3xn2+a, x_{n+1} = \frac{x_{n}(x_{n}^{2}+3a)}{3x_{n}^{2}+a},

where aa is a given positive number. Prove that xn→ax_{n} \to \sqrt{a} as n→∞n \to \infty.

?
Remark.
?

The law forming the numbers xnx_{n} here is more complicated than the previous one, but in return the numbers xnx_{n} approach a\sqrt{a} faster than before, especially if x0x_{0} is taken close to a\sqrt{a}.

Problem 867

Prove that the sequences formed, similarly to the earlier ones, by the formula xn+1=13(2xn+axn2)x_{n+1} = \frac{1}{3}\left(2x_{n}+\frac{a}{x_{n}^{2}}\right) or by the formula xn+1=xn(xn3+2a)2xn3+ax_{n+1} = \frac{x_{n}(x_{n}^{3}+2a)}{2x_{n}^{3}+a}, tend to a3\sqrt[3]{a}.

?
Problem 868

A sequence yny_{n} is defined as follows: y1=x2y_{1} = \frac{x}{2}, where 0<x<10 < x < 1; y2=x2+y122y_{2} = \frac{x}{2}+\frac{y_{1}^{2}}{2}, and in general yn=x2+yn−122y_{n} = \frac{x}{2}+\frac{y_{n-1}^{2}}{2}. Prove that yny_{n} is a bounded monotone quantity, and find lim⁡n→∞yn\lim_{n \to \infty } y_{n}.

?
Problem 869

y1=x2y_{1} = \frac{x}{2}, where 0<x<10 < x < 1; y2=x2−y122y_{2} = \frac{x}{2}-\frac{y_{1}^{2}}{2}; ; yn=x2−yn−122y_{n} = \frac{x}{2}-\frac{y_{n-1}^{2}}{2}. Show that lim⁡n→∞yn\lim_{n \to \infty } y_{n} exists and equals 1+x−1\sqrt{1+x}-1.

?
Section 3.3The Concept of a Function. Continuity. Graphical Representation of Functions[98/26%]
Problem 870

A solid rod ACAC is made up of pieces of length 11 dm, 22 dm, 11 dm and of weight 22 kg, 33 kg, and 11 kg. Find the weight of the segment ABAB of this rod as a function of its length xx, where xx is measured from the point AA — the start of the piece weighing 22 kg.

?
Problem 871

On what intervals can the functions yy, zz, uu, and vv, defined by the equalities

y=1−x2,z=ln⁡(x2−3x+2), y = \sqrt{1-x^{2}}, \qquad z = \ln (x^{2}-3x+2), u=ln⁡(x2−1),v=ln⁡(x+1)+ln⁡(x−1)? u = \ln (x^{2}-1), \qquad v = \ln (x+1)+\ln (x-1)?

be considered?

?
Problem 872

On what intervals can the functions

y=arccos⁡3x,z=arcsin⁡2x1+x2? y = \arccos \frac{3}{x}, \qquad z = \arcsin \frac{2x}{1+x^{2}}?

be considered?

?
Problem 873

On what intervals can the functions defined by the equalities

y={xln⁡x,x≠0;0,x=0;z={12xln⁡x2,x≠0;0,x=0. y = \begin{cases} x \ln x, & x \neq 0; \\ 0, & x = 0; \end{cases} \qquad z = \begin{cases} \frac{1}{2} x \ln x^{2}, & x \neq 0; \\ 0, & x = 0. \end{cases}

be considered?

?
Problem 874

A function is given by the equalities:

f(x)={2x,0≤x<1;3−x,1≤x≤2. f(x) = \begin{cases} 2x, & 0 \leq x < 1; \\ 3-x, & 1 \leq x \leq 2. \end{cases}

Is f(x)f(x) continuous on the whole segment 0≤x≤20 \leq x \leq 2?

?
Problem 875

For what aa is the function yy, given by the equalities

y={xln⁡(x2),x≠0;a,x=0, y = \begin{cases} x \ln (x^{2}), & x \neq 0; \\ a, & x = 0, \end{cases}

continuous on the interval (−∞,∞)(-\infty , \infty )?

?
Problem 876

Denoting by [a]\left[a\right] the integer part of aa, determine for which values of xx the function yy, defined by the equality

y=x−[x]. y = \sqrt{x}-\left[\sqrt{x}\right].

has a discontinuity.

?
Problem 877

For what value of xx does the function e1/xe^{1/x} have a discontinuity, and what is this discontinuity?

?
Problem 878

What discontinuity, and for what xx, does the function

y=11+21/(x−1). y = \frac{1}{1+2^{1/(x-1)}}.

have?

?
Problem 879

For what values of xx do the functions

u=xx2−4,v=x2+3x2−3,w=1x−x3? u = \frac{x}{x^{2}-4}, \quad v = \frac{x^{2}+3}{x^{2}-3}, \quad w = \frac{1}{x-x^{3}}?

have a discontinuity?

?
Problem 880

For what values of xx does the function tg⁡3xtg⁡2x\frac{\operatorname {tg} 3x}{\operatorname {tg} 2x} have a discontinuity?

?
Problem 881

For what values of xx does the function xsin⁡x\frac{x}{\sin x} have a discontinuity?

?
Problem 882

For what values of xx does the function ln⁡ln⁡(1+x2)\ln \ln (1+x^{2}) have a discontinuity?

?
Problem 883

For what values of xx does the function cos⁡πx2x2(x−1)\frac{\cos \frac{\pi x}{2}}{x^{2}(x-1)} have a discontinuity?

?
Problem 884

Does the function defined by the equalities

y={e−1/x2,x≠0;0,x=0? y = \begin{cases} e^{-1/x^{2}}, & x \neq 0; \\ 0, & x = 0? \end{cases}

have a discontinuity?

?
Problem 885

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=x24y = \frac{x^{2}}{4}.

?
Problem 886

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=−x22y = -\frac{x^{2}}{2}.

?
Problem 887

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=x2−3x4y = \frac{x^{2}-3x}{4}.

?
Problem 888

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=4x−x23y = \frac{4x-x^{2}}{3}.

?
Problem 889

Plot point by point the graphs of the functions y=ax2y = ax^{2} for a=4,2,1,12,14a = 4, 2, 1, \frac{1}{2}, \frac{1}{4}.

?
Problem 890

Plot point by point the graphs of the functions y=ax2+x−1y = ax^{2}+x-1 for a=1,14,120a = 1, \frac{1}{4}, \frac{1}{20}, and the line x=y+1x = y+1.

?
Problem 891

Using the identity

ax2+bx+c=a(x+b2a)2+4ac−b24a(a≠0), ax^{2}+bx+c = a\left(x+\frac{b}{2a}\right)^{2}+\frac{4ac-b^{2}}{4a} \quad (a \neq 0),

plot the graph of the function y=ax2+bx+cy = ax^{2}+bx+c.

?
Problem 892

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x310y = \frac{x^{3}}{10}.

?
Problem 893

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x3−9x10y = \frac{x^{3}-9x}{10}.

?
Problem 894

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x410y = \frac{x^{4}}{10}.

?
Problem 895

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x(x−1)(x−2)(x−3)24y = \frac{x(x-1)(x-2)(x-3)}{24}.

?
Problem 896

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x4+15x30y = \frac{x^{4}+15x}{30}.

?
Problem 897

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=4x−5x3+x510y = \frac{4x-5x^{3}+x^{5}}{10}.

?
Problem 898

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x6100y = \frac{x^{6}}{100}.

?
Problem 899

Plot the graph of the entire rational function of degree higher than the second (higher-order parabola) y=x7100y = \frac{x^{7}}{100}.

?
Problem 900

Plot the hyperbola that is the graph of the fractional function y=1xy = \frac{1}{x}.

?
Problem 901

Plot the hyperbola that is the graph of the fractional function y=xx−1y = \frac{x}{x-1}.

?
Problem 902

Plot the hyperbola that is the graph of the fractional function y=x2+1xy = \frac{x^{2}+1}{x}.

?
Problem 903

Plot the hyperbola that is the graph of the fractional function y=2x−1+1x+1y = 2x-1+\frac{1}{x+1}.

?
Problem 904

Plot the graph of the fractional function having an asymptote parallel to the OxOx axis: y=11+x2y = \frac{1}{1+x^{2}} (the witch of Agnesi).

?
Problem 905

Plot the graph of the fractional function having an asymptote parallel to the OxOx axis: y=x1+x2y = \frac{x}{1+x^{2}} (the serpentine of Newton).

?
Problem 906

Plot the graph of the fractional function having an asymptote parallel to the OxOx axis: y=x2−2x+1x2+1y = \frac{x^{2}-2x+1}{x^{2}+1}.

?
Problem 907

Plot the graph of the fractional function having asymptotes parallel to the axes: y=1x2y = \frac{1}{x^{2}}.

?
Problem 908

Plot the graph of the fractional function having asymptotes parallel to the axes: y=11−x2y = \frac{1}{1-x^{2}}.

?
Problem 909

Plot the graph of the fractional function having asymptotes parallel to the axes: y=x3−x2y = \frac{x}{3-x^{2}}.

?
Problem 910

Plot the graph of the fractional function having asymptotes parallel to the axes: y=1x+1x−1+1x−2y = \frac{1}{x}+\frac{1}{x-1}+\frac{1}{x-2}.

?
Problem 911

Plot the graph of the fractional function having asymptotes parallel to the axes: y=1x−1x−1+1x−2y = \frac{1}{x}-\frac{1}{x-1}+\frac{1}{x-2}.

?
Problem 912

Plot the graph of the fractional function having asymptotes parallel to the axes: y=2(3−x2)(5−x2)y = \frac{2}{(3-x^{2})(5-x^{2})}.

?
Problem 913

Plot the graph of the fractional function having a vertical and an oblique asymptote: y=x+1x2y = x+\frac{1}{x^{2}}.

?
Problem 914

Plot the graph of the fractional function having a vertical and an oblique asymptote: y=x+2xx2−1y = x+\frac{2x}{x^{2}-1}.

?
Problem 915

Plot the graph of the irrational function y=±x−2y = \pm \sqrt{x-2} (a parabola).

?
Problem 916

Plot the graph of the irrational function y=±1225−x2y = \pm \frac{1}{2} \sqrt{25-x^{2}} (an ellipse).

?
Problem 917

Plot the graph of the irrational function y=±12x2−1y = \pm \frac{1}{2} \sqrt{x^{2}-1} (a hyperbola).

?
Problem 918

Plot the graph of the irrational function y=±xx2y = \pm \frac{x\sqrt{x}}{2} (Neile's parabola).

?
Problem 919

Plot the graph of the irrational function y=±xx4−xy = \pm x \sqrt{\frac{x}{4-x}} (the cissoid of Diocles).

?
Problem 920

Plot the graph of the irrational function y=±x−1x+1y = \pm \sqrt{\frac{x-1}{x+1}}.

?
Problem 921

Plot the graph of the irrational function y=x2/3y = x^{2/3}.

?
Problem 922

Plot the graph of the irrational function y=±x3/4y = \pm x^{3/4}.

?
Problem 923

Plot the graph of the transcendental function y=axy = a^{x} for a>1a > 1 and for a<1a < 1.

?
Problem 924

Plot the graph of the transcendental function y=e−x2y = e^{-x^{2}}.

?
Problem 925

Plot the graph of the transcendental function y=e1/xy = e^{1/x}.

?
Problem 926

Plot the graph of the transcendental function y=e−1/x2y = e^{-1/x^{2}}.

?
Problem 927

Plot the graph of the transcendental function y=ln⁡xy = \ln x.

?
Problem 928

Plot the graph of the transcendental function y=Asin⁡xy = A \sin x for A=12,1A = \frac{1}{2}, 1, and 22.

?
Problem 929

Plot the graph of the transcendental function y=sin⁡xay = \sin \frac{x}{a} for a=12,1a = \frac{1}{2}, 1, and 22.

?
Problem 930

Plot the graph of the transcendental function y=cos⁡xy = \cos x.

?
Problem 931

Plot the graph of the transcendental function y=sin⁡(x+a)y = \sin (x+a).

?
Problem 932

Plot the graph of the transcendental function y=sin⁡xxy = \frac{\sin x}{x}.

?
Problem 933

Plot the graph of the transcendental function y=e−axsin⁡bxy = e^{-ax} \sin bx.

?
Problem 934

Plot the graph of the transcendental function y=sin⁡x2y = \sin x^{2}.

?
Problem 935

Plot the graph of the transcendental function y=tg⁡xy = \operatorname {tg} x.

?
Problem 936

Plot the graph of the transcendental function y=1cos⁡xy = \frac{1}{\cos x}.

?
Problem 937

Plot the graph of the transcendental function y=xsin⁡1xy = x \sin \frac{1}{x}.

?
Problem 938

Plot the graph of the function y=arcsin⁡xy = \arcsin x.

?
Problem 939

Plot the graph of the function y=arccos⁡xy = \arccos x.

?
Problem 940

Plot the graph of the function y=arctg⁡xy = \operatorname {arctg} x.

?
Problem 941

Plot the graph of the function y=arcctg⁡xy = \operatorname {arcctg} x.

?
Problem 942

Plot the graph of the function y=arctg⁡1xy = \operatorname {arctg} \frac{1}{x}.

?
Problem 943

Plot the graph of the function y=arcctg⁡1xy = \operatorname {arcctg} \frac{1}{x}.

?
Problem 944

Plot the graph of the function y=arcsin⁡(sin⁡x)y = \arcsin (\sin x).

?
Problem 945

Plot the graph of the function y=arccos⁡(cos⁡x)y = \arccos (\cos x).

?
Problem 946

Plot the graph of the function y=arctg⁡(tg⁡x)y = \operatorname {arctg}(\operatorname {tg} x).

?
Problem 947

Prove the equality

arctg⁡x+arctg⁡1x={π2,for x>0;−π2,for x<0. \operatorname {arctg} x+\operatorname {arctg} \frac{1}{x} = \begin{cases} \frac{\pi }{2}, & \text{for } x > 0; \\ -\frac{\pi }{2}, & \text{for } x < 0. \end{cases}
?
Problem 948

Prove the equality arctg⁡x+arctg⁡y=arctg⁡x+y1−xy+επ\operatorname {arctg} x+\operatorname {arctg} y = \operatorname {arctg} \frac{x+y}{1-xy}+\varepsilon \pi, where

{ε=0,if xy<1;ε=−1,if xy>1 and x<0;ε=1,if xy>1 and x>0. \begin{cases} \varepsilon = 0, & \text{if } xy < 1; \\ \varepsilon = -1, & \text{if } xy > 1 \text{ and } x < 0; \\ \varepsilon = 1, & \text{if } xy > 1 \text{ and } x > 0. \end{cases}
?
Problem 949

Prove the equality arcsin⁡x+arcsin⁡y=ηarcsin⁡(x1−y2+y1−x2)+επ\arcsin x+\arcsin y = \eta \arcsin \left(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right)+\varepsilon \pi, where

{η=1,ε=0,if xy<0 or x2+y2≤1;η=−1,ε=−1,if x2+y2>1 and x<0,y<0;η=−1,ε=1,if x2+y2>1 and x>0,y>0. \begin{cases} \eta = 1, \varepsilon = 0, & \text{if } xy < 0 \text{ or } x^{2}+y^{2} \leq 1; \\ \eta = -1, \varepsilon = -1, & \text{if } x^{2}+y^{2} > 1 \text{ and } x < 0, y < 0; \\ \eta = -1, \varepsilon = 1, & \text{if } x^{2}+y^{2} > 1 \text{ and } x > 0, y > 0. \end{cases}
?
Problem 950

Prove the equality

arctg⁡x+arctg⁡1−x1+x={π4,if −1<x<∞;−3π4,if −∞<x<−1. \operatorname {arctg} x+\operatorname {arctg} \frac{1-x}{1+x} = \begin{cases} \frac{\pi }{4}, & \text{if } -1 < x < \infty ; \\ -\frac{3\pi }{4}, & \text{if } -\infty < x < -1. \end{cases}
?
Problem 951

Prove the equality arccos⁡x+arccos⁡(x2+123−3x2)=π3\arccos x+\arccos \left(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^{2}}\right) = \frac{\pi }{3} for 12≤x≤1\frac{1}{2} \leq x \leq 1.

?
Problem 952

Prove the equality arcsin⁡sin⁡x+cos⁡x2=3π4−x\arcsin \frac{\sin x+\cos x}{\sqrt{2}} = \frac{3\pi }{4}-x, if π4<x<5π4\frac{\pi }{4} < x < \frac{5\pi }{4}.

?
Problem 953

Prove the equality π4=4arctg⁡15−arctg⁡1239\frac{\pi }{4} = 4 \operatorname {arctg} \frac{1}{5}-\operatorname {arctg} \frac{1}{239}.

?
Problem 954

Prove the equality 2arctg⁡x+arcsin⁡2x1+x2=π2 \operatorname {arctg} x+\arcsin \frac{2x}{1+x^{2}} = \pi for x>1x > 1.

?
Problem 955

Prove the equality 2x−12−1πarctg⁡(tg⁡2x−12π)=[x]\frac{2x-1}{2}-\frac{1}{\pi } \operatorname {arctg}\left(\operatorname {tg} \frac{2x-1}{2} \pi \right) = \left[x\right], i.e., the integer part of xx, if xx is not an integer.

?
Problem 956

Prove that the sum

arcsin⁡x+3arccos⁡x+arcsin⁡(2x1−x2) \arcsin x+3 \arccos x+\arcsin \left(2x\sqrt{1-x^{2}}\right)

does not depend on xx for x2<12x^{2} < \frac{1}{2}.

?
Problem 957

Solve the equation x3−7x+5=0x^{3}-7x+5 = 0 by plotting the graph of the function y=x3−7x+56y = \frac{x^{3}-7x+5}{6}.

?
Problem 958

Solve the equation x3+3x+2=0x^{3}+3x+2 = 0 by plotting the graph of the function y=x3+3x+210y = \frac{x^{3}+3x+2}{10}.

?
Problem 959

Solve the equation x4−7x−5=0x^{4}-7x-5 = 0 by plotting the graphs of two functions: y=x410y = \frac{x^{4}}{10} and y=7x+510y = \frac{7x+5}{10}.

?
Problem 960

Solve the equation x4−4x3+7x2−3x+2=0x^{4}-4x^{3}+7x^{2}-3x+2 = 0 graphically by plotting the graphs of two functions: y=x2−4x+75y = \frac{x^{2}-4x+7}{5} and y=3x−25x2y = \frac{3x-2}{5x^{2}}.

?
Problem 961

Solve the equation 2x=4x2^{x} = 4x by plotting the graphs of the functions y=2x4y = \frac{2^{x}}{4} and y=xy = x.

?
Problem 962

Solve graphically the system of two equations:

{x2+y=10,x+y2=4. \begin{cases} x^{2}+y = 10, \\ x+y^{2} = 4. \end{cases}
?
Problem 963

Solve the equation x=tg⁡xx = \operatorname {tg} x graphically.

?
Problem 964

The same for the equation ex=sin⁡xe^{x} = \sin x.

?
Problem 965

Find the first two positive roots of the equation xcos⁡x=1x \cos x = 1 by plotting the graphs of the functions y=cos⁡xy = \cos x and y=1xy = \frac{1}{x}.

?
Problem 966

A sequence of numbers is formed according to the following law:

x1=12e−x0,x2=12e−x1,x3=12e−x2,…, x_{1} = \frac{1}{2} e^{-x_{0}}, \qquad x_{2} = \frac{1}{2} e^{-x_{1}}, \qquad x_{3} = \frac{1}{2} e^{-x_{2}}, \qquad \ldots ,

where x0x_{0} is an arbitrary real number. Prove that as n→∞n \to \infty, xn→ξx_{n} \to \xi, where ξ\xi is the unique root of the equation 2ξ=e−ξ2\xi = e^{-\xi }.

?
Problem 967

The numbers of a sequence are formed by the following law:

xn+1=αxn+βγxn+δ, x_{n+1} = \frac{\alpha x_{n}+\beta }{\gamma x_{n}+\delta },

where α,β,γ,δ\alpha , \beta , \gamma , \delta are given real numbers such that αδ−βγ=1\alpha \delta -\beta \gamma = 1, γ≠0\gamma \neq 0; (α+δ)2≥4(\alpha +\delta )^{2} \geq 4. The number x0x_{0} is chosen so that γx0+δ≠0\gamma x_{0}+\delta \neq 0. Find lim⁡n→∞xn\lim_{n \to \infty } x_{n}.

?
Section 3.4Finding Derivatives[117/97%]
Problem 968

Find the derivative of the function y=2x3−5x2+7x−12y = 2x^{3}-5x^{2}+7x-12.

?
Problem 969

Find the derivative of the function y=x4−3x2+17y = x^{4}-3x^{2}+17.

?
Problem 970

Find the derivative of the function y=x−12x2+13x3−14x4y = x-\frac{1}{2}x^{2}+\frac{1}{3}x^{3}-\frac{1}{4}x^{4}.

?
Problem 971

Find the derivative of the function y=x3(x2−1)2y = x^{3}(x^{2}-1)^{2}.

?
Problem 972

Find the derivative of the function y=x−1x+1y = \frac{x-1}{x+1}.

?
Problem 973

Find the derivative of the function y=x1−x2y = \frac{x}{1-x^{2}}.

?
Problem 974

Find the derivative of the function y=3x−1x5y = \frac{3x-1}{x^{5}}.

?
Problem 975

Find the derivative of the function y=x2+x+1x2−x+1y = \frac{x^{2}+x+1}{x^{2}-x+1}.

?
Problem 976

Find the derivative of the function y=xy = \sqrt{x}.

?
Problem 977

Find the derivative of the function y=x3y = \sqrt[3]{x}.

?
Problem 978

Find the derivative of the function y=xxxy = \sqrt{x\sqrt{x\sqrt{x}}}.

?
Problem 979

Find the derivative of the function y=ex(x2−2x+2)y = e^{x}(x^{2}-2x+2).

?
Problem 980

Find the derivative of the function y=xsin⁡xy = x \sin x.

?
Problem 981

Find the derivative of the function y=x2sin⁡x+2xcos⁡x−2sin⁡xy = x^{2} \sin x+2x \cos x-2 \sin x.

?
Problem 982

Find the derivative of the function y=xln⁡x−xy = x \ln x-x.

?
Problem 983

Find the derivative of the function y=x33ln⁡x−19x3y = \frac{x^{3}}{3} \ln x-\frac{1}{9}x^{3}.

?
Problem 984

Find the derivative of the function y=x+ln⁡x−1xy = \sqrt{x}+\ln x-\frac{1}{\sqrt{x}}.

?
Problem 985

Find the derivative of the function y=1sin⁡xy = \frac{1}{\sin x}.

?
Problem 986

Find the derivative of the function y=1cos⁡xy = \frac{1}{\cos x}.

?
Problem 987

Find the derivative of the function y=1ln⁡xy = \frac{1}{\ln x}.

?
Problem 988

Find the derivative of the function y=ex−1ex+1y = \frac{e^{x}-1}{e^{x}+1}.

?
Problem 989

Find the derivative of the function y=ex(sin⁡x−cos⁡x)y = e^{x}(\sin x-\cos x).

?
Problem 990

Find the derivative of the function y=(ax+b)ny = (ax+b)^{n}.

?
Problem 991

Find the derivative of the function y=sin⁡3xy = \sin^{3} x.

?
Problem 992

Find the derivative of the function y=(x2−1)5y = (x^{2}-1)^{5}.

?
Problem 993

Find the derivative of the function y=cos⁡5xy = \cos^{5} x.

?
Problem 994

Find the derivative of the function y=3x−5y = \sqrt{3x-5}.

?
Problem 995

Find the derivative of the function y=sin⁡5xy = \sin 5x.

?
Problem 996

Find the derivative of the function y=a2−x2y = \sqrt{a^{2}-x^{2}}.

?
Problem 997

Find the derivative of the function y=e−xy = e^{-x}.

?
Problem 998

Find the derivative of the function y=x1−x2y = \frac{x}{\sqrt{1-x^{2}}}.

?
Problem 999

Find the derivative of the function y=ln⁡sin⁡xy = \ln \sin x.

?
Problem 1000

Find the derivative of the function y=1(1+x2)1+x2y = \frac{1}{(1+x^{2})\sqrt{1+x^{2}}}.

?
Problem 1001

Find the derivative of the function y=ln⁡tg⁡xy = \ln \operatorname {tg} x.

?
Problem 1002

Find the derivative of the function y=xx2+1y = x\sqrt{x^{2}+1}.

?
Problem 1003

Find the derivative of the function y=e−x2y = e^{-x^{2}}.

?
Problem 1004

Find the derivative of the function y=x+xy = \sqrt{x+\sqrt{x}}.

?
Problem 1005

Find the derivative of the function y=ln⁡(x3+x2)y = \ln (x^{3}+x^{2}).

?
Problem 1006

Find the derivative of the function y=(2x+1)23y = \sqrt[3]{(2x+1)^{2}}.

?
Problem 1007

Find the derivative of the function y=ln⁡(x+x2+a2)y = \ln \left(x+\sqrt{x^{2}+a^{2}}\right).

?
Problem 1008

Find the derivative of the function y=cos⁡ax⋅sin⁡bxy = \cos ax \cdot \sin bx.

?
Problem 1009

Find the derivative of the function y=ln⁡1−x1+xy = \ln \sqrt{\frac{1-x}{1+x}}.

?
Problem 1010

Find the derivative of the function y=eax⋅cos⁡bxy = e^{ax} \cdot \cos bx.

?
Problem 1011

Find the derivative of the function y=ln⁡1−sin⁡x1+sin⁡xy = \ln \sqrt{\frac{1-\sin x}{1+\sin x}}.

?
Problem 1012

Find the derivative of the function y=ex−e−xex+e−xy = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}.

?
Problem 1013

Find the derivative of the function y=ln⁡tg⁡(x2+π4)y = \ln \operatorname {tg}\left(\frac{x}{2}+\frac{\pi }{4}\right).

?
Problem 1014

Find the derivative of the function y=x44((ln⁡x)2−12ln⁡x+18)y = \frac{x^{4}}{4}\left((\ln x)^{2}-\frac{1}{2}\ln x+\frac{1}{8}\right).

?
Problem 1015

Find the derivative of the function y=ln⁡1+1−x2xy = \ln \frac{1+\sqrt{1-x^{2}}}{x}.

?
Problem 1016

Find the derivative of the function y=ln⁡x2+1+x4+3x2+1xy = \ln \frac{x^{2}+1+\sqrt{x^{4}+3x^{2}+1}}{x}.

?
Problem 1017

Find the derivative of the function y=ctg⁡πx+cos⁡πx2sin⁡3πxy = \operatorname {ctg} \pi x+\frac{\cos \pi x}{2 \sin^{3} \pi x}.

?
Problem 1018

Find the derivative of the function y=ex3(x23−2x3+2)y = e^{\sqrt[3]{x}}\left(\sqrt[3]{x^{2}}-2\sqrt[3]{x}+2\right).

?
Problem 1019

Find the derivative of the function y=(x+2a)x2−a2+a2ln⁡(x+x2−a2)y = (x+2a)\sqrt{x^{2}-a^{2}}+a^{2} \ln \left(x+\sqrt{x^{2}-a^{2}}\right).

?
Problem 1020

Find the derivative of the function y=sin⁡(2ln⁡x)−cos⁡(2ln⁡x)x2y = \frac{\sin (2\ln x)-\cos (2\ln x)}{x^{2}}.

?
Problem 1021

Find the derivative of the function y=12ln⁡(2⋅tg⁡x+1+2tg⁡2x)y = \frac{1}{\sqrt{2}} \ln \left(\sqrt{2} \cdot \operatorname {tg} x+\sqrt{1+2\operatorname {tg}^{2} x}\right).

?
Problem 1022

Find the derivative of the function y=ln⁡1+x2+1x−x2+1y = \ln \frac{1+\sqrt{x^{2}+1}}{x}-\sqrt{x^{2}+1}.

?
Problem 1023

Find the derivative of the function y=ln⁡3−x22−x2y = \ln \frac{3-x^{2}}{2-x^{2}}.

?
Problem 1024

Find the derivative of the function y=12a(ln⁡a2+x2a+x−aa+x)y = \frac{1}{2a}\left(\ln \frac{\sqrt{a^{2}+x^{2}}}{a+x}-\frac{a}{a+x}\right).

?
Problem 1025

Find the derivative of the function y=ln⁡11−x4y = \ln \frac{1}{\sqrt{1-x^{4}}}.

?
Problem 1026

Find the derivative of the function y=tg⁡x+tg⁡3x+35tg⁡5x+17tg⁡7xy = \operatorname {tg} x+\operatorname {tg}^{3} x+\frac{3}{5}\operatorname {tg}^{5} x+\frac{1}{7}\operatorname {tg}^{7} x.

?
Problem 1027

Find the derivative of the function y=−cos⁡x2sin⁡2x+12ln⁡tg⁡x2y = -\frac{\cos x}{2\sin^{2} x}+\frac{1}{2}\ln \operatorname {tg} \frac{x}{2}.

?
Problem 1028

Find the derivative of the function y=12tg⁡2x+ln⁡cos⁡xy = \frac{1}{2}\operatorname {tg}^{2} x+\ln \cos x.

?
Problem 1029

Find the derivative of the function y=sin⁡x2cos⁡2x−12ln⁡tg⁡(π4−x2)y = \frac{\sin x}{2\cos^{2} x}-\frac{1}{2}\ln \operatorname {tg}\left(\frac{\pi }{4}-\frac{x}{2}\right).

?
Problem 1030

Find the derivative of the function y=tg⁡x⋅ln⁡cos⁡x+tg⁡x−xy = \operatorname {tg} x \cdot \ln \cos x+\operatorname {tg} x-x.

?
Problem 1031

Find the derivative of the function y=ln⁡a+bex−aa+bex+ay = \ln \frac{\sqrt{a+be^{x}}-\sqrt{a}}{\sqrt{a+be^{x}}+\sqrt{a}}.

?
Problem 1032

Find the derivative of the function y=ln⁡ex+2+e2x+4ex+1ex+2−e2x+4ex+1y = \ln \frac{e^{x}+2+\sqrt{e^{2x}+4e^{x}+1}}{e^{x}+2-\sqrt{e^{2x}+4e^{x}+1}}.

?
Problem 1033

Find the derivative of the function y=(x−2)1+ex−ln⁡1+ex−11+ex+1y = (x-2)\sqrt{1+e^{x}}-\ln \frac{\sqrt{1+e^{x}}-1}{\sqrt{1+e^{x}}+1}.

?
Problem 1034

Find the derivative of the function y=ln⁡tg⁡x2−ctg⁡x⋅ln⁡(1+sin⁡x)−xy = \ln \operatorname {tg} \frac{x}{2}-\operatorname {ctg} x \cdot \ln (1+\sin x)-x.

?
Problem 1035

Find the derivative of the function y=ln⁡a+bcos⁡nx+b2−a2sin⁡nxa+bcos⁡nxy = \ln \frac{a+b\cos nx+\sqrt{b^{2}-a^{2}}\sin nx}{a+b\cos nx}.

?
Problem 1036

Find the derivative of the function y=(1−1−x)ln⁡x+21−x−2ln⁡(1+1−x)y = (1-\sqrt{1-x})\ln x+2\sqrt{1-x}-2\ln (1+\sqrt{1-x}).

?
Problem 1037

Find the derivative of the function y=x2ln⁡xa+bx2−1bln⁡a+bx2y = \frac{x^{2}\ln x}{a+bx^{2}}-\frac{1}{b}\ln \sqrt{a+bx^{2}}.

?
Problem 1038

Find the derivative of the function y=xln⁡(x+x+a)−x+ay = \sqrt{x}\ln \left(\sqrt{x}+\sqrt{x+a}\right)-\sqrt{x+a}.

?
Problem 1039

Find the derivative of the function y=1aarctg⁡xay = \frac{1}{a}\operatorname {arctg} \frac{x}{a}.

?
Problem 1040

Find the derivative of the function y=arcsin⁡xay = \arcsin \frac{x}{a}.

?
Problem 1041

Find the derivative of the function y=arcsin⁡1xy = \arcsin \frac{1}{x}.

?
Problem 1042

Find the derivative of the function y=(arcsin⁡x)2y = (\arcsin x)^{2}.

?
Problem 1043

Find the derivative of the function y=arcsin⁡sin⁡xy = \arcsin \sin x.

?
Problem 1044

Find the derivative of the function y=arccos⁡1xy = \arccos \frac{1}{x}.

?
Problem 1045

Find the derivative of the function y=x2−a2+aarcsin⁡axy = \sqrt{x^{2}-a^{2}}+a \arcsin \frac{a}{x}.

?
Problem 1046

Find the derivative of the function y=arccos⁡2−xx2y = \arccos \frac{2-x}{x\sqrt{2}}.

?
Problem 1047

Find the derivative of the function y=arcsin⁡1−x2y = \arcsin \sqrt{1-x^{2}}.

?
Problem 1048

Find the derivative of the function y=arctg⁡x+1x−1y = \operatorname {arctg} \frac{x+1}{x-1}; x≠1x \neq 1.

?
Problem 1049

Find the derivative of the function y=23arctg⁡2x+13y = \frac{2}{\sqrt{3}} \operatorname {arctg} \frac{2x+1}{\sqrt{3}}.

?
Problem 1050

Find the derivative of the function y=arccos⁡(3x−4x3)y = \arccos (3x-4x^{3}).

?
Problem 1051

Find the derivative of the function y=arcsin⁡2x1+x2y = \arcsin \frac{2x}{1+x^{2}}.

?
Problem 1052

Find the derivative of the function y=arcsin⁡1−x21+x2y = \arcsin \frac{1-x^{2}}{1+x^{2}}.

?
Problem 1053

Find the derivative of the function y=13arctg⁡x31−x2y = \frac{1}{\sqrt{3}} \operatorname {arctg} \frac{x\sqrt{3}}{1-x^{2}}.

?
Problem 1054

Find the derivative of the function y=1abarctg⁡(batg⁡x)y = \frac{1}{\sqrt{ab}} \operatorname {arctg}\left(\sqrt{\frac{b}{a}} \operatorname {tg} x\right).

?
Problem 1055

Find the derivative of the function y=23arctg⁡x+13arctg⁡x1−x2y = \frac{2}{3} \operatorname {arctg} x+\frac{1}{3} \operatorname {arctg} \frac{x}{1-x^{2}}.

?
Problem 1056

Find the derivative of the function y=142ln⁡x2+x2+1x2−x2+1+122arctg⁡x21−x2y = \frac{1}{4\sqrt{2}} \ln \frac{x^{2}+x\sqrt{2}+1}{x^{2}-x\sqrt{2}+1}+\frac{1}{2\sqrt{2}} \operatorname {arctg} \frac{x\sqrt{2}}{1-x^{2}}.

?
Problem 1057

Find the derivative of the function y=2a2−b2arctg⁡(a−ba+btg⁡x2)y = \frac{2}{\sqrt{a^{2}-b^{2}}} \operatorname {arctg}\left(\sqrt{\frac{a-b}{a+b}} \operatorname {tg} \frac{x}{2}\right); a>b≥0a > b \geq 0.

?
Problem 1058

Find the derivative of the function y=23arctg⁡(53tg⁡x2+43)y = \frac{2}{3} \operatorname {arctg}\left(\frac{5}{3} \operatorname {tg} \frac{x}{2}+\frac{4}{3}\right).

?
Problem 1059

Find the derivative of the function y=x1−x2arcsin⁡x+ln⁡1−x2y = \frac{x}{\sqrt{1-x^{2}}} \arcsin x+\ln \sqrt{1-x^{2}}.

?
Problem 1060

Find the derivative of the function y=ax−x2−aarctg⁡a−xxy = \sqrt{ax-x^{2}}-a \operatorname {arctg} \sqrt{\frac{a-x}{x}}.

?
Problem 1061

Find the derivative of the function y=ln⁡x2−x+1x−ln⁡(x2−x+1)x+3arctg⁡2x−13y = \ln \frac{\sqrt{x^{2}-x+1}}{x}-\frac{\ln (x^{2}-x+1)}{x}+\sqrt{3} \operatorname {arctg} \frac{2x-1}{\sqrt{3}}.

?
Problem 1062

Find the derivative of the function y=x−1−x⋅arcsin⁡xy = \sqrt{x}-\sqrt{1-x} \cdot \arcsin \sqrt{x}.

?
Problem 1063

Find the derivative of the function y=3x+24x2x−1+34arctg⁡x−1y = \frac{3x+2}{4x^{2}}\sqrt{x-1}+\frac{3}{4} \operatorname {arctg} \sqrt{x-1}.

?
Problem 1064

Find the derivative of the function y=ln⁡1+x2x−1x3arctg⁡x−12x2y = \ln \frac{\sqrt{1+x^{2}}}{x}-\frac{1}{x^{3}} \operatorname {arctg} x-\frac{1}{2x^{2}}.

?
Problem 1065

Find the derivative of the function y=(a−x)(x−b)−(a−b)arctg⁡a−xx−by = \sqrt{(a-x)(x-b)}-(a-b) \operatorname {arctg} \sqrt{\frac{a-x}{x-b}}.

?
Problem 1066

Find the derivative of the function y=arcsin⁡(sin⁡x−cos⁡x)+ln⁡(sin⁡x+cos⁡x+sin⁡2x)y = \arcsin (\sin x-\cos x)+\ln \left(\sin x+\cos x+\sqrt{\sin 2x}\right).

?
Problem 1067

Find the derivative of the function y=2arctg⁡2tg⁡x1−tg⁡x+ln⁡1+2tg⁡x+tg⁡x1−2tg⁡x+tg⁡xy = 2\operatorname {arctg} \frac{\sqrt{2\operatorname {tg} x}}{1-\operatorname {tg} x}+\ln \frac{1+\sqrt{2\operatorname {tg} x}+\operatorname {tg} x}{1-\sqrt{2\operatorname {tg} x}+\operatorname {tg} x}.

?
Problem 1068

Find the derivative of the function y=ln⁡(tg⁡x+2+tg⁡2x)+arctg⁡tg⁡x2+tg⁡2xy = \ln \left(\operatorname {tg} x+\sqrt{2+\operatorname {tg}^{2} x}\right)+\operatorname {arctg} \frac{\operatorname {tg} x}{\sqrt{2+\operatorname {tg}^{2} x}}.

?
Problem 1069

Find the derivative of the function y=xarctg⁡(1+x)+ln⁡(x+2x+x)−xy = x \operatorname {arctg}(1+\sqrt{x})+\ln (x+2\sqrt{x}+x)-\sqrt{x}.

?
Problem 1070

Find the derivative of the function y=xarcsin⁡xx+1+arctg⁡x−xy = x \arcsin \sqrt{\frac{x}{x+1}}+\operatorname {arctg} \sqrt{x}-\sqrt{x}.

?
Problem 1071

Find the derivative of the function y=arcsin⁡xx+ln⁡1+1−x1−1−xy = \frac{\arcsin \sqrt{x}}{\sqrt{x}}+\ln \sqrt{\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}}.

?
Problem 1072

Find the derivative of the function y=(x2+1)(arctg⁡x)2−2x⋅arctg⁡x+ln⁡(1+x2)y = (x^{2}+1)(\operatorname {arctg} x)^{2}-2x \cdot \operatorname {arctg} x+\ln (1+x^{2}).

?
Problem 1073

Find the derivative of the function y=ln⁡1+x44+x1+x44−x−2arctg⁡1+x44xy = \ln \frac{\sqrt[4]{1+x^{4}}+x}{\sqrt[4]{1+x^{4}}-x}-2\operatorname {arctg} \frac{\sqrt[4]{1+x^{4}}}{x}.

?
Problem 1074

Find the derivative of the function y=xxy = x^{x}.

?
Problem 1075

Find the derivative of the function y=xsin⁡xy = x^{\sin x}.

?
Problem 1076

Find the derivative of the function y=∣x∣y = \left|x\right|.

?
Problem 1077

Find the derivative of the function y=x2sin⁡1xy = x^{2} \sin \frac{1}{x}.

?
Problem 1078

From the equality 1+x+x2+…+xn=xn+1−1x−11+x+x^{2}+\ldots +x^{n} = \frac{x^{n+1}-1}{x-1}, find a formula for the sum 1+2x+3x2+…+nxn−11+2x+3x^{2}+\ldots +nx^{n-1}.

?
Problem 1079

From the equality

12+cos⁡x+cos⁡2x+…+cos⁡nx=sin⁡(n+12)x2sin⁡x2 \frac{1}{2}+\cos x+\cos 2x+\ldots +\cos nx = \frac{\sin \left(n+\frac{1}{2}\right)x}{2\sin \frac{x}{2}}

find a formula for sin⁡x+2sin⁡2x+…+nsin⁡nx\sin x+2\sin 2x+\ldots +n\sin nx.

?
Problem 1080

From the equality

cos⁡x+cos⁡3x+…+cos⁡(2n−1)x=sin⁡2nx2sin⁡x \cos x+\cos 3x+\ldots +\cos (2n-1)x = \frac{\sin 2nx}{2\sin x}

obtain a formula for the sum

sin⁡x+3sin⁡3x+…+(2n−1)sin⁡(2n−1)x. \sin x+3\sin 3x+\ldots +(2n-1)\sin (2n-1)x.
?
Problem 1081

From the formula

2m−1sin⁡x⋅sin⁡(x+πm)⋅sin⁡(x+2πm)⋯sin⁡(x+(m−1)πm)=sin⁡mx 2^{m-1} \sin x \cdot \sin \left(x+\frac{\pi }{m}\right) \cdot \sin \left(x+\frac{2\pi }{m}\right) \cdots \sin \left(x+\frac{(m-1)\pi }{m}\right) = \sin mx

derive the formula

ctg⁡x+ctg⁡(x+πm)+ctg⁡(x+2πm)+…+ctg⁡(x+m−1mπ)=mctg⁡mx. \operatorname {ctg} x+\operatorname {ctg}\left(x+\frac{\pi }{m}\right)+\operatorname {ctg}\left(x+\frac{2\pi }{m}\right)+\ldots +\operatorname {ctg}\left(x+\frac{m-1}{m}\pi \right) = m \operatorname {ctg} mx.

Deduce from this that

1sin⁡2x+1sin⁡2(x+πm)+…+1sin⁡2(x+m−1mπ)=m2sin⁡2mx. \frac{1}{\sin ^{2} x}+\frac{1}{\sin ^{2}\left(x+\frac{\pi }{m}\right)}+\ldots +\frac{1}{\sin ^{2}\left(x+\frac{m-1}{m}\pi \right)} = \frac{m^{2}}{\sin ^{2} mx}.
?
Problem 1082

Derive directly from the definition of the derivative the derivatives of the functions: arcsin⁡x\arcsin x; arctg⁡x\operatorname {arctg} x (see problems 948 and 949).

?
Problem 1083

Verify that if xy=1xy = 1, then

dx1+x4+dy1+y4=0. \frac{dx}{\sqrt{1+x^{4}}}+\frac{dy}{\sqrt{1+y^{4}}} = 0.
?
Problem 1084

If x1−y4=y2x\sqrt{1-y^{4}} = y\sqrt{2}, then

dy1−y4=12⋅dx1+x4. \frac{dy}{\sqrt{1-y^{4}}} = \frac{1}{\sqrt{2}} \cdot \frac{dx}{\sqrt{1+x^{4}}}.

Prove this.

?
Section 3.5The Geometric Meaning of the Derivative[29/66%]
Problem 1085

Tangent lines are drawn to the parabola y=4x−x24y = \frac{4x-x^{2}}{4} at the points (0,0)(0,0), (2,1)(2,1), (4,0)(4,0). Find their angles of inclination to the OxOx axis.

?
Remark.
?

If y=f(x)y = f(x) is the equation of a curve, and the function f(x)f(x) has a derivative y′=f′(x)y' = f'(x), then the curve has a tangent line at the point with abscissa xx. If α\alpha is the angle of inclination of the tangent line, i.e., the angle from the positive direction of the OxOx axis to the tangent line, then tg⁡α=f′(x)\operatorname {tg} \alpha = f'(x). By virtue of this, the equation of the tangent line to the curve has the form

Y−y=y′(X−x), Y-y = y'(X-x),

and the equation of the normal is as follows:

Y−y=−1y′(X−x). Y-y = -\frac{1}{y'}(X-x).

Here (x,y)(x,y) is the point of tangency, and (X,Y)(X,Y) is a point on the tangent line or normal.

Problem 1086

Find the angle of inclination of the tangent line to the hyperbola xy=a2xy = a^{2} at the point (a,a)(a,a).

?
Problem 1087

At what angle does the curve y=ln⁡xy = \ln x intersect the OxOx axis?

?
Problem 1088

The same question for the sinusoid y=sin⁡xy = \sin x.

?
Problem 1089

For what value of AA does the sinusoid y=Asin⁡xay = A \sin \frac{x}{a} intersect OyOy at an angle of 45°45°?

?
Problem 1090

For what aa does the curve y=axy = a^{x} intersect the OyOy axis at an angle of π4\frac{\pi }{4}?

?
Problem 1091

At what angle do the curves

y=sin⁡x3,y=x1+x2,y=x3+x2. y = \sin x \sqrt{3}, \quad y = \frac{x}{1+x^{2}}, \quad y = \frac{x}{\sqrt{3+x^{2}}}.

intersect the OyOy axis?

?
Problem 1092

For what value of aa does the curve y=ax−x34y = \frac{ax-x^{3}}{4} intersect the OxOx axis at an angle of 45°45°?

?
Problem 1093

Prove that for the parabola y=ax2y = ax^{2} the subtangent equals half the abscissa.

?
Remark.
?

If M(x,y)M(x,y) is the point of tangency of the curve y=f(x)y=f(x), PP is its projection onto the OxOx axis, and the tangent line and normal to the curve at the point MM meet OxOx at the points TT and NN respectively, then the resulting segments have the names: TPTP — the subtangent, PNPN — the subnormal, TMTM — the tangent segment, MNMN — the normal segment. They are expressed in terms of yy and y′y', taken at the point of tangency, by the formulas:

TP=yy′,PN=yy′,TM=yy′1+y′2,MN=y1+y′2. TP = \frac{y}{y'}, \quad PN = yy', \quad TM = \frac{y}{y'}\sqrt{1+y'^{2}}, \quad MN = y\sqrt{1+y'^{2}}.
Problem 1094

Prove that for the parabola y2=2pxy^{2} = 2px the subnormal is constant, independent of the choice of the point of tangency.

?
Problem 1095

Prove that for the curve y=axy = a^{x} the subtangent has constant length.

?
Problem 1096

Prove that for the parabola y2=2pxy^{2} = 2px the subtangent equals twice the abscissa of the point of tangency.

?
Problem 1097

Prove that for the catenary y=ach⁡xay = a \operatorname {ch} \frac{x}{a} the ordinate is the geometric mean between the normal segment and the quantity aa.

?
Problem 1098

Prove that for the curves y=axny = ax^{n} the ratio of the subtangent to the abscissa of the point of tangency is a constant quantity.

?
Problem 1099

Find the equation of the tangent line to the parabola y=3x−x2y = 3x-x^{2} at the point (1,2)(1,2).

?
Problem 1100

Find the tangent line to the parabola

y=x2−3x+33, y = \frac{x^{2}-3x+3}{3},

parallel to the line y=xy = x.

?
Problem 1101

Prove that the tangent line to the hyperbola xy=a2xy = a^{2} forms with the coordinate axes a triangle of constant area.

?
Problem 1102

Prove that for the astroid

x2/3+y2/3=a2/3 x^{2/3}+y^{2/3} = a^{2/3}

the length of the segment of the tangent line between the axes is a constant quantity.

?
Problem 1103

For what relation between the coefficients does the parabola y=ax2+bx+cy = ax^{2}+bx+c touch the OxOx axis?

?
Problem 1104

The same question for the parabola of 3rd order y=x3+px+qy = x^{3}+px+q.

?
Problem 1105

For what value of aa does the curve y=axy = a^{x} touch the line y=xy = x? Find the point of tangency.

?
Problem 1106

A sequence of numbers is formed according to the following law:

x1=a,x2=aa,x3=aaa,… x_{1} = a, \quad x_{2} = a^{a}, \quad x_{3} = a^{a^{a}}, \ldots

and in general xn=axn−1x_{n} = a^{x_{n-1}}. Prove, by considering the curves y=axy = a^{x} and y=xy = x, that for e−e<a<e1/ee^{-e} < a < e^{1/e} the quantity xnx_{n} tends to a definite limit as n→∞n \to \infty.

?
Problem 1107

An ellipse is given by the equations

{x=acos⁡t,y=bsin⁡t. \begin{cases} x = a\cos t, \\ y = b\sin t. \end{cases}

Find the angle of the tangent line to it with the OxOx axis.

?
Problem 1108

An astroid is given by the equations

{x=acos⁡3t,y=asin⁡3t. \begin{cases} x = a\cos ^{3} t, \\ y = a\sin ^{3} t. \end{cases}

Find the angle of the tangent line to it with the OxOx axis at the point where t=3π4t = \frac{3\pi }{4}.

?
Problem 1109

A cycloid is given by the equations

{x=a(t−sin⁡t),y=a(1−cos⁡t). \begin{cases} x = a(t-\sin t), \\ y = a(1-\cos t). \end{cases}

Find the angle of inclination of the tangent line to it.

?
Problem 1110

A cycloid is formed by a circle rolling along a line. Prove that the normal to the cycloid passes through the point of tangency of the circle and the line.

?
Problem 1111

Find the tangent line to the curve

{x=t2−3t+4,y=t2−4t+4 \begin{cases} x = t^{2}-3t+4, \\ y = t^{2}-4t+4 \end{cases}

at the point (2,1)(2,1).

?
Problem 1112

On the curve

{x=2t3−9t2+12t−1,y=t2+t+1 \begin{cases} x = 2t^{3}-9t^{2}+12t-1, \\ y = t^{2}+t+1 \end{cases}

find the points at which the tangent lines are parallel to the OyOy axis.

?
Problem 1113

Find the slope of the tangent line to the curve

{x=t4−2t3−t2+4t−2,y=t4+2t3−t2−4t−2, \begin{cases} x = t^{4}-2t^{3}-t^{2}+4t-2, \\ y = t^{4}+2t^{3}-t^{2}-4t-2, \end{cases}

if (0,0)(0,0) is the point of tangency.

?
Section 3.6Higher-Order Derivatives[85/62%]
Problem 1114

Show that the curve y=ln⁡xy = \ln x is convex upward.

?
Remark.
?

The derivative of the first derivative, i.e., (y′)′(y')' or (f′(x))′(f'(x))', is called the second derivative and is denoted by y′′y'' or f′′(x)f''(x). The derivative of the second derivative is called the third derivative, and so on. To avoid confusion with an exponent, the order of the derivative is denoted by a number placed in parentheses. Roman numerals are also sometimes used. If at a given point of the curve y=f(x)y = f(x) the quantity y′′>0y'' > 0, then the curve is convex downward, i.e., toward the negative direction of the OyOy axis. If y′′<0y'' < 0, then in the neighborhood of the given point the curve is convex upward. If y′′y'' changes sign as xx passes through a given value, then the point on the curve at that xx is an inflection point.

Problem 1115

Show that the curve y=axy = a^{x} is convex downward.

?
Problem 1116

Show that the sinusoid y=sin⁡xy = \sin x is convex upward where y>0y > 0, and downward where y<0y < 0.

?
Problem 1117

Show that the tangent curve y=tg⁡xy = \operatorname {tg} x is convex downward where y>0y > 0, and upward where y<0y < 0.

?
Problem 1118

Investigate the direction of convexity of the curve y=ax3+bxy = ax^{3}+bx for a>0a > 0.

?
Problem 1119

Find the inflection points of the curve y=e−x2y = e^{-x^{2}}.

?
Problem 1120

The same question for the curve y=11+x2y = \frac{1}{1+x^{2}}.

?
Problem 1121

Find y(5)y^{(5)} for the function y=x5y = x^{5}.

?
Remark.
?

Higher-order derivatives are usually found by successive differentiation. Sometimes a pattern is revealed that makes it possible to write the higher-order derivative without finding the preceding ones. Among such cases is the Leibniz formula for the derivative of order nn of the product of two functions:

(uv)(n)=uv(n)+Cn1u′v(n−1)+Cn2u′′v(n−2)+…+u(n)v. (uv)^{(n)} = uv^{(n)}+C_{n}^{1}u'v^{(n-1)}+C_{n}^{2}u''v^{(n-2)}+\ldots +u^{(n)}v.

Here Cn1,Cn2,…C_{n}^{1}, C_{n}^{2}, \ldots are the binomial coefficients.

Problem 1122

Find y(7)y^{(7)} for the function y=x6y = x^{6}.

?
Problem 1123

Find y(6)y^{(6)} for the function y=(3x+5)2(2x2+3)(x+7)2y = (3x+5)^{2}(2x^{2}+3)(x+7)^{2}.

?
Problem 1124

Find y′′′y''' for the function y=x35y = \sqrt[5]{x^{3}}.

?
Problem 1125

Find y′′′y''' for the function y=x5ln⁡xy = x^{5} \ln x.

?
Problem 1126

Find y′′′y''' for the function y=a3xy = a^{3x}.

?
Problem 1127

Find yIVy^{\mathrm{IV}} for the function y=axmy = \frac{a}{x^{m}}.

?
Problem 1128

Find y′′′y''' for the function y=x3x−1y = \frac{x^{3}}{x-1}.

?
Problem 1129

Find yIVy^{\mathrm{IV}} for the function y=x2e2xy = x^{2}e^{2x}.

?
Problem 1130

Find yIVy^{\mathrm{IV}} for the function y=x2cos⁡3xy = x^{2}\cos 3x.

?
Problem 1131

Find y(50)y^{(50)} for the function y=x2e2xy = x^{2}e^{2x}.

?
Problem 1132

Find y(40)y^{(40)} for the function y=x2sin⁡xy = x^{2}\sin x.

?
Problem 1133

Find yIVy^{\mathrm{IV}} for the function y=exsin⁡xy = e^{x}\sin x.

?
Problem 1134

Find y′′′y''' for the function y=arctg⁡(x+1+x2)y = \operatorname {arctg}\left(x+\sqrt{1+x^{2}}\right).

?
Problem 1135

Find yIVy^{\mathrm{IV}} for the function y=xx2+1+ln⁡(x+x2+1)y = x\sqrt{x^{2}+1}+\ln \left(x+\sqrt{x^{2}+1}\right).

?
Problem 1136

Setting y=exsin⁡xy = e^{x}\sin x, z=excos⁡xz = e^{x}\cos x, prove the equalities y′′=2zy'' = 2z, z′′=−2yz'' = -2y.

?
Problem 1137

Prove that the function y=Ce−x+C1e−2xy = Ce^{-x}+C_{1}e^{-2x} satisfies the equation y′′+3y′+2y=0y''+3y'+2y = 0.

?
Problem 1138

Setting yy equal to one of the functions: sin⁡(marcsin⁡x)\sin (m \arcsin x), cos⁡(marcsin⁡x)\cos (m \arcsin x), sin⁡(marccos⁡x)\sin (m \arccos x), cos⁡(marccos⁡x)\cos (m \arccos x), prove that (1−x2)y′′−xy′+m2y=0(1-x^{2})y''-xy'+m^{2}y = 0.

?
Problem 1139

Prove that the function y=e−xcos⁡xy = e^{-x}\cos x satisfies the equation yIV+4y=0y^{\mathrm{IV}}+4y = 0.

?
Problem 1140

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=1+x1−xy = \frac{1+x}{1-x}.

?
Problem 1141

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=xa+bxy = \frac{x}{a+bx}.

?
Problem 1142

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=αx+βγx+δy = \frac{\alpha x+\beta }{\gamma x+\delta }.

?
Problem 1143

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=1x(x+1)y = \frac{1}{x(x+1)}.

?
Problem 1144

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=xy = \sqrt{x}.

?
Problem 1145

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=sin⁡2xy = \sin^{2} x.

?
Problem 1146

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=x3emxy = x^{3}e^{mx}.

?
Problem 1147

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=x2sin⁡axy = x^{2}\sin ax.

?
Problem 1148

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=x3ln⁡xy = x^{3}\ln x.

?
Problem 1149

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=ln⁡(ax+b)y = \ln (ax+b).

?
Problem 1150

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=ln⁡1+x1−xy = \ln \frac{1+x}{1-x}.

?
Problem 1151

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=sin⁡3axy = \sin^{3} ax.

?
Problem 1152

Find a general expression for the quantity y(n)y^{(n)} for any nn: y=ln⁡(1+x)1+xy = \frac{\ln (1+x)}{1+x}.

?
Problem 1153

Prove the equality (eaxsin⁡(bx+c))(n)=eax(a2+b2)n/2sin⁡(bx+nφ+c)\left(e^{ax}\sin (bx+c)\right)^{(n)} = e^{ax}(a^{2}+b^{2})^{n/2}\sin (bx+n\varphi +c), where

sin⁡φ=ba2+b2,cos⁡φ=aa2+b2. \sin \varphi = \frac{b}{\sqrt{a^{2}+b^{2}}}, \qquad \cos \varphi = \frac{a}{\sqrt{a^{2}+b^{2}}}.
?
Problem 1154

Prove the equality (x4+1x3−x)(n)=(−1)n−1n!(1xn+1−1(x−1)n+1−1(x+1)n+1)\left(\frac{x^{4}+1}{x^{3}-x}\right)^{(n)} = (-1)^{n-1}n!\left(\frac{1}{x^{n+1}}-\frac{1}{(x-1)^{n+1}}-\frac{1}{(x+1)^{n+1}}\right); n>1n > 1.

?
Problem 1155

Prove the equality (sin⁡4x+cos⁡4x)(n)=4n−1cos⁡(4x+nπ2)(\sin^{4} x+\cos^{4} x)^{(n)} = 4^{n-1}\cos \left(4x+\frac{n\pi }{2}\right).

?
Problem 1156

Setting y=cos⁡(mln⁡x)y = \cos (m \ln x), prove that

x2y(n+2)+(2n+1)xy(n+1)+(n2+m2)y(n)=0. x^{2}y^{(n+2)}+(2n+1)xy^{(n+1)}+(n^{2}+m^{2})y^{(n)} = 0.
?
Problem 1157

Setting y=arctg⁡xy = \operatorname {arctg} x, prove that

y(n)=(n−1)!cos⁡ny⋅sin⁡n(y+π2). y^{(n)} = (n-1)! \cos ^{n} y \cdot \sin n\left(y+\frac{\pi }{2}\right).
?
Problem 1158

Setting y=xmln⁡xy = x^{m} \ln x, prove that

y(n)=m(m−1)…(m−n+1)xm−n×(ln⁡x+1m+1m−1+…+1m−n+1), y^{(n)} = m(m-1) \ldots (m-n+1) x^{m-n} \times \left(\ln x+\frac{1}{m}+\frac{1}{m-1}+\ldots +\frac{1}{m-n+1}\right),

for n≤mn \leq m. If n=m+1n = m+1, then

(xmln⁡x)(m+1)=m!x. (x^{m} \ln x)^{(m+1)} = \frac{m!}{x}.
?
Problem 1159

Prove that the derivative y(n)y^{(n)} of the composite function y=f(u)y = f(u), where u=x2u = x^{2}, is expressed by the formula

y(n)=2nxnf(n)+2n−2n(n−1)1xn−2f(n−1)+2n−4n(n−1)(n−2)(n−3)1⋅2xn−4f(n−2)+…. y^{(n)} = 2^{n}x^{n}f^{(n)}+2^{n-2}\frac{n(n-1)}{1}x^{n-2}f^{(n-1)}+2^{n-4}\frac{n(n-1)(n-2)(n-3)}{1 \cdot 2}x^{n-4}f^{(n-2)}+\ldots .
?
Problem 1160

The derivative of order nn of e−x2e^{-x^{2}} has the form e−x2Hn(x)e^{-x^{2}}H_{n}(x), where Hn(x)H_{n}(x) is a polynomial called the Chebyshev--Hermite polynomial. Prove the validity of the equalities:

?
(a)

Hn+1(x)+2xHn(x)+2nHn−1(x)=0H_{n+1}(x)+2xH_{n}(x)+2nH_{n-1}(x) = 0.

(b)

Hn(x)−Hn−1′(x)+2xHn−1(x)=0H_{n}(x)-H_{n-1}'(x)+2xH_{n-1}(x) = 0.

(c)

Hn′′(x)−2xHn′(x)+2nHn(x)=0H_{n}''(x)-2xH_{n}'(x)+2nH_{n}(x) = 0.

Problem 1161

For the Chebyshev--Hermite polynomials, prove the formula

Hn=(−1)n((2x)n−n(n−1)1(2x)n−2+n(n−1)(n−2)(n−3)1⋅2(2x)n−4−…). H_{n} = (-1)^{n}\left((2x)^{n}-\frac{n(n-1)}{1}(2x)^{n-2}+\frac{n(n-1)(n-2)(n-3)}{1 \cdot 2}(2x)^{n-4}-\ldots \right).
?
Problem 1162

The Laguerre polynomials Ln(x)L_{n}(x) are defined by the formula

(xne−x)(n)=e−xLn(x). (x^{n}e^{-x})^{(n)} = e^{-x}L_{n}(x).

Prove the equalities:

xLn′′(x)+(1−x)Ln′(x)+nLn(x)=0, xL_{n}''(x)+(1-x)L_{n}'(x)+nL_{n}(x) = 0, Ln+1(x)−(2n+1−x)Ln(x)+n2Ln−1(x)=0, L_{n+1}(x)-(2n+1-x)L_{n}(x)+n^{2}L_{n-1}(x) = 0, Ln(x)=(−1)n(xn−n21xn−1+n2(n−1)21⋅2xn−2−…). L_{n}(x) = (-1)^{n}\left(x^{n}-\frac{n^{2}}{1}x^{n-1}+\frac{n^{2}(n-1)^{2}}{1 \cdot 2}x^{n-2}-\ldots \right).
?
Problem 1163

The Legendre polynomials are defined by the equality

Pn(x)=12nn!((x2−1)n)(n). P_{n}(x) = \frac{1}{2^{n}n!}\left((x^{2}-1)^{n}\right)^{(n)}.

Prove the equalities:

(x2−1)Pn′′(x)+2xPn′(x)−n(n+1)Pn(x)=0, (x^{2}-1)P_{n}''(x)+2xP_{n}'(x)-n(n+1)P_{n}(x) = 0, (n+1)Pn+1(x)−(2n+1)xPn(x)+nPn−1(x)=0. (n+1)P_{n+1}(x)-(2n+1)xP_{n}(x)+nP_{n-1}(x) = 0.
?
Problem 1164

The Chebyshev polynomials Tn(x)T_{n}(x) are defined by the equality

Tn(x)=12n−1cos⁡(narccos⁡x)=12n−1((x+x2−1)n+(x−x2−1)n) T_{n}(x) = \frac{1}{2^{n-1}}\cos (n \arccos x) = \frac{1}{2^{n-1}}\left(\left(x+\sqrt{x^{2}-1}\right)^{n}+\left(x-\sqrt{x^{2}-1}\right)^{n}\right)

(the polynomials least deviating from zero). Prove the equality

(x2−1)dm+2Tn(x)dxm+2+(2m+1)xdm+1Tn(x)dxm+1+(m2−n2)dmTn(x)dxm=0. (x^{2}-1)\frac{d^{m+2}T_{n}(x)}{dx^{m+2}}+(2m+1)x\frac{d^{m+1}T_{n}(x)}{dx^{m+1}}+(m^{2}-n^{2})\frac{d^{m}T_{n}(x)}{dx^{m}} = 0.
?
Problem 1165

The derivative of order nn of arcsin⁡x\arcsin x has the form

pn(x)(1−x2)1−2n2, p_{n}(x)(1-x^{2})^{\frac{1-2n}{2}},

where pn(x)p_{n}(x) is a polynomial. Prove the equality

(1−x2)pn′′(x)+(2n−3)xpn′(x)−(n−1)2pn(x)=0. (1-x^{2})p_{n}''(x)+(2n-3)xp_{n}'(x)-(n-1)^{2}p_{n}(x) = 0.
?
Problem 1166

For the polynomials of the previous problem, prove the equalities:

p2m+1(x)=A(1+∑ν=1mm2(m−1)2…(m−ν+1)21⋅2⋅3⋯(2ν)(2x)2ν), p_{2m+1}(x) = A\left(1+\sum _{\nu =1}^{m} \frac{m^{2}(m-1)^{2} \ldots (m-\nu +1)^{2}}{1 \cdot 2 \cdot 3 \cdots (2\nu )}(2x)^{2\nu }\right), p2m(x)=Ax(1+∑ν=1m−1(m−1)2(m−2)2…(m−ν)21⋅2⋅3⋯(2ν+1)(2x)2ν). p_{2m}(x) = Ax\left(1+\sum _{\nu =1}^{m-1} \frac{(m-1)^{2}(m-2)^{2} \ldots (m-\nu )^{2}}{1 \cdot 2 \cdot 3 \cdots (2\nu +1)}(2x)^{2\nu }\right).

Here A=(1⋅3⋅5⋯(2m−1))2A = \left(1 \cdot 3 \cdot 5 \cdots (2m-1)\right)^{2}.

?
Problem 1167

The equality

dne1/xdxn=(−1)npn(x)x2ne1/x, \frac{d^{n}e^{1/x}}{dx^{n}} = \frac{(-1)^{n}p_{n}(x)}{x^{2n}}e^{1/x},

holds, where pn(x)p_{n}(x) is a polynomial. Prove the equalities:

?
(a)

x2pn′′−((2n−2)x+1)pn′+n(n−1)pn=0x^{2}p_{n}''-\left((2n-2)x+1\right)p_{n}'+n(n-1)p_{n} = 0,

(b)

pn=1+n1(n−1)x+n(n−1)1⋅2(n−1)(n−2)x2+n(n−1)(n−2)1⋅2⋅3(n−1)(n−2)(n−3)x3+…p_{n} = 1+\frac{n}{1}(n-1)x+\frac{n(n-1)}{1 \cdot 2}(n-1)(n-2)x^{2}+\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3}(n-1)(n-2)(n-3)x^{3}+\ldots.

Problem 1168

Prove the equality dn(xn−1e1/x)dxn=(−1)nxn+1e1/x\frac{d^{n}(x^{n-1}e^{1/x})}{dx^{n}} = \frac{(-1)^{n}}{x^{n+1}}e^{1/x}.

?
Problem 1169

In the equality (11+x2)(n)=pn(x)(1+x2)n+1\left(\frac{1}{1+x^{2}}\right)^{(n)} = \frac{p_{n}(x)}{(1+x^{2})^{n+1}} the numerator pn(x)p_{n}(x) is a polynomial of degree nn. Prove the equalities:

?
(a)

pn+1+(2n+2)xpn+n(n+1)(1+x2)pn−1=0p_{n+1}+(2n+2)xp_{n}+n(n+1)(1+x^{2})p_{n-1} = 0,

(b)

dpndx+n(n+1)pn−1=0\frac{dp_{n}}{dx}+n(n+1)p_{n-1} = 0,

(c)

(1+x2)pn′′−2nxpn′+n(n+1)pn=0(1+x^{2})p_{n}''-2nxp_{n}'+n(n+1)p_{n} = 0.

Problem 1170

For the preceding polynomials, prove the formulas:

p2m=(−1)m(2m)!((1+xi)2m+1+(1−xi)2m+12), p_{2m} = (-1)^{m}(2m)!\left(\frac{(1+xi)^{2m+1}+(1-xi)^{2m+1}}{2}\right), p2m−1=(−1)m(2m−1)!((1+xi)2m−(1−xi)2m2i). p_{2m-1} = (-1)^{m}(2m-1)!\left(\frac{(1+xi)^{2m}-(1-xi)^{2m}}{2i}\right).
?
Problem 1171

A curve is given by the equations

{x=eαtcos⁡βt,y=eαtsin⁡βt. \begin{cases} x = e^{\alpha t}\cos \beta t, \\ y = e^{\alpha t}\sin \beta t. \end{cases}

Find yx′′y_{x}''.

?
Remark.
?

When the relation between the variables is given parametrically, x=φ(t)x = \varphi (t), y=f(t)y = f(t), the derivatives yx′y_{x}', yx′′y_{x}'', yx′′′y_{x}''', should be found based on the invariance of the first differential: yx′=dydxy_{x}' = \frac{dy}{dx}, yx′′=dyx′dxy_{x}'' = \frac{dy_{x}'}{dx}, yx′′′=dyx′′dxy_{x}''' = \frac{dy_{x}''}{dx}.

Problem 1172

A curve is given by the equations

{x=a(cos⁡t−ln⁡ctg⁡t2),y=asin⁡t. \begin{cases} x = a\left(\cos t-\ln \operatorname {ctg} \frac{t}{2}\right), \\ y = a \sin t. \end{cases}

Find yx′′′y_{x}'''.

?
Problem 1173

A curve is given by the equations

{x=acos⁡3t,y=asin⁡3t. \begin{cases} x = a\cos ^{3} t, \\ y = a\sin ^{3} t. \end{cases}

Find yx′′′y_{x}'''.

?
Problem 1174

A curve is given by the equations

{x=acos⁡tcos⁡2t,y=asin⁡tcos⁡2t. \begin{cases} x = \frac{a\cos t}{\sqrt{\cos 2t}}, \\ y = \frac{a\sin t}{\sqrt{\cos 2t}}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1175

A curve is given by the equations

{x=acos⁡φcos⁡nφn,y=asin⁡φcos⁡nφn. \begin{cases} x = a\cos \varphi \sqrt[n]{\cos n\varphi }, \\ y = a\sin \varphi \sqrt[n]{\cos n\varphi }. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1176

A curve is given by the equations

{x=a(1+cos⁡t)cos⁡t,y=a(1−cos⁡t)sin⁡t. \begin{cases} x = a(1+\cos t)\cos t, \\ y = a(1-\cos t)\sin t. \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1177

A curve is given by the equations

{x=acos⁡t1+2cos⁡t,y=asin⁡t1+2cos⁡t. \begin{cases} x = \frac{a\cos t}{1+2\cos t}, \\ y = \frac{a\sin t}{1+2\cos t}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1178

A curve is given by the equations

{x=acos⁡5t,y=asin⁡5t. \begin{cases} x = a\cos ^{5} t, \\ y = a\sin ^{5} t. \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1179

A curve is given by the equations

{x=a(t+sin⁡t)+bsin⁡t2,y=a(2+cos⁡t)+bcos⁡t2. \begin{cases} x = a(t+\sin t)+b\sin \frac{t}{2}, \\ y = a(2+\cos t)+b\cos \frac{t}{2}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1180

A curve is given by the equations

{x=(2n+1)cos⁡t−cos⁡(2n+1)t,y=(2n+1)sin⁡t−sin⁡(2n+1)t. \begin{cases} x = (2n+1)\cos t-\cos (2n+1)t, \\ y = (2n+1)\sin t-\sin (2n+1)t. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1181

A curve is given by the equations

{x=acos⁡t,y=bsin⁡t. \begin{cases} x = a\cos t, \\ y = b\sin t. \end{cases}

Find d4ydx4\frac{d^{4}y}{dx^{4}}.

?
Problem 1182

A curve is given by the equations

{x=asin⁡t(2−sin⁡2t),y=bsin⁡2t⋅cos⁡t. \begin{cases} x = a\sin t(2-\sin ^{2} t), \\ y = b\sin ^{2} t \cdot \cos t. \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1183

A curve is given by the equations

{x=a(t−sin⁡t),y=acos⁡t. \begin{cases} x = a(t-\sin t), \\ y = a\cos t. \end{cases}

Find d4ydx4\frac{d^{4}y}{dx^{4}}.

?
Problem 1184

A curve is given by the equations

{x=acos⁡t+(at+b)sin⁡t,y=asin⁡t−(at+b)cos⁡t. \begin{cases} x = a\cos t+(at+b)\sin t, \\ y = a\sin t-(at+b)\cos t. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1185

A curve is given by the equations

{x=a(cos⁡t+tsin⁡t),y=a(sin⁡t−tcos⁡t). \begin{cases} x = a(\cos t+t\sin t), \\ y = a(\sin t-t\cos t). \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1186

A curve is given by the equations

{x=a(et(sin⁡t+cos⁡t)−1),y=a(et(sin⁡t−cos⁡t)+1). \begin{cases} x = a\left(e^{t}(\sin t+\cos t)-1\right), \\ y = a\left(e^{t}(\sin t-\cos t)+1\right). \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1187

A curve is given by the equations

{x=aet/2cos⁡t−sin⁡t,y=at−aln⁡cos⁡t−sin⁡t. \begin{cases} x = a \frac{e^{t/2}}{\sqrt{\cos t-\sin t}}, \\ y = at-a\ln \sqrt{\cos t-\sin t}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1188

A curve is given by the equations

{x=2tcos⁡t+(t2−2)sin⁡t,y=2tsin⁡t−(t2−2)cos⁡t. \begin{cases} x = 2t\cos t+(t^{2}-2)\sin t, \\ y = 2t\sin t-(t^{2}-2)\cos t. \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1189

A curve is given by the equations

{x=a(sin⁡tcos⁡2t−ln⁡tg⁡(π4+t2)),y=acos⁡3t. \begin{cases} x = a\left(\frac{\sin t}{\cos ^{2} t}-\ln \operatorname {tg}\left(\frac{\pi }{4}+\frac{t}{2}\right)\right), \\ y = \frac{a}{\cos ^{3} t}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1190

A curve is given by the equations

{x=a(sin⁡tcos⁡2t+ln⁡tg⁡(π4+t2)),y=acos⁡3t. \begin{cases} x = a\left(\frac{\sin t}{\cos ^{2} t}+\ln \operatorname {tg}\left(\frac{\pi }{4}+\frac{t}{2}\right)\right), \\ y = \frac{a}{\cos ^{3} t}. \end{cases}

Find d3ydx3\frac{d^{3}y}{dx^{3}}.

?
Problem 1191

A curve is given by the equations

{x=t+t31+t4,y=t−t31+t4. \begin{cases} x = \frac{t+t^{3}}{1+t^{4}}, \\ y = \frac{t-t^{3}}{1+t^{4}}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1192

A curve is given by the equations

{x=a1+t2,y=bt−bt2(1+t2). \begin{cases} x = \frac{a}{1+t^{2}}, \\ y = bt-\frac{bt}{2(1+t^{2})}. \end{cases}

Find d2ydx2\frac{d^{2}y}{dx^{2}}.

?
Problem 1193

Prove that when x2y2+x2+y2−1=0x^{2}y^{2}+x^{2}+y^{2}-1 = 0, the equality dx1−x4+dy1−y4=0\frac{dx}{\sqrt{1-x^{4}}}+\frac{dy}{\sqrt{1-y^{4}}} = 0 holds.

?
Problem 1194

Prove that when a+b(x+y)+cxy=m(x−y)a+b(x+y)+cxy = m(x-y), the equality holds:

dxa+2bx+cx2=dya+2by+cy2. \frac{dx}{a+2bx+cx^{2}} = \frac{dy}{a+2by+cy^{2}}.
?
Problem 1195

Verify that if y=1−x21+x2y = \sqrt{\frac{1-x^{2}}{1+x^{2}}}, then

dy1−x4+dx1−y4=0. \frac{dy}{\sqrt{1-x^{4}}}+\frac{dx}{\sqrt{1-y^{4}}} = 0.
?
Problem 1196

Verify that if A(a−cxy)=b+c(x+y)A(a-cxy) = b+c(x+y), then

dxa+bx+cx2=dya+by+cy2=0. \frac{dx}{a+bx+cx^{2}} = \frac{dy}{a+by+cy^{2}} = 0.
?
Problem 1197

Verify that if xx and yy are related by the equation

xa+by+cy2+ya+bx+cx2=A(x−y), x\sqrt{a+by+cy^{2}}+y\sqrt{a+bx+cx^{2}} = A(x-y),

then

dxxa+bx+cx2=dyya+by+cy2. \frac{dx}{x\sqrt{a+bx+cx^{2}}} = \frac{dy}{y\sqrt{a+by+cy^{2}}}.
?
Problem 1198

If P(x)P(x) and Q(x)Q(x) are polynomials such that

1−P2(x)=Q(x)1−x2, \sqrt{1-P^{2}(x)} = Q(x)\sqrt{1-x^{2}},

then dP(x)1−P2(x)=ndx1−x2\frac{dP(x)}{\sqrt{1-P^{2}(x)}} = n \frac{dx}{\sqrt{1-x^{2}}}, where nn is an integer. Prove this.

?
Section 3.7Functions of Several Variables. Their Derivatives and Differentials[102/60%]
Problem 1199

Show that

lim⁡y→0lim⁡x→0x2−y2x2+y2=−1,lim⁡x→0lim⁡y→0x2−y2x2+y2=1. \lim _{y \to 0} \lim _{x \to 0} \frac{x^{2}-y^{2}}{x^{2}+y^{2}} = -1, \qquad \lim _{x \to 0} \lim _{y \to 0} \frac{x^{2}-y^{2}}{x^{2}+y^{2}} = 1.
?
Problem 1200

Show that

lim⁡x→∞lim⁡y→∞x2+y21+(x−y)4=lim⁡y→∞lim⁡x→∞x2+y21+(x−y)4, \lim _{x \to \infty } \lim _{y \to \infty } \frac{x^{2}+y^{2}}{1+(x-y)^{4}} = \lim _{y \to \infty } \lim _{x \to \infty } \frac{x^{2}+y^{2}}{1+(x-y)^{4}},

while the quantity

lim⁡x,y→∞x2+y21+(x−y)4 \lim _{x,y \to \infty } \frac{x^{2}+y^{2}}{1+(x-y)^{4}}

nevertheless has no meaning.

?
Problem 1201

Show that if x→∞x \to \infty and y→∞y \to \infty in such a way that the ratio yx\frac{y}{x} has any given constant value, then 11+(x2−y)→0\frac{1}{1+(x^{2}-y)} \to 0. At the same time, this quantity may fail to tend to any limit if the ratio yx\frac{y}{x} is not constant.

?
Problem 1202

Show that f(x)f(x), defined by the equality

f(x)=lim⁡n→∞lim⁡m→∞cos⁡mn! 2πx, f(x) = \lim _{n \to \infty } \lim _{m \to \infty } \cos ^{m} n! \, 2\pi x,

equals one if xx is a rational number, and zero if xx is an irrational number; mm and nn are natural numbers.

?
Problem 1203

Show that in any neighborhood of the point (0,0)(0,0) the function f(x,y)f(x,y), defined by the equalities

f(0,0)=0,f(x,y)=x2−y2x2+y2for x2+y2>0, f(0,0) = 0, \qquad f(x,y) = \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \quad \text{for } x^{2}+y^{2} > 0,

takes every value in the interval (−1,1)(-1,1). Here, by a neighborhood of the point (0,0)(0,0) is meant any region (a disk, a rectangle, etc.) containing the point (0,0)(0,0) in its interior.

?
Problem 1204

A function is defined by the equalities:

f(x,y)=xyx2+y2for x2+y2>0,f(0,0)=0. f(x,y) = \frac{xy}{x^{2}+y^{2}} \quad \text{for } x^{2}+y^{2} > 0, \qquad f(0,0) = 0.

Show that this function is continuous with respect to each of the variables xx and yy separately, without being, in a neighborhood of the point (0,0)(0,0), a continuous function of both variables under their joint variation.

?
Problem 1205

Show that the function, defined for x2+y2>0x^{2}+y^{2} > 0 by the equality f(x,y)=ex/(x2+y2)f(x,y) = e^{x/(x^{2}+y^{2})}, takes every positive value in any region containing the point (0,0)(0,0) in its interior.

?
Problem 1206

Find the partial derivatives of the function u=x3+y3−3xyu = x^{3}+y^{3}-3xy.

?
Problem 1207

Find the partial derivatives of the function u=2x3−3x2y2+3y3u = 2x^{3}-3x^{2}y^{2}+3y^{3}.

?
Problem 1208

Find the partial derivatives of the function u=xy+xz+yzu = xy+xz+yz.

?
Problem 1209

Find the partial derivatives of the function u=xyu = x^{y}.

?
Problem 1210

Find the partial derivatives of the function u=x3+yz2+3xy−x+zu = x^{3}+yz^{2}+3xy-x+z.

?
Problem 1211

Find the partial derivatives of the function u=exyu = e^{xy}.

?
Problem 1212

Find the partial derivatives of the function u=arctg⁡xyu = \operatorname {arctg} \frac{x}{y}.

?
Problem 1213

Find the partial derivatives of the function u=xy−3x+5yu = xy-\frac{3}{x}+\frac{5}{y}.

?
Problem 1214

Find the partial derivatives of the function u=x2+y2+z2+xy+xz+yzu = x^{2}+y^{2}+z^{2}+xy+xz+yz.

?
Problem 1215

Find the partial derivatives of the function u=(xy)zu = (xy)^{z}.

?
Problem 1216

Find the partial derivatives of the function u=zxyu = z^{xy}.

?
Problem 1217

Find the total differential of the function u=sin⁡(x2+y2)u = \sin (x^{2}+y^{2}).

?
Problem 1218

Find the total differential of the function u=arctg⁡xyu = \operatorname {arctg} \frac{x}{y}.

?
Problem 1219

Find the total differential of the function u=ln⁡tg⁡xyu = \ln \operatorname {tg} \frac{x}{y}.

?
Problem 1220

Find the total differential of the function u=x2+y2+z2u = \sqrt{x^{2}+y^{2}+z^{2}}.

?
Problem 1221

Find the total differential of the function u=ln⁡(x+y+z)u = \ln (x+y+z).

?
Problem 1222

Find the total differential of the function u=xyu = x^{y}.

?
Problem 1223

Verify Euler's theorem on an example by direct computation of the derivatives: u=x2+y2+z2ln⁡xyu = \sqrt{x^{2}+y^{2}+z^{2}} \ln \frac{x}{y}.

?
Remark.
?

If, when all the variables are increased tt-fold, a function is multiplied by tnt^{n}, then it is called a homogeneous function of degree nn. Thus, for example, if

f(tx,ty,tz)=tnf(x,y,z), f(tx,ty,tz) = t^{n}f(x,y,z),

then f(x,y,z)f(x,y,z) is a homogeneous function of degree nn. By Euler's theorem, if f(x1,x2,…,xm)f(x_{1},x_{2},\ldots ,x_{m}) is a homogeneous function of degree nn, then the equality

x1∂f∂x1+x2∂f∂x2+…+xm∂f∂xm=nf(x1,x2,…,xm). x_{1}\frac{\partial f}{\partial x_{1}}+x_{2}\frac{\partial f}{\partial x_{2}}+\ldots +x_{m}\frac{\partial f}{\partial x_{m}} = nf(x_{1},x_{2},\ldots ,x_{m}).

holds.

Problem 1224

Verify Euler's theorem by direct computation of the derivatives: u=xyex/zu = \frac{x}{y}e^{x/z}.

?
Problem 1225

Verify Euler's theorem by direct computation of the derivatives: u=sin⁡x+y+zx2+y2+z2u = \sin \frac{x+y+z}{\sqrt{x^{2}+y^{2}+z^{2}}}.

?
Problem 1226

Verify Euler's theorem by direct computation of the derivatives: u=xx2+y2+z2u = \frac{x}{x^{2}+y^{2}+z^{2}}.

?
Problem 1227

Verify Euler's theorem by direct computation of the derivatives: u=x2+y2+z2u = \sqrt{x^{2}+y^{2}+z^{2}}.

?
Problem 1228

Verify Euler's theorem by direct computation of the derivatives: u=arctg⁡xyu = \operatorname {arctg} \frac{x}{y}.

?
Problem 1229

Find the derivative of the function f=4x3−2xy+3y3f = 4x^{3}-2xy+3y^{3} at the point (1,2)(1,2) in the direction π3\frac{\pi }{3} to the OxOx axis.

?
Problem 1230

Find the derivative of the function f=5x+10x2y+y5f = 5x+10x^{2}y+y^{5} at the point (1,2)(1,2) in the direction toward the point (5,−1)(5,-1).

?
Problem 1231

Find the derivative of the function f=x2+y2f = x^{2}+y^{2} at the point (1,2)(1,2) in the direction of the line 3x+4y=113x+4y = 11:

?
(a)

in the direction of increasing xx;

(b)

in the direction of decreasing xx.

Problem 1232

Write the expression for the derivative of the function f(x,y)f(x,y):

?
(a)

in the direction of the tangent to the curve φ(x,y)=C\varphi (x,y) = C;

(b)

in the direction of the normal to the curve φ(x,y)=C\varphi (x,y) = C.

Problem 1233

Find the derivative of the function f=x3+2xy2+3yz2f = x^{3}+2xy^{2}+3yz^{2} at the point (3,3,1)(3,3,1) in the direction of the vector 2i→+2j→+k→2\overrightarrow {i}+2\overrightarrow {j}+\overrightarrow {k}.

?
Problem 1234

Find the derivative of the function F=x2a2+y2b2+z2c2F = \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} in the direction of the outward normal to the surface F=constF = \mathrm{const}.

?
Problem 1235

Find ∂2u∂x2\frac{\partial^{2} u}{\partial x^{2}}, ∂2u∂x∂y\frac{\partial^{2} u}{\partial x \partial y}, ∂2u∂y2\frac{\partial^{2} u}{\partial y^{2}}; u=12ln⁡(x2+y2)u = \frac{1}{2}\ln (x^{2}+y^{2}).

?
Problem 1236

Find ∂2u∂x2\frac{\partial^{2} u}{\partial x^{2}}, ∂2u∂x∂y\frac{\partial^{2} u}{\partial x \partial y}, ∂2u∂y2\frac{\partial^{2} u}{\partial y^{2}}; u=arctg⁡x+y1−xyu = \operatorname {arctg} \frac{x+y}{1-xy}.

?
Problem 1237

Find ∂2u∂x2\frac{\partial^{2} u}{\partial x^{2}}, ∂2u∂x∂y\frac{\partial^{2} u}{\partial x \partial y}, ∂2u∂y2\frac{\partial^{2} u}{\partial y^{2}}; u=xsin⁡(x+y)+ycos⁡(x+y)u = x\sin (x+y)+y\cos (x+y).

?
Problem 1238

Find ∂2u∂x∂y\frac{\partial^{2} u}{\partial x \partial y}, if u=xyu = x^{y}.

?
Problem 1239

Find ∂3u∂x2∂y\frac{\partial^{3} u}{\partial x^{2} \partial y}, if u=ln⁡(x+y)u = \ln (x+y).

?
Problem 1240

Find ∂4u∂x2∂y2\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}, if u=ln⁡(x2+y2)u = \ln (x^{2}+y^{2}).

?
Problem 1241

Find ∂2u∂x2\frac{\partial^{2} u}{\partial x^{2}}, ∂2u∂y2\frac{\partial^{2} u}{\partial y^{2}}, ∂2u∂z2\frac{\partial^{2} u}{\partial z^{2}}, if u=x2+y2+z2u = \sqrt{x^{2}+y^{2}+z^{2}}.

?
Problem 1242

Find ∂3u∂x∂y∂z\frac{\partial^{3} u}{\partial x \partial y \partial z}, if u=exyzu = e^{xyz}.

?
Problem 1243

Find d2ud^{2}u, if u=x2−xy+2y2+3x−5y+7u = x^{2}-xy+2y^{2}+3x-5y+7.

?
Problem 1244

Find d2ud^{2}u, if u=x2y2u = x^{2}y^{2}.

?
Problem 1245

Find d2ud^{2}u, if u=exyu = e^{xy}.

?
Problem 1246

Find d2ud^{2}u, if u=12ln⁡(x2+y2)u = \frac{1}{2}\ln (x^{2}+y^{2}).

?
Problem 1247

Find d2ud^{2}u, if u=sin⁡(x+y+z)u = \sin (x+y+z).

?
Problem 1248

Find d4ud^{4}u, if u=x4+4x3y+2xy2z−3xyz2+z4u = x^{4}+4x^{3}y+2xy^{2}z-3xyz^{2}+z^{4}.

?
Problem 1249

Find d4ud^{4}u, if u=x4−3x2y2+y4+5xyz−x3−y3+x2−xy+y2+2x−5yu = x^{4}-3x^{2}y^{2}+y^{4}+5xyz-x^{3}-y^{3}+x^{2}-xy+y^{2}+2x-5y.

?
Problem 1250

Find d4ud^{4}u, if u=x4+3x3y+z4−x2y+z3u = x^{4}+3x^{3}y+z^{4}-x^{2}y+z^{3}.

?
Problem 1251

Find d3ud^{3}u, if u=xyzu = xyz.

?
Problem 1252

Find d4ud^{4}u, if u=ln⁡(2x+3y−z)u = \ln (2x+3y-z).

?
Problem 1253

Find the partial derivatives by first computing the total differentials of the needed order: u=sin⁡(2x+y)u = \sin (2x+y); find ∂3u∂x3\frac{\partial^{3} u}{\partial x^{3}} and ∂3u∂x∂y2\frac{\partial^{3} u}{\partial x \partial y^{2}}.

?
Problem 1254

Find the partial derivatives by first computing the total differentials of the needed order: u=cos⁡(x+y)u = \cos (x+y); find ∂4u∂x2∂y2\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}.

?
Problem 1255

Find the partial derivatives by first computing the total differentials of the needed order: u=ln⁡(ax+by+cz)u = \ln (ax+by+cz); find ∂4u∂x4\frac{\partial^{4} u}{\partial x^{4}} and ∂4u∂x2∂y2\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}.

?
Problem 1256

Find the partial derivatives by first computing the total differentials of the needed order: u=ex+2y+3zu = e^{x+2y+3z}; find ∂3u∂x3\frac{\partial^{3} u}{\partial x^{3}} and ∂3u∂x∂y∂z\frac{\partial^{3} u}{\partial x \partial y \partial z}.

?
Problem 1257

Find the total differentials of the first two orders for the composite function u=φ(t)u = \varphi (t); t=xyt = xy.

?
Problem 1258

Find the total differentials of the first two orders for the composite function u=φ(t)u = \varphi (t); t=x2+y2t = x^{2}+y^{2}.

?
Problem 1259

Find the total differentials of the first two orders for the composite function u=φ(t)u = \varphi (t); t=x2+y2+z2t = x^{2}+y^{2}+z^{2}.

?
Problem 1260

Find the total differentials of the first two orders for the composite function u=φ(ξ,η)u = \varphi (\xi ,\eta ); ξ=ax+by+cz\xi = ax+by+cz, η=a1x+b1y+c1z\eta = a_{1}x+b_{1}y+c_{1}z.

?
Problem 1261

Find the total differentials of the first two orders for the composite function u=φ(ξ,η,ζ)u = \varphi (\xi ,\eta ,\zeta ); ξ=ax\xi = ax, η=by\eta = by, ζ=cz\zeta = cz.

?
Problem 1262

Find the derivatives of the first two orders of the function u=φ(ξ,η)u = \varphi (\xi ,\eta ); ξ=x+y\xi = x+y, η=x−y\eta = x-y.

?
Problem 1263

Find the derivatives of the first two orders of the function u=φ(ξ,η)u = \varphi (\xi ,\eta ); ξ=x2+y2\xi = x^{2}+y^{2}, η=xy\eta = xy.

?
Problem 1264

Find the derivatives of order nn of u=φ(t)u = \varphi (t), where t=ax+by+czt = ax+by+cz.

?
Problem 1265

Find ∂nu∂xn\frac{\partial^{n} u}{\partial x^{n}}, if u=φ(ξ,η,ζ)u = \varphi (\xi ,\eta ,\zeta ), ξ=ax\xi = ax, η=by\eta = by, ζ=cz\zeta = cz.

?
Problem 1266

Show that if x2=ηξx^{2} = \eta \xi, y2=ζξy^{2} = \zeta \xi, z2=ξηz^{2} = \xi \eta, then

x∂u∂x+y∂u∂y+z∂u∂z=ξ∂u∂ξ+η∂u∂η+ζ∂u∂ζ. x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z} = \xi \frac{\partial u}{\partial \xi }+\eta \frac{\partial u}{\partial \eta }+\zeta \frac{\partial u}{\partial \zeta }.
?
Problem 1267

Setting x=ρcos⁡φx = \rho \cos \varphi, y=ρsin⁡φy = \rho \sin \varphi, find

∣∂x∂ρ∂x∂φ∂y∂ρ∂y∂φ∣. \begin{vmatrix} \frac{\partial x}{\partial \rho } & \frac{\partial x}{\partial \varphi } \\ \frac{\partial y}{\partial \rho } & \frac{\partial y}{\partial \varphi } \end{vmatrix}.
?
Problem 1268

Setting x=ρsin⁡φcos⁡ψx = \rho \sin \varphi \cos \psi, y=ρsin⁡φsin⁡ψy = \rho \sin \varphi \sin \psi, z=ρcos⁡φz = \rho \cos \varphi, compute the value of the functional determinant (Jacobian)

D(x,y,z)D(ρ,φ,ψ)=∣∂x∂ρ∂x∂φ∂x∂ψ∂y∂ρ∂y∂φ∂y∂ψ∂z∂ρ∂z∂φ∂z∂ψ∣. \frac{D(x,y,z)}{D(\rho ,\varphi ,\psi )} = \begin{vmatrix} \frac{\partial x}{\partial \rho } & \frac{\partial x}{\partial \varphi } & \frac{\partial x}{\partial \psi } \\ \frac{\partial y}{\partial \rho } & \frac{\partial y}{\partial \varphi } & \frac{\partial y}{\partial \psi } \\ \frac{\partial z}{\partial \rho } & \frac{\partial z}{\partial \varphi } & \frac{\partial z}{\partial \psi } \end{vmatrix}.
?
Problem 1269

Setting r2=a2+ρ2−2aρcos⁡φr^{2} = a^{2}+\rho^{2}-2a\rho \cos \varphi, find D(ρ,φ,ψ)D(ρ,r,ψ)\frac{D(\rho ,\varphi ,\psi )}{D(\rho ,r,\psi )}.

?
Problem 1270

Setting x=ξηζx = \xi \eta \zeta, y=ξη−ξηζy = \xi \eta -\xi \eta \zeta, z=η−ξηz = \eta -\xi \eta, find the Jacobian D(x,y,z)D(ξ,η,ζ)\frac{D(x,y,z)}{D(\xi ,\eta ,\zeta )}.

?
Problem 1271

Prove that for x=cos⁡φx = \cos \varphi, y=sin⁡φcos⁡θy = \sin \varphi \cos \theta, z=sin⁡φsin⁡θcos⁡ψz = \sin \varphi \sin \theta \cos \psi the Jacobian equals −sin⁡3φsin⁡2θsin⁡ψ-\sin^{3}\varphi \sin^{2}\theta \sin \psi.

?
Problem 1272

Prove that for

u1=x11−r2,u2=x21−r2,u3=x31−r2, u_{1} = \frac{x_{1}}{\sqrt{1-r^{2}}}, \qquad u_{2} = \frac{x_{2}}{\sqrt{1-r^{2}}}, \qquad u_{3} = \frac{x_{3}}{\sqrt{1-r^{2}}},

where r2=x12+x22+x32r^{2} = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}; the equality

D(u1,u2,u3)D(x1,x2,x3)=(1−r2)−5/2. \frac{D(u_{1},u_{2},u_{3})}{D(x_{1},x_{2},x_{3})} = (1-r^{2})^{-5/2}.

holds.

?
Problem 1273

Prove that

D(∂v∂x1,∂v∂x2,…,∂v∂xn)D(y1,y2,…,yn)=D(∂v∂y1,∂v∂y2,…,∂v∂yn)D(x1,x2,…,xn). \frac{D\left(\frac{\partial v}{\partial x_{1}},\frac{\partial v}{\partial x_{2}},\ldots ,\frac{\partial v}{\partial x_{n}}\right)}{D(y_{1},y_{2},\ldots ,y_{n})} = \frac{D\left(\frac{\partial v}{\partial y_{1}},\frac{\partial v}{\partial y_{2}},\ldots ,\frac{\partial v}{\partial y_{n}}\right)}{D(x_{1},x_{2},\ldots ,x_{n})}.

Here v=v(x1,x2,…,xn;y1,y2,…,yn)v = v(x^{1},x^{2},\ldots ,x_{n};y^{1},y_{2},\ldots ,y_{n}) is a function with continuous second derivatives.

?
Problem 1274

Prove that for homogeneous functions φ(x,y,z)\varphi (x,y,z) of degree nn the equality

(x∂∂x+y∂∂y+z∂∂z)mφ=n(n−1)…(n−m+1)φ. \left(x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+z\frac{\partial }{\partial z}\right)^{m} \varphi = n(n-1) \ldots (n-m+1)\varphi .

holds.

?
Problem 1275

Verify that ∂u∂t=a2∂2u∂x2\frac{\partial u}{\partial t} = a^{2}\frac{\partial^{2} u}{\partial x^{2}}, if u=1te−x2/(4a2t)u = \frac{1}{\sqrt{t}} e^{-x^{2}/(4a^{2}t)}.

?
Problem 1276

Verify that x∂u∂x+y∂u∂y+z∂u∂z=0x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z} = 0, if u=(xy)z/yu = \left(\frac{x}{y}\right)^{z/y}.

?
Problem 1277

Compute ∂2u∂x2−2∂2u∂x∂y+∂2u∂y2\frac{\partial^{2} u}{\partial x^{2}}-2\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}, if u=φ(x+y)u = \varphi (x+y).

?
Problem 1278

Compute x2∂2u∂x2−2xy∂2u∂x∂y+y2∂2u∂y2x^{2}\frac{\partial^{2} u}{\partial x^{2}}-2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}}, if u=f(x2+y2)u = f\left(\sqrt{x^{2}+y^{2}}\right).

?
Problem 1279

Compute ∂2u∂x2⋅∂2u∂y2−(∂2u∂x∂y)2\frac{\partial^{2} u}{\partial x^{2}} \cdot \frac{\partial^{2} u}{\partial y^{2}}-\left(\frac{\partial^{2} u}{\partial x \partial y}\right)^{2}, if u=φ(xy)u = \varphi (xy).

?
Problem 1280

The function called the potential of the ball x2+y2+z2=a2x^{2}+y^{2}+z^{2} = a^{2} is defined by the equalities:

u={2πa2−2π3(x2+y2+z2),for x2+y2+z2<a2;4πa33x2+y2+z2,for x2+y2+z2≥a2. u = \begin{cases} 2\pi a^{2}-\frac{2\pi }{3}(x^{2}+y^{2}+z^{2}), & \text{for } x^{2}+y^{2}+z^{2} < a^{2}; \\ \frac{4\pi a^{3}}{3\sqrt{x^{2}+y^{2}+z^{2}}}, & \text{for } x^{2}+y^{2}+z^{2} \geq a^{2}. \end{cases}

Verify that uu and its first derivatives are continuous for any xx, yy, and zz, and that the Laplacian

Δu=∂2u∂x2+∂2u∂y2+∂2u∂z2=0or −4π, \Delta u = \frac{\partial ^{2} u}{\partial x^{2}}+\frac{\partial ^{2} u}{\partial y^{2}}+\frac{\partial ^{2} u}{\partial z^{2}} = 0 \quad \text{or } -4\pi ,

depending on whether the point M(x,y,z)M(x,y,z) lies outside the ball x2+y2+z2=a2x^{2}+y^{2}+z^{2} = a^{2} or inside it.

?
Problem 1281

Verify the equality

(x∂∂x+y∂∂y+z∂∂z)2u=0,u=x2+y2+z2. \left(x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+z\frac{\partial }{\partial z}\right)^{2} u = 0, \qquad u = \sqrt{x^{2}+y^{2}+z^{2}}.
?
Problem 1282

Verify the equality ∂u∂x+∂u∂y+∂u∂z=3x+y+z\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} = \frac{3}{x+y+z}; u=ln⁡(x3+y3+z3−3xyz)u = \ln (x^{3}+y^{3}+z^{3}-3xyz).

?
Problem 1283

Verify the equality 1x∂z∂x+1y∂z∂y=zy2\frac{1}{x}\frac{\partial z}{\partial x}+\frac{1}{y}\frac{\partial z}{\partial y} = \frac{z}{y^{2}}; z=yφ(x2−y2)z = y\varphi (x^{2}-y^{2}).

?
Problem 1284

Verify the equality x∂u∂x+y∂u∂y=xy+ux\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} = xy+u; u=xy+xφ(yx)u = xy+x\varphi \left(\frac{y}{x}\right).

?
Problem 1285

Verify the equality ∂2ln⁡r∂x2+∂2ln⁡r∂y2+∂2ln⁡r∂z2=1r2\frac{\partial^{2} \ln r}{\partial x^{2}}+\frac{\partial^{2} \ln r}{\partial y^{2}}+\frac{\partial^{2} \ln r}{\partial z^{2}} = \frac{1}{r^{2}}; r=x2+y2+z2r = \sqrt{x^{2}+y^{2}+z^{2}}.

?
Problem 1286

Verify the equality ∂2u∂x2−∂2u∂y2−2α∂u∂y=α2u\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial y^{2}}-2\alpha \frac{\partial u}{\partial y} = \alpha^{2}u; u=e−αxφ(x−y)u = e^{-\alpha x}\varphi (x-y).

?
Problem 1287

Verify the equality ∂2u∂x2−∂2u∂y2=−2φ′′\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial y^{2}} = -2\varphi ''; u=φ(y−x)−xφ′(y−x)u = \varphi (y-x)-x\varphi '(y-x).

?
Problem 1288

Verify the equality (x2−y2)∂z∂x+xy∂z∂y=xyz(x^{2}-y^{2})\frac{\partial z}{\partial x}+xy\frac{\partial z}{\partial y} = xyz; z=eyφ(yex2/(2y2))z = e^{y}\varphi \left(ye^{x^{2}/(2y^{2})}\right).

?
Problem 1289

Verify the equality x∂u∂x+y∂u∂y=u−x2−y2x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} = u-x^{2}-y^{2}; u=xφ(yx)−x2−y2u = x\varphi \left(\frac{y}{x}\right)-x^{2}-y^{2}.

?
Problem 1290

Verify the equality ∂2u∂t2=a2∂2u∂x2\frac{\partial^{2} u}{\partial t^{2}} = a^{2}\frac{\partial^{2} u}{\partial x^{2}}; u=φ(x+at)+ψ(x−at)u = \varphi (x+at)+\psi (x-at).

?
Problem 1291

Verify the equality ∂2u∂x2+∂2u∂y2=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} = 0; u=ln⁡(x2+y2)u = \ln (x^{2}+y^{2}).

?
Problem 1292

Verify the equality ∂2u∂x2+∂2u∂y2+∂2u∂z2=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}} = 0; u=1x2+y2+z2u = \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}.

?
Problem 1293

Verify the equality ∂2u∂x2+∂2u∂y2+∂2u∂z2=a2u\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}} = a^{2}u; u=1r(Ae−ar+Bear)u = \frac{1}{r}(Ae^{-ar}+Be^{ar}), where r=x2+y2+z2r = \sqrt{x^{2}+y^{2}+z^{2}}.

?
Problem 1294

Verify the equality ∂2ln⁡z∂x∂y=2z\frac{\partial^{2} \ln z}{\partial x \partial y} = 2z; z=φ′(x)ψ′(x)(φ(x)+ψ(y))2z = \frac{\varphi '(x)\psi '(x)}{(\varphi (x)+\psi (y))^{2}}.

?
Problem 1295

Verify the equality x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2=0x^{2}\frac{\partial^{2} u}{\partial x^{2}}+2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}} = 0, if u=xφ(yx)u = x\varphi \left(\frac{y}{x}\right).

?
Problem 1296

Verify the equality x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2+x∂u∂x+y∂u∂y=n2ux^{2}\frac{\partial^{2} u}{\partial x^{2}}+2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}}+x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} = n^{2}u, where u=xnφ(yx)+y−nψ(yx)u = x^{n}\varphi \left(\frac{y}{x}\right)+y^{-n}\psi \left(\frac{y}{x}\right).

?
Problem 1297

Verify the equality x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2=n(n−1)ux^{2}\frac{\partial^{2} u}{\partial x^{2}}+2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}} = n(n-1)u, where u=xnφ(yx)+x1−nψ(yx)u = x^{n}\varphi \left(\frac{y}{x}\right)+x^{1-n}\psi \left(\frac{y}{x}\right).

?
Problem 1298

Verify the equality ∂u∂x⋅∂2u∂x∂y=∂u∂y⋅∂2u∂x2\frac{\partial u}{\partial x} \cdot \frac{\partial^{2} u}{\partial x \partial y} = \frac{\partial u}{\partial y} \cdot \frac{\partial^{2} u}{\partial x^{2}}, if u=f(x+φ(y))u = f\left(x+\varphi (y)\right).

?
Problem 1299

Verify the equality ∂2u∂x2−2∂2u∂x∂y+∂2u∂y2=0\frac{\partial^{2} u}{\partial x^{2}}-2\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}} = 0, if u=xφ(x+y)+yψ(x+y)u = x\varphi (x+y)+y\psi (x+y).

?
Problem 1300

Verify the equality

a2(u∂2u∂x2−(∂u∂x)2)=b2(u∂2u∂y2−(∂u∂y)2), a^{2}\left(u\frac{\partial ^{2} u}{\partial x^{2}}-\left(\frac{\partial u}{\partial x}\right)^{2}\right) = b^{2}\left(u\frac{\partial ^{2} u}{\partial y^{2}}-\left(\frac{\partial u}{\partial y}\right)^{2}\right),

where u=φ(ay+bx)ψ(bx−ay)u = \varphi (ay+bx)\psi (bx-ay).

?
Section 3.8Differentiation of Implicit Functions[39/79%]
Problem 1301

x3+y3−3axy=0x^{3}+y^{3}-3axy = 0. Find y′y' at x=yx=y.

?
Remark.
?

An equation f(x,y)=0f(x,y) = 0, having as a solution a pair of numbers (x0,y0)(x_{0},y_{0}), defines in a neighborhood of the number x0x_{0} the quantity yy as a function of xx, if the partial derivative ∂f∂y\frac{\partial f}{\partial y} is nonzero at the point (x0,y0)(x_{0},y_{0}) and continuous in a neighborhood of it. The successive derivatives of yy with respect to xx can be obtained from the equalities:

∂f∂x+∂f∂yy′=0, \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}y' = 0, ∂2f∂x2+2∂2f∂x∂yy′+∂2f∂y2y′2+∂f∂yy′′=0, \frac{\partial ^{2} f}{\partial x^{2}}+2\frac{\partial ^{2} f}{\partial x \partial y}y'+\frac{\partial ^{2} f}{\partial y^{2}}y'^{2}+\frac{\partial f}{\partial y}y'' = 0, ∂3f∂x3+3∂3f∂x2∂yy′+3∂3f∂x∂y2y′2+∂3f∂y3y′3+3∂2f∂x∂yy′′+3∂2f∂y2y′y′′+∂f∂yy′′′=0. \frac{\partial ^{3} f}{\partial x^{3}}+3\frac{\partial ^{3} f}{\partial x^{2} \partial y}y'+3\frac{\partial ^{3} f}{\partial x \partial y^{2}}y'^{2}+\frac{\partial ^{3} f}{\partial y^{3}}y'^{3}+3\frac{\partial ^{2} f}{\partial x \partial y}y''+3\frac{\partial ^{2} f}{\partial y^{2}}y'y''+\frac{\partial f}{\partial y}y''' = 0.

In a similar way one can also compute the derivatives of implicit functions given by a single equation in several variables.

If several implicit functions are given by several equations, then in computing the derivatives it is sometimes convenient to use differentials. Here one should keep in mind that the higher-order differentials of those variables taken as independent are equal to zero.

Problem 1302

x3+y3−3axy=0x^{3}+y^{3}-3axy = 0. Find xx for which y′=0y' = 0.

?
Problem 1303

xy=yxx^{y} = y^{x}. Find y′y' for x≠yx \neq y.

?
Problem 1304

xsin⁡y−cos⁡y+cos⁡2y=0x\sin y-\cos y+\cos 2y = 0. Find y′y'.

?
Problem 1305

x=y−αsin⁡yx = y-\alpha \sin y. Find y′y' and y′′y''.

?
Problem 1306

x2+2xy+y2−4x+2y−2=0x^{2}+2xy+y^{2}-4x+2y-2 = 0. Find y′′′y''' at x=1x=1, y=1y=1.

?
Problem 1307

x+y=ex−yx+y = e^{x-y}. Find y′′y''.

?
Problem 1308

ln⁡x2+y2=arctg⁡yx\ln \sqrt{x^{2}+y^{2}} = \operatorname {arctg} \frac{y}{x}. Find y′′y''.

?
Problem 1309

(x−a)2+(y−b)2=R2(x-a)^{2}+(y-b)^{2} = R^{2}. Regarding xx here as the independent variable and yy as a function, take the fourth derivative of both sides of the equation.

?
Problem 1310

For the equation x(x2+y2)−a(x2−y2)=0x(x^{2}+y^{2})-a(x^{2}-y^{2}) = 0 at x=0x=0, y=0y=0, the formula ∂f∂x+∂f∂yy′=0\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}y' = 0 does not make it possible to find y′y'; however, y′y' can be found here too, from the formula

∂2f∂x2+2∂2f∂x∂yy′+∂2f∂y2y′2+∂f∂yy′′=0. \frac{\partial ^{2} f}{\partial x^{2}}+2\frac{\partial ^{2} f}{\partial x \partial y}y'+\frac{\partial ^{2} f}{\partial y^{2}}y'^{2}+\frac{\partial f}{\partial y}y'' = 0.

Find y′y'.

?
Problem 1311

(x2+y2−bx)2=a2(x2+y2)(x^{2}+y^{2}-bx)^{2} = a^{2}(x^{2}+y^{2}). Find y′y' at x=0x=0, y=0y=0.

?
Problem 1312

The same question for the equation x3+y3−3xy=0x^{3}+y^{3}-3xy = 0.

?
Problem 1313

The equations x2−y2+z2=1x^{2}-y^{2}+z^{2} = 1, y2−2x+z=0y^{2}-2x+z = 0 are given. Find y′y' and z′′z'' at x=1x=1, y=1y=1, z=1z=1.

?
Problem 1314
{x3+y3+z3=a3,x2+y2+z2=b2. \begin{cases} x^{3}+y^{3}+z^{3} = a^{3}, \\ x^{2}+y^{2}+z^{2} = b^{2}. \end{cases}

Find y′y' and z′z'.

?
Problem 1315

From the equations x2+y2−z2=0x^{2}+y^{2}-z^{2} = 0, x2+2y2+3z2=1x^{2}+2y^{2}+3z^{2} = 1, find d2yd^{2}y and d2zd^{2}z, if xx is the independent variable.

?
Problem 1316

From the equations x2+y2=2z2x^{2}+y^{2} = 2z^{2}, x2+2y2+z2=4x^{2}+2y^{2}+z^{2} = 4, find dxdz\frac{dx}{dz} and d2ydz2\frac{d^{2}y}{dz^{2}} at the point (1,−1,1)(1,-1,1), if zz is the independent variable.

?
Problem 1317

x+y+z=ax+y+z = a, x3+y3+z3=3xyzx^{3}+y^{3}+z^{3} = 3xyz. Find the derivatives of yy and zz.

?
Problem 1318

At the point (1,1,−2)(1,1,-2), find the first and second derivatives of yy and zz, if x+y+z=0x+y+z = 0, x3+y3−z3=10x^{3}+y^{3}-z^{3} = 10.

?
Problem 1319

x2+y2+z2=2zx^{2}+y^{2}+z^{2} = 2z. Find ∂2z∂x2\frac{\partial^{2} z}{\partial x^{2}}.

?
Problem 1320

x3+y3+z3−3z=0x^{3}+y^{3}+z^{3}-3z = 0. Find ∂2z∂x∂y\frac{\partial^{2} z}{\partial x \partial y}.

?
Problem 1321

x2−2y2+z2−4x+2z−5=0x^{2}-2y^{2}+z^{2}-4x+2z-5 = 0. Find ∂z∂x\frac{\partial z}{\partial x} and ∂2z∂x∂y\frac{\partial^{2} z}{\partial x \partial y}.

?
Problem 1322

xcos⁡y+ycos⁡z+zcos⁡x=ax\cos y+y\cos z+z\cos x = a. Find ∂z∂x\frac{\partial z}{\partial x} and ∂z∂y\frac{\partial z}{\partial y}.

?
Problem 1323

xy+xz+yz=1xy+xz+yz = 1. Find dzdz and d2zd^{2}z.

?
Problem 1324

Find d2zd^{2}z, if x2a2+y2b2+z2c2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1.

?
Problem 1325

Find d2zd^{2}z, if cos⁡2x+cos⁡2y+cos⁡2z=1\cos^{2} x+\cos^{2} y+\cos^{2} z = 1.

?
Problem 1326

Find d2zd^{2}z at the point x=ax=a, y=ay=a, z=0z=0, if x3+z3−3axz=y3x^{3}+z^{3}-3axz = y^{3}.

?
Problem 1327

Find the second partial derivatives of zz, if this function of xx and yy is defined by the equation y=xφ(z)+ψ(z)y = x\varphi (z)+\psi (z).

?
Problem 1328

Show that x∂u∂x+y∂u∂y+z∂u∂z=0x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z} = 0, if uv=3x−2y+zuv = 3x-2y+z, v2=x2+y2+z2v^{2} = x^{2}+y^{2}+z^{2}.

?
Problem 1329

xu+yv=0xu+yv = 0, uv−xy=5uv-xy = 5; at x=1x=1, y=−1y=-1 we take u=v=2u=v=2. Find ∂2u∂x2\frac{\partial^{2} u}{\partial x^{2}} and ∂2u∂x∂y\frac{\partial^{2} u}{\partial x \partial y}.

?
Problem 1330

x=φ(t)x = \varphi (t), y=ψ(t)y = \psi (t), z=kt2z = kt^{2}. Find the derivatives of xx and yy with respect to zz.

?
Problem 1331

x=acos⁡usin⁡vx = a\cos u \sin v, y=bcos⁡ucos⁡vy = b\cos u \cos v, z=csin⁡uz = c\sin u. Find ∂z∂x\frac{\partial z}{\partial x} and ∂z∂y\frac{\partial z}{\partial y}.

?
Problem 1332

Show that if F(x,y,z)=0F(x,y,z) = 0, then ∂z∂x⋅∂x∂y⋅∂y∂z=−1\frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} = -1.

?
Problem 1333

The function uu is given by the formula u=x3y2zu = x^{3}y^{2}z, where the variables xx, yy, and zz are related by the equation x2+y2+z2=3xyzx^{2}+y^{2}+z^{2} = 3xyz. Determine ∂u∂x\frac{\partial u}{\partial x} at the point (1,1,1)(1,1,1), when:

?
(a)

the independent variables are xx and yy;

(b)

the independent variables are xx and zz.

Problem 1334

Show that zz, defined as a function of xx and yy by the equation x−az=φ(y−bz)x-az = \varphi (y-bz), where φ\varphi is any differentiable function, satisfies the partial differential equation

a∂z∂x+b∂z∂y=1, a\frac{\partial z}{\partial x}+b\frac{\partial z}{\partial y} = 1,

called the equation of cylindrical surfaces.

?
Problem 1335

Show that zz, given as a function of xx and yy by the equation z=xφ(zy)z = x\varphi \left(\frac{z}{y}\right), satisfies the equation of conical surfaces

x∂z∂x+y∂z∂y=z. x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y} = z.
?
Problem 1336

Show that, subject to the equations

z=αx+yφ(α)+ψ(α), z = \alpha x+y\varphi (\alpha )+\psi (\alpha ), 0=x+yφ′(α)+ψ′(α), 0 = x+y\varphi '(\alpha )+\psi '(\alpha ),

zz satisfies the equation

∂2z∂x2∂2z∂y2−(∂2z∂x∂y)2=0. \frac{\partial ^{2} z}{\partial x^{2}}\frac{\partial ^{2} z}{\partial y^{2}}-\left(\frac{\partial ^{2} z}{\partial x \partial y}\right)^{2} = 0.
?
Problem 1337

Show that for y=xφ(z)+ψ(z)y = x\varphi (z)+\psi (z) the equation

∂2z∂x2(∂z∂y)2−2∂z∂x∂z∂y∂2z∂x∂y+∂2z∂y2(∂z∂x)2=0. \frac{\partial ^{2} z}{\partial x^{2}}\left(\frac{\partial z}{\partial y}\right)^{2}-2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\frac{\partial ^{2} z}{\partial x \partial y}+\frac{\partial ^{2} z}{\partial y^{2}}\left(\frac{\partial z}{\partial x}\right)^{2} = 0.

is satisfied.

?
Problem 1338

The function zz of xx and yy is given by the equations:

(z−φ(α))2=x2(y2−α2), (z-\varphi (\alpha ))^{2} = x^{2}(y^{2}-\alpha ^{2}), (z−φ(α))φ′(α)=αx2. (z-\varphi (\alpha ))\varphi '(\alpha ) = \alpha x^{2}.

Show that

∂z∂x∂z∂y=xy. \frac{\partial z}{\partial x}\frac{\partial z}{\partial y} = xy.
?
Problem 1339

The equations are given:

z=ψ(α)(x+α)2ψ′(α)+1x+α,y+ln⁡((x+α)2ψ′(α))=0. z = \frac{\psi (\alpha )}{(x+\alpha )^{2}\psi '(\alpha )}+\frac{1}{x+\alpha }, \qquad y+\ln \left((x+\alpha )^{2}\psi '(\alpha )\right) = 0.

Show that

(z−∂z∂y)2+∂z∂x=0. \left(z-\frac{\partial z}{\partial y}\right)^{2}+\frac{\partial z}{\partial x} = 0.
?
Section 3.9Change of Variables[45/82%]
Problem 1340

Take yy as the new independent variable and transform the equation y′′−xy′3+eyy′3=0y''-xy'^{3}+e^{y}y'^{3} = 0.

?
Remark.
?

Sometimes it becomes necessary to swap the roles of the variables, taking as the independent variable those that were previously considered functions. Because of this, expressions containing derivatives usually change form.

If the independent variable xx is replaced by a new independent variable tt by means of the equality x=φ(t)x = \varphi (t), it is most convenient to use the formulas

yx′=1φ′(t)dydt,yx′′=1φ′(t)(1φ′(t)⋅dydt)′,yx′′′=1φ′(1φ′(1φ′⋅dydt)′)′, … y_{x}' = \frac{1}{\varphi '(t)}\frac{dy}{dt}, \qquad y_{x}'' = \frac{1}{\varphi '(t)}\left(\frac{1}{\varphi '(t)} \cdot \frac{dy}{dt}\right)', \qquad y_{x}''' = \frac{1}{\varphi '}\left(\frac{1}{\varphi '}\left(\frac{1}{\varphi '} \cdot \frac{dy}{dt}\right)'\right)', \ \ldots

Here, on the left-hand sides the derivatives are taken with respect to xx, while on the right-hand sides the derivatives are taken with respect to tt.

In particular, if yy is taken as the new variable, then, setting x=φ(y)x = \varphi (y), we can apply the previous formulas, where now

dydt=1;φ(t)=x;φ′(t)=x′;… \frac{dy}{dt} = 1; \quad \varphi (t) = x; \quad \varphi '(t) = x'; \quad \ldots

In this way we obtain the equalities:

y′=1x′,y′′=−x′′x′3,y′′′=3x′′2−x′x′′′x′5, … y' = \frac{1}{x'}, \quad y'' = -\frac{x''}{x'^{3}}, \quad y''' = \frac{3x''^{2}-x'x'''}{x'^{5}}, \ \ldots
Problem 1341

Transform the equation y′y′′′−3y′′2=0y'y'''-3y''^{2} = 0, taking the independent variable xx as a function of yy.

?
Problem 1342

Transform in the same way the equation

y′2yIV−10y′y′′y′′′+15y′′3=0. y'^{2}y^{\mathrm{IV}}-10y'y''y'''+15y''^{3} = 0.
?
Problem 1343

In the equation (1−x2)d2ydx2−xdydx+a2y=0(1-x^{2})\frac{d^{2}y}{dx^{2}}-x\frac{dy}{dx}+a^{2}y = 0, set x=cos⁡tx = \cos t.

?
Problem 1344

In the equation x2y′′+3xy′+y=0x^{2}y''+3xy'+y = 0, set x=etx = e^{t}.

?
Problem 1345

In the equation x3y′′′+2x2y′′−xy′+y=0x^{3}y'''+2x^{2}y''-xy'+y = 0, set t=ln⁡xt = \ln x.

?
Problem 1346

In the equation (x+a)3y′′′+3(a+x)2y′′+(a+x)y′+by=0(x+a)^{3}y'''+3(a+x)^{2}y''+(a+x)y'+by = 0, set ln⁡(a+x)=t\ln (a+x) = t.

?
Problem 1347

In the equation (1+x2)2y′′+2x(1+x2)y′+y=0(1+x^{2})^{2}y''+2x(1+x^{2})y'+y = 0, set x=tg⁡tx = \operatorname {tg} t.

?
Problem 1348

Show that by the substitution x=12ln⁡tg⁡2tx = \frac{1}{2}\ln \operatorname {tg} 2t the equation

d2ydx2+2e2x−e−2xe2x+e−2xdydx+4m2y(e2x+e−2x)2=0 \frac{d^{2}y}{dx^{2}}+2\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}\frac{dy}{dx}+\frac{4m^{2}y}{(e^{2x}+e^{-2x})^{2}} = 0

is transformed into the equation y′′+4m2y=0y''+4m^{2}y = 0.

?
Problem 1349

By the substitution x=1−t2x = \sqrt{1-t^{2}}, transform the equation

(x−x3)y′′+(1−3x2)y′−xy=0. (x-x^{3})y''+(1-3x^{2})y'-xy = 0.

Prove that the new equation will be of the same form as the previous one.

?
Problem 1350

Transform the equation

(1−x2)2y′′′−2x(1−x2)y′+2xy1−x=0, (1-x^{2})^{2}y'''-2x(1-x^{2})y'+\frac{2xy}{1-x} = 0,

by setting x=e2t−1e2t+1x = \frac{e^{2t}-1}{e^{2t}+1}.

?
Remark.
?

In the following problems it is convenient to make use of formulas, independent of the choice of argument, expressing the derivatives in terms of differentials:

y′=dydx,y′′=dx d2y−dy d2xdx3, … y' = \frac{dy}{dx}, \qquad y'' = \frac{dx \, d^{2}y-dy \, d^{2}x}{dx^{3}}, \ \ldots
Problem 1351

Transform the expression xy′−y1+y′2\frac{xy'-y}{\sqrt{1+y'^{2}}} to polar coordinates, setting x=rcos⁡φx = r\cos \varphi, y=rsin⁡φy = r\sin \varphi.

?
Problem 1352

The same question for the expression x+yy′xy′−y\frac{x+yy'}{xy'-y}.

?
Problem 1353

The same question for the expression (1+y′2)3/2y′′\frac{(1+y'^{2})^{3/2}}{y''}.

?
Problem 1354

In the equation (1−x2)2(a−y′′)=by(1-x^{2})^{2}(a-y'') = by, set

x=th⁡ξ,y=aηch⁡ξ, x = \operatorname {th} \xi , \qquad y = \frac{a\eta }{\operatorname {ch} \xi },

where ξ\xi is the argument and η\eta is the function.

?
Problem 1355

In the equation 2y′′+(x+y)(1−y′)3=02y''+(x+y)(1-y')^{3} = 0, set x−y=ux-y = u, x+y=vx+y = v and take uu as the argument and vv as the function.

?
Problem 1356

Transform the equation y′′=A(x−α)2(x−β)2yy'' = \frac{A}{(x-\alpha )^{2}(x-\beta )^{2}}y, setting u=yx−βu = \frac{y}{x-\beta }, t=ln⁡x−αx−βt = \ln \frac{x-\alpha }{x-\beta } and taking tt as the argument and uu as the function.

?
Problem 1357

Transform the equation

xd2ydx2−dydx+xy=0, x\frac{d^{2}y}{dx^{2}}-\frac{dy}{dx}+xy = 0,

taking as the new argument t=x24t = \frac{x^{2}}{4}.

?
Problem 1358

Transform the equation

xyd2ydx2−x(dydx)2+ydydx=0, xy\frac{d^{2}y}{dx^{2}}-x\left(\frac{dy}{dx}\right)^{2}+y\frac{dy}{dx} = 0,

taking as the argument yy and as the new function z=ln⁡yxz = \ln \frac{y}{x}.

?
Problem 1359

The variable xx is a function of tt. In place of it, a new function is introduced by means of the fractional-linear substitution

y=αx+βγx+δ, y = \frac{\alpha x+\beta }{\gamma x+\delta },

where αδ−βγ≠0\alpha \delta -\beta \gamma \neq 0. Prove the equality

x′′′x′−32(x′′x′)2=y′′′y′−32(y′′y′)2 \frac{x'''}{x'}-\frac{3}{2}\left(\frac{x''}{x'}\right)^{2} = \frac{y'''}{y'}-\frac{3}{2}\left(\frac{y''}{y'}\right)^{2}

(Schwarz).

?
Problem 1360

In the equation 9y′′2yV−45y′′y′′′yIV+40y′′′3=09y''^{2}y^{\mathrm{V}}-45y''y'''y^{\mathrm{IV}}+40y'''^{3} = 0 the variables xx and yy are subjected to a homographic transformation by the formulas:

y=a1X+b1Y+c1aX+bY+c,x=a2X+b2Y+c2aX+bY+c. y = \frac{a_{1}X+b_{1}Y+c_{1}}{aX+bY+c}, \qquad x = \frac{a_{2}X+b_{2}Y+c_{2}}{aX+bY+c}.

Prove that the form of the equation does not change.

?
Remark.
?

In the following problems it is required to transform an expression containing partial derivatives with respect to xx and yy by introducing new independent variables. If the new variables are given by the equalities u=φ(x,y)u = \varphi (x,y), v=ψ(x,y)v = \psi (x,y), then by the rule for differentiating a composite function we have

∂z∂x=∂z∂u∂φ∂x+∂z∂v∂ψ∂x,∂z∂y=∂z∂u∂φ∂y+∂z∂v∂ψ∂y. \frac{\partial z}{\partial x} = \frac{\partial z}{\partial u}\frac{\partial \varphi }{\partial x}+\frac{\partial z}{\partial v}\frac{\partial \psi }{\partial x}, \qquad \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u}\frac{\partial \varphi }{\partial y}+\frac{\partial z}{\partial v}\frac{\partial \psi }{\partial y}.

These equalities make it possible to pass from derivatives with respect to xx and yy to derivatives with respect to uu and vv. Continuing in the same way, one can also express the higher-order derivatives.

Problem 1361

In the equation y∂z∂x−x∂z∂y=0y\frac{\partial z}{\partial x}-x\frac{\partial z}{\partial y} = 0, introduce new independent variables uu and vv, setting u=xu=x, v=x2+y2v=x^{2}+y^{2}.

?
Problem 1362

In the equation x∂z∂x+y∂z∂y−z=0x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}-z = 0, set u=xu=x, v=yxv=\frac{y}{x} and take uu and vv as the new independent variables.

?
Problem 1363

The same question for the equation

(x+mz)∂z∂x+(y+nz)∂z∂y=0, (x+mz)\frac{\partial z}{\partial x}+(y+nz)\frac{\partial z}{\partial y} = 0,

where u=xu=x, v=y+nzx+mzv=\frac{y+nz}{x+mz}.

?
Problem 1364

Transform to polar coordinates, setting x=rcos⁡φx = r\cos \varphi, y=rsin⁡φy = r\sin \varphi, the expression w=x∂u∂y−y∂u∂xw = x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}.

?
Problem 1365

Transform to polar coordinates, setting x=rcos⁡φx = r\cos \varphi, y=rsin⁡φy = r\sin \varphi, the expression w=x∂u∂x−y∂u∂yw = x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}.

?
Problem 1366

Transform to polar coordinates, setting x=rcos⁡φx = r\cos \varphi, y=rsin⁡φy = r\sin \varphi, the expression w=(∂u∂x)2+(∂u∂y)2w = \left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}.

?
Problem 1367

Transform to polar coordinates, setting x=rcos⁡φx = r\cos \varphi, y=rsin⁡φy = r\sin \varphi, the expression w=∂2u∂x2+∂2u∂y2w = \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}.

?
Problem 1368

Transform to polar coordinates, setting x=rcos⁡φx = r\cos \varphi, y=rsin⁡φy = r\sin \varphi, the expression

w=y2∂2z∂x2−2xy∂2z∂x∂y+x2∂2z∂y2−x∂z∂x−y∂z∂y. w = y^{2}\frac{\partial ^{2} z}{\partial x^{2}}-2xy\frac{\partial ^{2} z}{\partial x \partial y}+x^{2}\frac{\partial ^{2} z}{\partial y^{2}}-x\frac{\partial z}{\partial x}-y\frac{\partial z}{\partial y}.
?
Problem 1369

Taking uu and vv as the new independent variables, transform the equation ∂2z∂x2−a2∂2z∂y2=0\frac{\partial^{2} z}{\partial x^{2}}-a^{2}\frac{\partial^{2} z}{\partial y^{2}} = 0; u=y+axu = y+ax, v=y−axv = y-ax.

?
Problem 1370

Taking uu and vv as the new independent variables, transform the equation ∂2z∂x2+∂2z∂y2+m2z=0\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}+m^{2}z = 0; 2x=u2−v22x = u^{2}-v^{2}, y=uvy = uv.

?
Problem 1371

Taking uu and vv as the new independent variables, transform the equation ∂2z∂x2+2xy2∂z∂x+2(y−y3)∂z∂y+x2y2z=0\frac{\partial^{2} z}{\partial x^{2}}+2xy^{2}\frac{\partial z}{\partial x}+2(y-y^{3})\frac{\partial z}{\partial y}+x^{2}y^{2}z = 0; x=uvx = uv, y=1vy = \frac{1}{v}.

?
Problem 1372

Using a substitution of the form u=x+αyu = x+\alpha y, v=x+βyv = x+\beta y, transform the equation

∂2z∂x2−(a+b)∂2z∂x∂y+ab∂2z∂y2=0 \frac{\partial ^{2} z}{\partial x^{2}}-(a+b)\frac{\partial ^{2} z}{\partial x \partial y}+ab\frac{\partial ^{2} z}{\partial y^{2}} = 0

into the form

∂2z∂u∂v=0. \frac{\partial ^{2} z}{\partial u \partial v} = 0.
?
Problem 1373

Taking uu and vv as the new independent variables, and ww as the new function, transform to the new variables the equation ∂z∂y+12y∂2z∂y2=1x\frac{\partial z}{\partial y}+\frac{1}{2}y\frac{\partial^{2} z}{\partial y^{2}} = \frac{1}{x}; u=xyu = \frac{x}{y}, v=xv = x, w=xz−yw = xz-y.

?
Problem 1374

Taking uu and vv as the new independent variables, and ww as the new function, transform to the new variables the equation ∂2z∂x2+∂2z∂y2−2∂2z∂x∂y=0\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}-2\frac{\partial^{2} z}{\partial x \partial y} = 0; u=x+yu = x+y, v=yxv = \frac{y}{x}, w=zxw = \frac{z}{x}.

?
Problem 1375

Taking uu and vv as the new independent variables, and ww as the new function, transform to the new variables the equation ∂2z∂x2+2∂2z∂x∂y+∂2z∂y2=0\frac{\partial^{2} z}{\partial x^{2}}+2\frac{\partial^{2} z}{\partial x \partial y}+\frac{\partial^{2} z}{\partial y^{2}} = 0; u=x+yu = x+y, v=x−yv = x-y, w=xy−zw = xy-z.

?
Problem 1376

What form does the equation ∂2u∂x2+∂2u∂y2+u=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+u = 0 take, if u=φ(r)u = \varphi (r), where r=x2+y2r = \sqrt{x^{2}+y^{2}}?

?
Problem 1377

The same question for the equation

∂4u∂x4+2∂4u∂x2∂y2+∂4u∂y4=0. \frac{\partial ^{4} u}{\partial x^{4}}+2\frac{\partial ^{4} u}{\partial x^{2} \partial y^{2}}+\frac{\partial ^{4} u}{\partial y^{4}} = 0.
?
Problem 1378

What form does the equation ∂2w∂x∂y+aw=0\frac{\partial^{2} w}{\partial x \partial y}+aw = 0 take, if w=φ(u)w = \varphi (u), where u=(x−x0)(y−y0)u = (x-x_{0})(y-y_{0})?

?
Problem 1379

What form does the equation ∂2u∂x2+∂2u∂y2+∂2u∂z2+u=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}+u = 0 take, if u=φ(r)u = \varphi (r), r=x2+y2+z2r = \sqrt{x^{2}+y^{2}+z^{2}}?

?
Problem 1380

Transform the expressions

Δ1v=(∂v∂x)2+(∂v∂y)2+(∂v∂z)2 \Delta _{1}v = \left(\frac{\partial v}{\partial x}\right)^{2}+\left(\frac{\partial v}{\partial y}\right)^{2}+\left(\frac{\partial v}{\partial z}\right)^{2}

and

Δ2v=∂2v∂x2+∂2v∂y2+∂2v∂z2 \Delta _{2}v = \frac{\partial ^{2} v}{\partial x^{2}}+\frac{\partial ^{2} v}{\partial y^{2}}+\frac{\partial ^{2} v}{\partial z^{2}}

to spherical coordinates, setting

x=rsin⁡θcos⁡φ,y=rsin⁡θsin⁡φ,z=vcos⁡θ. x = r\sin \theta \cos \varphi , \qquad y = r\sin \theta \sin \varphi , \qquad z = v\cos \theta .
?
Problem 1381

If x=φ(u,v)x = \varphi (u,v), y=ψ(u,v)y = \psi (u,v), and moreover ∂φ∂u=∂ψ∂v\frac{\partial \varphi }{\partial u} = \frac{\partial \psi }{\partial v}, ∂φ∂v=−∂ψ∂u\frac{\partial \varphi }{\partial v} = -\frac{\partial \psi }{\partial u}, then

∂2w∂u2+∂2w∂v2=(∂2w∂x2+∂2w∂y2)((∂φ∂u)2+(∂φ∂v)2). \frac{\partial ^{2} w}{\partial u^{2}}+\frac{\partial ^{2} w}{\partial v^{2}} = \left(\frac{\partial ^{2} w}{\partial x^{2}}+\frac{\partial ^{2} w}{\partial y^{2}}\right)\left(\left(\frac{\partial \varphi }{\partial u}\right)^{2}+\left(\frac{\partial \varphi }{\partial v}\right)^{2}\right).

Prove this.

?
Problem 1382

Prove that, introducing new variables by the formulas

X=dydx,Y=xdydx−y, X = \frac{dy}{dx}, \qquad Y = x\frac{dy}{dx}-y,

we obtain x=dYdXx = \frac{dY}{dX}, y=XdYdX−Yy = X\frac{dY}{dX}-Y.

?
Problem 1383

The Ampère transformation consists in introducing new variables by the formulas:

X=x,Y=∂z∂y,Z=z−y∂z∂y. X = x, \qquad Y = \frac{\partial z}{\partial y}, \qquad Z = z-y\frac{\partial z}{\partial y}.

Prove that this yields the equalities:

∂Z∂X=∂z∂x,∂Z∂Y=−y,z=Z−Y∂Z∂Y. \frac{\partial Z}{\partial X} = \frac{\partial z}{\partial x}, \qquad \frac{\partial Z}{\partial Y} = -y, \qquad z = Z-Y\frac{\partial Z}{\partial Y}.
?
Problem 1384

The Legendre transformation consists in introducing, in place of the variables xx, yy, and zz, new ones by the formulas X=pX = p, Y=qY = q, Z=px+qy−zZ = px+qy-z, where p=∂z∂xp = \frac{\partial z}{\partial x} and q=∂z∂yq = \frac{\partial z}{\partial y}. Prove that for the derivatives of the new variables one obtains the equalities:

∂Z∂X=x,∂Z∂Y=y, \frac{\partial Z}{\partial X} = x, \qquad \frac{\partial Z}{\partial Y} = y, ∂2Z∂X2=trt−s2,∂2Z∂X∂Y=−srt−s2,∂2Z∂Y2=rrt−s2, \frac{\partial ^{2} Z}{\partial X^{2}} = \frac{t}{rt-s^{2}}, \qquad \frac{\partial ^{2} Z}{\partial X \partial Y} = -\frac{s}{rt-s^{2}}, \qquad \frac{\partial ^{2} Z}{\partial Y^{2}} = \frac{r}{rt-s^{2}},

where

r=∂2z∂x2,s=∂2z∂x∂y,t=∂2z∂y2. r = \frac{\partial ^{2} z}{\partial x^{2}}, \qquad s = \frac{\partial ^{2} z}{\partial x \partial y}, \qquad t = \frac{\partial ^{2} z}{\partial y^{2}}.
?