Math Problems Vault

Problem 1.14.1

Find the angles at which the graph of the function $y = f(x)$ intersects the $x$ -axis:

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(1)

$y = \sin 3 x$

(2)

$y = \operatorname {tg} x$

(3)

$y = \ln \left|x\right|$ ;

(4)

$y = 1-e^{x}$

(5)

$y = \operatorname {arctg} \alpha x, \alpha > 0$

(6)

$y = (x-1)^{3}(x-2)^{2}(x-3)$ ;

(7)

$y = \frac{(x-1)(x+2)}{(x+1)(x-2)}$

(8)

$x = (t-1)^{2}(t-2), y = (t-1)^{2}(t-3), 2 < t < +\infty$ ;

(9)

$x = \frac{3 a t}{t^{3}+1}, y = \frac{3 a t^{2}}{t^{3}+1},-1 < t < \frac{1}{2}$

(10)

$x^{2}+y^{2}+2 y-9 = 0, y > -1$ .

Problem 1.14.2

Find the points at which the tangents to the graph of the function $y = f(x)$ are parallel to the $x$ -axis:

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(1)

$y = 2 x^{3}-3 x^{2}-12 x+7$ ;

(2)

$y = 3 x^{4}+4 x^{3}-12 x^{2}-1$ ;

(3)

$y = 3 x^{4}-28 x^{3}-6 x^{2}+84 x+1$ ;

(4)

$y = \cos 2 x-5 \cos x$ ;

(5)

$y = \left(3-x^{2}\right) e^{x}$ ;

(6)

$y = (2-x)^{3} /(x-3)^{2}$ ;

(7)

$y = \left|x-5\right|(x-3)^{3}$ ;

(8)

$x = t^{3} /\left(1+t^{2}\right), y = \left(t^{3}-2 t^{2}\right) /\left(1+t^{2}\right)$ ;

(9)

$x^{2}+2 y^{2}+4 x-4 y = 0, y > 1$ .

Problem 1.14.3

Write the equation of the tangent to the graph of the function $y = f(x)$ at the indicated point:

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(1)

$y = \sqrt{5-x^{2}}, x = 1$

(2)

$y = \operatorname {arctg} 2 x, x = 0$

(3)

$y = \ln \frac{x^{2}-2 x+1}{x^{2}+x+1}, x = 0$

(4)

$y = 4 \operatorname {ctg} x-\frac{\cos x}{\sin^{2} x}, x = \frac{\pi }{2}$

(5)

$y = \sqrt[3]{x-1}, x = 1$

(6)

$y = \left(x^{3}+2 x^{2}\right) /(x-1)^{2}, x = -2$ ;

(7)

$y = \left|x-1\right| \sqrt[3]{x+2}, x = 6$

(8)

$x^{2}+y^{2}-2 x+6 y = 0, y > -3, x = 0$ ;

(9)

$x = t e^{t}, y = t e^{-t}, t > -1, t = t_{0} > -1$ ;

(10)

$x = a(t-\sin t), y = a(1-\cos t), t = t_{0} \neq 2 \pi k, k \in \mathbb {Z}$ .

Problem 1.14.4

Write the equation of the tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ at the point $\left(x_{0} ; y_{0}\right)$ .

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Problem 1.14.5

Find the points of intersection with the $x$ -axis of the tangents to the ellipses

\frac{x^{2}}{100}+\frac{y^{2}}{b^{2}} = 1

at points with abscissa $x_{0}$ .

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Problem 1.14.6

Find the angle between the tangents to the curve

2 x^{2}-4 x y+y^{2}-2 x+6 y-3 = 0,

passing through the point $(3 ; 4)$ .

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Problem 1.14.7

Write the equation of the normal to the graph of the function $y = f(x)$ at the indicated point:

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(1)

$y = \cos 2 x-2 \sin x, x = \pi$

(2)

$y = \operatorname {arcctg}(1 / x), x = 1$ ;

(3)

$y = x^{3} /(2-x)^{2}, x = 6$ ;

(4)

$y = x / \sqrt[3]{x+1}, x = -3$ ;

(5)

$y^{2} = 2 p x, y \geqslant 0, x = x_{0}$ ;

(6)

$x = e^{2 t} \cos^{2} t, y = e^{2 t} \sin^{2} t,\left|t\right| < \pi / 4, t = \pi / 6$ .

Problem 1.14.8

Write the equation of the normal to the ellipse $3 x^{2}+2 x y+2 y^{2}+3 x- -4 y = 0$ at the point $(-2 ; 1)$ .

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Problem 1.14.9

Write the equation of the normal to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1$ at the point $\left(x_{0} ; y_{0}\right)$ .

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Problem 1.14.10

Write the equation of the tangent and the normal to the curve at the point $M$ :

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(1)

$x^{3}+y^{2}+2 x-6 = 0, M(-1 ; 3)$

(2)

$4 x^{3}-3 x y^{2}+6 x^{2}-5 x y-8 y^{2}+9 x+14 = 0, M(-2 ; 3)$ ;

(3)

$y^{4}-4 x^{4}-6 x y = 0, M(1 ; 2)$ ;

(4)

$x^{5}+y^{5}-2 x y = 0, M(1 ; 1)$ ;

(5)

$(4-x) y^{2} = x^{3}, M\left(x_{0} ; y_{0}\right)$ ;

(6)

$\left(\frac{x}{a}\right)^{n}+\left(\frac{y}{b}\right)^{n} = 2, n \in \mathbb {N}, M(a ; b)$ ;

(7)

$x = t^{2}, y = t^{3}, M(4 ; 8)$ ;

(8)

$x = t \cos t, y = t \sin t, M\left(\frac{\pi \sqrt{2}}{8} ; \frac{\pi \sqrt{2}}{8}\right)$ ;

(9)

$x = \sqrt{2} \cos^{3} t, y = \sqrt{2} \sin^{3} t, M(1 / 2 ; 1 / 2)$ ;

(10)

$x = (1+t) / t^{3}, y = 3 /\left(2 t^{2}\right)+1 /(2 t), M(2 ; 2)$ ;

(11)

$x = (2 t-1) / t^{2}, y = \left(3 t^{2}-1\right) / t^{3}, M(1 ; 2)$ .

Problem 1.14.11

Determine at which points and at what angle the graphs of the functions intersect:

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(1)

$f_{1}(x) = x-x^{3}, f_{2}(x) = 5 x$

(2)

$f_{1}(x) = \sqrt{2} \sin x, f_{2}(x) = \sqrt{2} \cos x$

(3)

$f_{1}(x) = x^{2}, f_{2}(x) = x^{3}$ ;

(4)

$f_{1}(x) = 1 / x, f_{2}(x) = \sqrt{x}$ ;

(5)

$f_{1}(x) = x^{3}, f_{2}(x) = 1 / x^{2}$ ;

(6)

$f_{1}(x) = \ln x, f_{2}(x) = x^{2} /(2 e)$ ;

(7)

$f_{1}(x) = x^{2}-4 x+4, f_{2}(x) = -x^{2}+6 x-4$ ;

(8)

$f_{1}(x) = 4 x^{2}+2 x-8, f_{2}(x) = x^{3}-x+10$ ;

(9)

$f_{1}(x) = \varphi (x), f_{2}(x) = \varphi (x) \sin \pi x, \varphi (x)$ is an everywhere differentiable function.

Problem 1.14.12

Determine at which points and at what angle the curves intersect:

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(1)

$y = x^{2}$ and $x = y^{2}$ ;

(2)

$y = \frac{2}{3} x^{5}-\frac{1}{9} x^{3}$ and $x = 1$ ;

(3)

$y = e^{x / 2}$ and $x = 2$ ;

(4)

$x^{2}+y^{2} = 5$ and $y^{2} = 4 x$ ;

(5)

$y^{2} = 2 x^{3}$ and $64 x-48 y-11 = 0$ ;

(6)

$x^{3}+y^{3}-x y-7 = 0$ and $y = x+1$ ;

(7)

$x = \frac{t^{3}}{1+t^{2}}, y = \frac{t^{3}-2 t^{2}}{1+t^{2}}$ and $y+\ln \frac{5 x}{8} = 0$ ;

(8)

$x = \left(1+\frac{1}{t}\right)^{t}, y = \left(1+\frac{1}{t}\right)^{t+1}$ and $y = x$ .

Problem 1.14.13

Prove that the families of parabolas $y^{2} = p^{2}-2 p x$ and $y^{2} = 2 q x + q^{2}, p \neq 0, q \neq 0$ , form an orthogonal net, i.e., the curves of these families intersect at right angles.

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Problem 1.14.14

Prove that the families of hyperbolas $x^{2}-y^{2} = a$ and $x y = b$ form an orthogonal net.

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Problem 1.14.15

Prove that the family of ellipses

x = \frac{1}{2}\left(a+\frac{1}{a}\right) \cos t, \quad y = \frac{1}{2}\left(a-\frac{1}{a}\right) \sin t, \quad 0 \leqslant t < 2 \pi

and the family of hyperbolas

x = \frac{1}{2}\left(t+\frac{1}{t}\right) \cos \alpha , \quad y = \frac{1}{2}\left(t-\frac{1}{t}\right) \sin \alpha , \quad 0 < t < +\infty

( $a$ and $\alpha$ are constants, with $a > 0, \alpha \neq \pi k / 2, k \in \mathbb {Z}$ ), form an orthogonal net.

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Problem 1.14.16

Determine the angle between the left- and right-hand tangents at the point $M$ of the graph of the function $y = f(x)$ :

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(1)

$y = \left|x\right|, M(0 ; 0)$ ;

(2)

$y = \sqrt[3]{x^{2}}, M(0 ; 0)$

(3)

$y = \arccos (\cos x), M(\pi ; \pi )$ ;

(4)

$y = \sqrt{1-e^{-3 x^{2}}}, M(0 ; 0)$ ;

(5)

$y = \sqrt{2 x^{3}+9 x^{2}}, M(0 ; 0)$ ;

(6)

$y = \frac{1}{\sqrt{3}} \arcsin \frac{2 x}{1+x^{2}}, M\left(1 ; \frac{\pi }{2 \sqrt{3}}\right)$ ;

(7)

$x^{3}+y^{3} = 3 x^{2}, M(0 ; 0)$ ;

(8)

$y = \begin{cases} \frac{x}{1+e^{1 / x}}, & \text{ if } x \neq 0, \\ 0, & \text{ if } x = 0,\end{cases} \quad M(0 ; 0)$ ;

(9)

$y = \frac{\sqrt{1+\left|x-2\right|}}{1+x}, M\left(2 ; \frac{1}{3}\right)$ ;

(10)

$x = t-\frac{t}{1+t^{2}}, y = \frac{2+t^{2}}{1+t^{2}}, M(0 ; 2)$ .

Problem 1.14.17

(1)(2)(3)(4)

Compute, at the point $(1 ; 2)$ of the parabola $y^{2} = 4 x$ , the lengths of the segments:

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(1)

of the tangent;

(2)

of the normal;

(3)

of the subtangent;

(4)

of the subnormal.

Problem 1.14.18

Find the length of the subtangent of the graph of the function $y = a^{x}, a > 0$ , $a \neq 1$ , at each of its points.

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Problem 1.14.19

Find the lengths of the subtangent and the subnormal of the graph of the function $y = a x^{n}, n \in \mathbb {N}, a \neq 0$ , at each of its points.

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Problem 1.14.20

(1)(2)

Find the lengths of the subtangent and the subnormal:

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(1)

of the parabola $y^{2} = 2 p x$ ;

(2)

of the hyperbola $x^{2}-y^{2} = a^{2}$ .

Problem 1.14.21

Prove that the subtangents of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ and the circle $x^{2}+y^{2} = a^{2}$ at points with equal abscissas are equal.

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Problem 1.14.22

Prove that at points with equal abscissas the subnormals of the graphs of the functions

y = f(x) \quad \text{ and } \quad y = \sqrt{f^{2}(x)+a^{2}}

where $f(x)$ is a differentiable function, are equal.

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Problem 1.14.23

(1)(2)

Find the lengths of the subtangent and the subnormal for the curve:

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(1)

$y^{2} = x^{3}$

(2)

$x y^{2} = 1$ .

Problem 1.14.24

Find the lengths of the segments of the tangent, the normal, the subtangent, and the subnormal of the cycloid

x = a(t-\sin t), \quad y = a(1-\cos t)

at each of its points not lying on the $x$ -axis.

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Problem 1.14.25

Find the lengths of the segments of the tangent and the normal of the tractrix $x = a(\ln \operatorname {tg}(t / 2)+\cos t), \quad y = a \sin t, \quad 0 < t < \pi , \quad a > 0$ .

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Problem 1.14.26

Prove that for the curve

x = 2 a\left(\ln \sin t-\sin ^{2} t\right), \quad y = a \sin 2 t, \quad 0 < t < \pi , \quad a > 0

the sum of the lengths of the subtangent and the subnormal is constant and equal to $2 a$ .

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Problem 1.14.27

Prove that if $r = f(\varphi )$ is the equation of a curve in the polar coordinate system, and $\omega$ is the angle formed by the tangent and the polar radius of the point of tangency, then $\operatorname {tg} \omega = r /\left|r^{\prime }\right|$ .

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Problem 1.14.28

Find the angle between the tangent and the polar radius of the point of tangency for the following curves:

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(1)

the spiral of Archimedes $r = a \varphi$ ;

(2)

the hyperbolic spiral $r = a / \varphi$ ;

(3)

the logarithmic spiral $r = a e^{b \varphi }$ ;

(4)

the cardioid $r = a(\cos \varphi +1)$ ;

(5)

the arc of the lemniscate of Bernoulli $r^{2} = a^{2} \cos 2 \varphi , 0 \leqslant \varphi < \pi / 4$ .

Problem 1.14.29

At what angle do the curves $r = \varphi$ and $r = 1 / \varphi$ intersect at the point $(1 ; 1)$ ?

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Problem 1.14.30

Prove that the angle between the tangent to the spiral of Archimedes $r = a \varphi$ and the polar radius of the point of tangency tends to $\pi / 2$ as $\varphi \rightarrow +\infty$ .

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Problem 1.14.31

Find the distance from the pole to an arbitrary tangent of the curve $r = a e^{b \varphi }$ .

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Problem 1.14.32

Write, in Cartesian and in polar coordinates, the equation of the normal to the cardioid $r = a(1+\cos \varphi )$ at the point with polar angle $\varphi = \pi / 6$ .

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Problem 1.14.33

The Moscow-Tbilisi train covers a distance of 2400 km in 44 h 14 min. Determine the average speed of the train.

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Problem 1.14.34

Determine the average rate of change of the function $y = \sin (1 / x)$ on the interval $[2 / \pi ; 6 / \pi ]$ .

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Problem 1.14.35

A body of mass $m = 1,5$ moves rectilinearly according to the law $S(t) = t^{2}+t+1$ . Find the kinetic energy of the body 5 s after the start of motion (the mass $m$ is given in kilograms, the distance $S$ in meters).

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Problem 1.14.36

A point moves along the parabola $y = 8 x-x^{2}$ so that its abscissa changes according to the law $x = \sqrt{t}$ ( $x$ is measured in meters, $t$ in seconds). What is the rate of change of the ordinate of the point 9 s after the start of motion?

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Problem 1.14.37

The radius of a ball increases uniformly at a rate of $5 \mathrm{~ cm} / \mathrm{c}$ . What is the rate of change of the volume of the ball at the moment when its radius becomes equal to 50 cm?

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Problem 1.14.38

A wheel rotates so that the angle of rotation is proportional to the square of time. The first revolution was completed in $8 \mathrm{~ s}$ . Find the angular velocity 64 s after the start of motion.

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Problem 1.14.39

Two points move along the $x$ -axis, having the laws of motion

x = 100+5 t \quad \text{ and } \quad x = t^{2} / 2.

At what speed are they receding from each other at the moment they meet ( $x$ is measured in meters, $t$ in seconds)?

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Problem 1.14.40

A ferry is pulled toward the shore by a rope that is wound onto a winch at a speed of $3 \mathrm{m} /$ min. Determine the speed of the ferry at the moment when it is 25 m from the shore, if the winch is located on the shore 4 m above the surface of the water.

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Problem 1.14.41

The law of motion of a material point thrown at an angle $\alpha$ to the horizontal with initial speed $v_{0}$ , neglecting air resistance, has the form

x = \left(v_{0} \cos \alpha \right) t, \quad y = \left(v_{0} \sin \alpha \right) t-\left(g t^{2}\right) / 2

where $t$ is the time, $g$ is the acceleration due to gravity. Determine the coordinates of the velocity vector and the magnitude of the velocity.

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Problem 1.14.42

A point moves along the spiral of Archimedes $r = a \varphi$ so that the angular velocity of rotation of its polar radius is constant and equal to $6^{\circ }$ per second. Determine the rate of elongation of the polar radius $r$ , if $a = 10 \mathrm{~ m}$ .

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Problem 1.14.43

The distance $r$ of a satellite of the Earth from its center can be approximately expressed by the formula

r = a\left(1-\varepsilon \cos \frac{2 \pi \left(t-t_{0}\right)}{P}-\frac{\varepsilon ^{2}}{2}\left(\cos \frac{4 \pi (t-t_{0})}{P}-1\right)\right)

where $t$ is the time, $a$ is the semi-major axis of the elliptical orbit, $\varepsilon$ is its eccentricity, $P$ is the period of revolution of the satellite, $t_{0}$ is the time of passage through perigee. Find the magnitude of the rate of change of the distance $r$ (the so-called radial velocity of the satellite).

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Problem 1.14.44

The amount of heat $Q$ in J needed to heat 1 kg of water from $0^{\circ } \mathrm{C}$ to $t^{\circ } \mathrm{C}$ is given by the formula

Q = t+2 \cdot 10^{-5} t^{2}+3 \cdot 10^{-7} t^{3}

Determine the heat capacity of water at $t = 100^{\circ } \mathrm{C}$ .

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Problem 1.14.45

The mass $m(t)$ of a radioactive substance changes according to the law

m = m_{0} 2^{\left(t_{0}-t\right) / T}

where $t$ is the time, $m_{0}$ is the mass at time $t_{0}$ , $T$ is the half-life. Prove that the rate of decay of the radioactive substance is proportional to the amount of the substance. Find the coefficient of proportionality.

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