Derivatives, differentiation rules, and applications such as tangents, extrema, and related rates.
These practice problems cover the derivative, from the limit definition through the applications a first calculus course expects. You will differentiate explicitly and implicitly, approximate with differentials and Taylor polynomials, and use derivatives for tangents, extrema, related rates, and curve sketching. Every problem includes a full step-by-step solution.
Definition of the Derivative is the difference quotient, for readers who still need to see where the rules come from. Differentiation Techniques covers the product, quotient, and chain rules, together with implicit and logarithmic differentiation. Approximation and Higher Derivatives treats differentials and derivatives of higher order. Applications of Derivatives is the home of related rates, maxima and minima, and optimization. Theorems and Indeterminate Forms includes the Mean Value Theorem, Rolle's theorem, and L'Hôpital's rule. Curve Sketching and Tangents, Normals, and Asymptotes ask you to recover the shape of a graph from derivative information. Taylor Series builds a local polynomial from derivatives at a point.
If the limit definition is unfamiliar, go back to Limits, which is the easier set. After the differentiation rules are reliable, move on to Integrals. For convergence of series, remainders, and Fourier expansions, continue with Series, which is the harder companion to the Taylor chapter here.
Problem 1
EasyDefinition of the Derivative
Find the slope of the secant line passing through the points $M_1(1,\,y_1)$ and $M_2(2,\,y_2)$ on the hyperbola $y = \frac{1}{x}$.
The law of motion of a point is $s = \sin t$, where $s$ is the distance in centimetres and $t$ is the time in seconds. Find the average velocity of the point over the time interval from $t = 0$ to $t = \dfrac{\pi}{2}$.
Given $f(x) = \begin{cases} \arctan\!\left(1+\dfrac{1}{x}\right), & x \neq 0, \\[6pt] 0, & x = 0, \end{cases}$ find (a) $f'_{-}(0)$ and $f'_{+}(0)$; (b) $\lim_{x\to 0^{+}}f'(x)$ and $\lim_{x\to 0^{-}}f'(x)$. Does $f'(0)$ exist?
Show that the following functions have no finite derivative at the indicated points: a) $y = \sqrt{x}$ at $x = 0$; b) $y = \sqrt[n]{x-1}$ at $x = 1$ ($n \geq 2$); c) $y = \tan x$ at $x = \dfrac{\pi}{2}+2k\pi$; d) $y = |\sin x|$ at $x = k\pi$, $k = 0,\pm 1,\pm 2,\ldots$.
The law of motion of a point is $s = 3t^{2} + 2t + 1$, where $s$ is the distance in metres and $t$ the time in seconds. Find the velocity of the point at $t = 2$.
Find $y'$ for $y = \sqrt{a^{2}+b^{2}-2ab\cos\varphi}$, then evaluate $y'\!\left(\dfrac{\pi}{2}\right)$, $y'(\pi)$ and $y'\!\left(\dfrac{3\pi}{2}\right)$. Here $a, b \gt 0$.
Prove that if $D(x) = \det\bigl[f_{ij}(x)\bigr]_{n\times n}$, then $D'(x) = \sum_{k=1}^{n} D_{k}(x)$ where $D_{k}(x)$ is obtained from $D(x)$ by replacing the $k$-th row with its derivative $\bigl(f'_{k1},\,f'_{k2},\ldots,f'_{kn}\bigr)$, all other rows unchanged.
Given $f(x) = \begin{cases} 1+x, & x \lt 0, \\ x, & 0 \lt x \lt 1, \\ 2-x, & 1 \lt x \lt 2, \\ 3x-x^{2}, & x \gt 2, \end{cases}$ 1° sketch the graph; 2° for which $x$ is $f$ continuous? 3° for which $x$ does $f'(x)$ exist? 4° for which $x$ is $f'$ continuous?
Given $f(x) = \dfrac{\sqrt{x+1}-1}{x}$ for $x \neq 0$: 1° find $\displaystyle\lim_{x\to 0}f'(x)$; 2° if $f$ is extended by $f(0) = \tfrac12$, find $f'(0)$.
Given $f(x) = \begin{cases} (x-2)\arctan\dfrac{1}{x-2}, & x \neq 2, \\[6pt] 0, & x = 2, \end{cases}$ is $f$ differentiable for every $x$? Find $f'(2)$.
Prove that if $f$ is differentiable and periodic with period $\omega$, so that $f(x+\omega) = f(x)$ for all $x$, then $f'$ is periodic with the same period $\omega$.
Prove the following identities by showing that the derivative of the left-hand side is zero and then evaluating at one point: (a) $\arcsin x + \arccos x = \dfrac{\pi}{2}$ on $[0,1]$; (b) $\arctan x + \operatorname{arccot} x = \dfrac{\pi}{2}$ on $[0,\infty)$; (c) $\arctan x + \arctan\dfrac{1}{x} = \pm\dfrac{\pi}{2}$; (d) $2\arctan x + \arcsin\dfrac{2x}{1+x^{2}} = \pi$ on $[1,\infty)$.
Show that $y$ as a function of $x$, given parametrically by $x = 2t+3t^{2}$, $y = t^{2}+2t^{3}$, satisfies $2y'^{3} + y'^{2} - y = 0$, where $y' = \dfrac{dy}{dx}$.
Given $x = \dfrac{at}{1+t^{3}}$, $y = \dfrac{a\bigl(4t^{3}+1\bigr)}{6\bigl(1+t^{3}\bigr)^{2}}$, find $y'$ and determine the algebraic equation relating $x$ and $y'$.
Railway rails are made of equal iron segments of length $l_{0} = 20\ \text{m}$ at $0^{\circ}\text{C}$. Find the minimum gap between two adjacent rails so that they do not touch when the ambient temperature rises from $t_{1} = 20^{\circ}\text{C}$ to $t_{2} = 40^{\circ}\text{C}$. The coefficient of linear thermal expansion of iron is $\alpha = 1.2\cdot10^{-5}\ {}^{\circ}\text{C}^{-1}$.
Because of overloading, the power of a heater connected to a network of voltage $U_{1} = 220\ \text{V}$ drops from $P_{1} = 1000\ \text{W}$ to $P_{2} = 800\ \text{W}$. Find the drop in the network voltage, assuming the resistance $R$ stays constant.
Calculate the value of the differential of each function over the indicated change of the variable. 1° $y = \dfrac{1}{(\tan x+1)^{2}}$, from $x = \dfrac{\pi}{6}$ to $x = \dfrac{61\pi}{360}$; 2° $y = \sin 2\varphi$, from $\varphi = \dfrac{\pi}{6}$ to $\varphi = \dfrac{61\pi}{360}$; 3° $y = \cos^{2}\varphi$, from $\varphi = 60^{\circ}$ to $\varphi = 60^{\circ}30'$.
For $y = x^{2}$ at $x = 1$, compute the increment $\Delta y$ and the differential $dy$ for 1° $\Delta x = 0.1$; 2° $\Delta x = 0.01$; 3° $\Delta x = 0.001$. Determine the absolute and relative errors committed when $\Delta y$ is replaced by $dy$.
For $y = x^{3}-x$ at $x = 2$, compute $\Delta y$ and $dy$ for 1° $\Delta x = 1$; 2° $\Delta x = 0.1$; 3° $\Delta x = -0.01$. Find the absolute and relative errors committed in replacing $\Delta y$ by $dy$.
If the $n$-th significant digit of an approximate value $x$ is in error by at most half a unit in that place, find a bound for the error of the mantissa of $\log_{10}x$.
A motion is given by $x = 5t^{2}$ (metres, seconds). At $t = 2\ \text{s}$, compute $\Delta x$ and $dx$ for 1° $\Delta t = 1\ \text{s}$; 2° $\Delta t = 0.1\ \text{s}$; 3° $\Delta t = 0.001\ \text{s}$.
The side of a square is $a = 8$ length units. By how much does its area increase if the side increases by 1° $\Delta a = 1$; 2° $\Delta a = 0.5$; 3° $\Delta a = 0.1$ length units? Replace the increment by the differential and estimate the relative error.
Find the increment $\Delta V$ of the volume of a sphere when the radius changes by $\Delta r$. Compute $\Delta V$ and $dV$ for $r = 3$ with 1° $\Delta r = 0.5$; 2° $\Delta r = 0.1$; 3° $\Delta r = 0.01$, and determine the error made in replacing the increment by the differential.
An isosceles triangle has equal sides $a = 4\ \text{cm}$ and included angle $\alpha = 60^{\circ}$. By how much must $\alpha$ increase so that the area increases by $0.16\ \text{cm}^{2}$?
A circular sector has radius $R = 100\ \text{cm}$ and central angle $\alpha = 60^{\circ}$. By how much does its area decrease if the central angle decreases by $30'$?
By how much does the volume of a regular hexagonal pyramid with lateral edge $l = 5\ \text{cm}$ increase when the base edge grows from $a = 4\ \text{cm}$ to $a = 4.1\ \text{cm}$?
The length of a telegraph wire is $s = 2b\left(1+\dfrac{2f^{2}}{3b^{2}}\right)$, where $2b$ is the distance between the supports and $f$ is the maximum sag. By how much does the sag increase when the length increases by $ds$ because of heating?
In reading a scale reflected in a mirror by a telescope, the deflection $x$ is measured at a fixed scale distance $s$. How do small errors in the measured deflection $x$ affect the value of the angle $\alpha$?
Prove the following approximate formulas for sufficiently small $\alpha$: 1° $\dfrac{1}{1+\alpha}\approx 1-\alpha$; 2° $\sqrt{1+\alpha}\approx 1+\tfrac12\alpha$; 3° $\dfrac{1}{(1+\alpha)^{2}}\approx 1-2\alpha$.
Prove the following approximate formulas for sufficiently small $\alpha$: 1° $\bigl(1\pm\alpha\bigr)^{3}\approx 1\pm3\alpha$; 2° $\dfrac{1}{\sqrt{1+\alpha}}\approx 1-\tfrac12\alpha$.
Prove the following approximate formulas for sufficiently small $\alpha$: 1° $\cos\alpha\approx1$; 2° $\sin\alpha\approx\alpha$; 3° $\tan\alpha\approx\alpha$.
The function $f(x) = x^{2}$ is given. At some point $x$ the increment of the independent variable is $\Delta x = 0.2$, and the corresponding differential is $df = 0.8$. Find the value of $x$.
A function $y = f(x)$ is given. At some point the increment of the independent variable is $\Delta x = 0.2$ and the corresponding differential is $df = 0.8$. Find the value of the derivative at that point.
The period of oscillation of a pendulum is $T = 2\pi\sqrt{\dfrac{l}{981}}$ seconds, with $l$ in centimetres. By how much must the length $l = 20\ \text{m}$ be changed so that the period decreases by $0.1\ \text{s}$?
In uniform motion the distance $s$ is measured with an accuracy of $0.2\%$ and the time $t$ with an accuracy of $0.4\%$. With what accuracy can the velocity be computed?
Prove that if $u,v,w,t$ are functions of $x$ possessing first and second derivatives, then \[ \begin{vmatrix} u & v & w \\ \bigl(ut\bigr)' & \bigl(vt\bigr)' & \bigl(wt\bigr)' \\ \bigl(ut\bigr)'' & \bigl(vt\bigr)'' & \bigl(wt\bigr)'' \end{vmatrix} = t^{2}\begin{vmatrix} u & v & w \\ u' & v' & w' \\ u'' & v'' & w'' \end{vmatrix}. \].
Let $y = f(x)$ possess derivatives of all orders. Prove that \[ y'\sum_{v=1}^{n}\bigl(-1\bigr)^{v}\binom{n}{v}y^{-v}\frac{d^{n}\bigl(y^{v}\bigr)}{dx^{n}} = \frac{y}{n+1}\sum_{v=1}^{n+1}\bigl(-1\bigr)^{v+1}\binom{n+1}{v}y^{-v}\frac{d^{n+1}\bigl(y^{v}\bigr)}{dx^{n+1}}. \tag{$*$} \].
Determine: 1° $\dfrac{d^{n}\bigl(x^{2}-1\bigr)^{n}}{dx^{n}}$; 2° its value at $x = 1$ and $x = -1$; 3° $\dfrac{d^{\,n-1}\bigl(x^{2}-1\bigr)^{n}}{dx^{\,n-1}}$ at $x = 1$ and $x = -1$.
A point moves rectilinearly according to $s = \dfrac{2}{9}\sin\dfrac{\pi t}{2}+s_{0}$ (cm, s). Find the acceleration $w$ at the end of the first second.
A point $M$ moves along the circle $x^{2}+y^{2} = a^{2}$ with constant angular velocity $\omega$, starting at $(a,0)$ when $t = 0$. Find the law of motion of its projection $M_{1}$ onto $Ox$, together with the velocity and acceleration of $M_{1}$: at the initial moment, at the passage through the origin, and their maximum values.
Derive formulas for the second and third derivatives of the inverse function $x = f^{-1}(y)$ of $y = f(x)$, expressing $\dfrac{d^{2}x}{dy^{2}}$ and $\dfrac{d^{3}x}{dy^{3}}$ through $y'_{x},\ y''_{xx},\ y'''_{xxx}$.
Let $x = x(y)$ be the inverse of $y = y(x)$. Prove that \[ \frac{d^{3}y}{dx^{3}}\left(\frac{dx}{dy}\right)^{2}+\frac{d^{3}x}{dy^{3}}\left(\frac{dy}{dx}\right)^{2}+3\,\frac{d^{2}y}{dx^{2}}\cdot\frac{d^{2}x}{dy^{2}} = 0. \].
A body moves along the $Ox$ axis according to $x = \dfrac{t^{3}}{3}-\dfrac{3}{2}t^{2}-4t$. Find the velocity and acceleration, and determine the moments at which the body changes direction.
A body of mass $a$ kg moves in a straight line according to $s = 1+t+t^{2}$. Find its kinetic energy $\dfrac{mv^{2}}{2}$ five seconds after the start of motion.
A train starts with speed $54\ \text{km/h}$ and covers $600\ \text{m}$ in the first $30\ \text{s}$. Assuming uniform acceleration, find its speed and acceleration at the end of the 30th second. The track has radius of curvature $R$.
A point oscillates according to $x = a\cos\omega t$. Find the velocity and acceleration at $x = \pm a$ and at $x = 0$, and show that $\dfrac{d^{2}x}{dt^{2}} = -\omega^{2}x$.
A point moves harmonically, $x = a\sin kt$. Determine the amplitude $a$ and the circular frequency $k$ given that $v = v_{1}$ when $x = x_{1}$, and $v = v_{2}$ when $x = x_{2}$.
A wheel rotates so that the angle of rotation is proportional to the square of the time, $\varphi = ct^{2}$. The first revolution is completed in $8\ \text{s}$. Find the angular acceleration $32\ \text{s}$ after the start.
The ends of a rod $AB = 10\ \text{cm}$ slide along the coordinate axes, $A$ along $Ox$ and $B$ along $Oy$. The point $A$ moves away from the origin at $\dfrac{dx}{dt} = 2\ \text{cm/s}$. Find the speed of $B$ at the moment when $OA = 6\ \text{cm}$.
A point $A(x,y)$ moves uniformly round the circle $x^{2}+y^{2} = r^{2}$, completing one revolution in $T$ seconds, starting from $A_{0}(r,0)$ at $t = 0$. Find the velocity and acceleration of its projection onto $Ox$.
A point moves rectilinearly so that its velocity is proportional to the square root of the distance travelled, $v = c\sqrt{s}$. Show that the motion occurs under a force of constant magnitude.
A person of height $h = 1.7\ \text{m}$ walks away from a lamp $H = 3\ \text{m}$ above the ground at $6.34\ \text{km/h}$. How fast does the tip of the shadow of their head move?
Drops fall from the mouth of a vertical pipe one after another at intervals of $\tau = 0.1\ \text{s}$, each starting from rest. Find the difference in the velocities of the first and second drops one second after the first leaves.
A wheel of radius $a$ rolls along a straight line, turning through $\varphi = t+\dfrac{t^{2}}{2}$ radians in $t$ seconds. Find the velocity and acceleration of its centre.
Find the ratio of the velocity of the centre of a sphere of radius $R$, rolling without slipping on two parallel rails a distance $d$ apart, to the angular velocity of the sphere.
A body is thrown vertically upward with initial speed $v_{0}$. Find (a) its height after $t$ seconds, (b) its velocity and acceleration, (c) the time to reach the greatest height and that height.
A heavy point is launched in the vertical plane $Oxy$ at an angle $\alpha$ to the horizon with initial speed $v_{0}$. Neglecting air resistance, find the equations of motion, the velocity, the acceleration and the trajectory.
A projectile leaves with speed $v_{0} = 400\ \text{m/s}$ at an angle $\alpha$ to the horizon. Find the height at which it strikes a vertical wall $3000\ \text{m}$ away, and its speed at that moment. Air resistance is neglected.
A rocket starting from rest ejects gas at exhaust speed $v_{1} = 300\ \text{m/s}$, at a rate of $m_{1} = 90\ \text{g/s}$, and its initial mass is $m_{2} = 990\ \text{g}$. After how many seconds does its acceleration reach $w = 30\ \text{m/s}^{2}$?
A hemispherical bowl of radius $R$ is filled with water at the constant rate of $a$ litres per second. Find the rate at which the depth increases at depth $h$, and show that it is inversely proportional to the area of the free surface.
Barometric pressure obeys $\ln\dfrac{p}{p_{0}} = ch$, where $p_{0}$ is the pressure at sea level and $h$ the altitude. At $h = 5540\ \text{m}$ the pressure is half the normal value. Find the rate of change of pressure with altitude.
A straight conductor slides with constant velocity $v$ across a circular conducting ring of radius $R$, in a uniform magnetic field of induction $B$ perpendicular to the plane of the ring. Find the electromotive force in the circuit when the conductor is at distance $d$ from the centre.
Show that $f(x) = x^{4}-2x^{2}$ satisfies the conditions of Rolle's theorem on $\bigl[0,\sqrt2\,\bigr]$ and find the corresponding $\xi$. Do the same on $\bigl[-\sqrt2,\sqrt2\,\bigr]$.
If the equation $a_{0}x^{n}+a_{1}x^{\,n-1}+\cdots+a_{n-1}x = 0$ has a positive root $x = x_{0}$, show that the equation $na_{0}x^{\,n-1}+\bigl(n-1\bigr)a_{1}x^{\,n-2}+\cdots+a_{n-1} = 0$ has at least one positive root smaller than $x_{0}$.
Sketch the graph of $f(x) = \begin{cases} x, & |x|\lt 2,\\ 1, & |x|>2, \end{cases}$ and explain why Lagrange's theorem cannot be applied to it on $[0,2]$.
On the curve $y = x^{2}-1$ the points $A\bigl(0,y_{1}\bigr)$ and $B\bigl(1,y_{2}\bigr)$ are given. Using Lagrange's theorem, find the point $M$ of the arc $AB$ for which the area of the triangle $AMB$ is greatest, and compute that area.
Let $f$ satisfy the conditions of Lagrange's theorem on $[a,b]$, and set \[ F(x) = \begin{vmatrix} x & f(x) & 1 \\ b & f(b) & 1 \\ a & f(a) & 1 \end{vmatrix}. \] Prove Lagrange's theorem by applying Rolle's theorem to $F$, and explain the geometric meaning of $F$.
Let $f$ satisfy the conditions of Cauchy's theorem on $[0,x]$ and suppose $f(0) = f'(0) = \cdots = f^{(n-1)}(0) = 0$. Prove that $\dfrac{f(x)}{x^{n}} = \dfrac{f^{(n)}\bigl(\theta x\bigr)}{n!}$ with $0\lt \theta\lt 1$.
In the formula $f(x+h)-f(x) = h\,f'\bigl(x+\theta h\bigr)$, $0\lt \theta\lt 1$, find $\theta$ as a function of $x$ and $h$ for $f(x) = \sqrt{x}$, and compute $\lim_{h\to0}\theta$.
Given the curve $y = \arctan\bigl(\ln x\bigr)$. 1° Find the angle at which the curve intersects the $x$-axis. 2° Calculate $\displaystyle\lim_{x\to0^{+}}y'$ and $\displaystyle\lim_{x\to\infty}y'$.
Find the order of the infinitesimal $\alpha(x) = e-\bigl(1+x\bigr)^{1/x}$ with respect to $x$ as $x\to0$. Recall that $\alpha(x)$ is of order $n$ with respect to $x$ if $\displaystyle\lim_{x\to0}\frac{\alpha(x)}{x^{n}} = C\neq0$.
Given the function $f(x) = \begin{cases} \dfrac{1}{1+x^{2}}, & x\lt \dfrac12,\\[8pt] \dfrac25x+\dfrac35, & \dfrac12\lt x\lt 1,\\[8pt] x^{2}-6x+6, & x\gt 1, \end{cases}$ investigate 1° its continuity; 2° the derivative and its continuity; 3° the extrema, and the least and greatest values on $\left[\dfrac35,\dfrac52\right]$; 4° sketch the graph. (The function is not defined at $x = \tfrac12$ and $x = 1$.)
Find the relation between the coefficients $k$ and $m$ for which the function $\displaystyle f(x) = \sqrt2\,\frac{\sin^{2}x}{k\sqrt2-m\sin x}$ has an extremum at $x = \dfrac{\pi}{4}$.
Find the infimum and supremum of $\displaystyle f(x) = \left(1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}\right)e^{-x} = S_{n}(x)\,e^{-x}$ on $\bigl(0,+\infty\bigr)$, where $S_{n}$ is the $n$-th partial sum of $e^{x}$.
Determine $a$ and $b$ so that the point $\left(2,\,\dfrac52\right)$ is an inflection point of the curve $x^{2}y+ax+by = 0$. Has the curve any further inflection points?
In the function $\displaystyle f(x) = \frac{x+a}{\sqrt{x^{2}+b}}$, determine the parameters $a$ and $b$ so that $f$ has an extremum at $x = -1$ and an inflection point at $x = -2$.
Determine $a$ and $b$ so that the graph of $\displaystyle f(x) = \frac{\sin^{2}x}{a-b\cos x}$ has an extremum at the point $\left(\dfrac{\pi}{3},\,\dfrac14\right)$.
On the curve $y = x^{2}-1$ find the point at which 1° the normal is parallel to the line $y = x+1$; 2° the normal is perpendicular to the line $y = -x$.
Tangents to the curves $ay = x^{2}$ and $y = x^{2}+x$, drawn at points with the same abscissa, intersect at a point $S$. Determine $a$ so that $S$ lies on the line $y = 2x$.
Normals to the parabola $y = x^{2}$ are drawn at the points with abscissas $x$ and $x+h$. Find the limiting position of their intersection point as $h\to0$.
Given the parabolas $y^{2}+4x-4 = 0$ and $8x^{2}-2x-3y-6 = 0$, draw the normals to the first at their points of intersection and show that three of these normals meet at a single point.
The tangent to $y = x^{3}$ at a point $P$ meets the $x$-axis at $S$; the orthogonal projection of $P$ on the $x$-axis is $T$. Show that $\overline{OS} = 2\,\overline{ST}$, where $O$ is the origin.
The tangent to the curve $y = \bigl(\sqrt{a}-\sqrt{x}\bigr)^{2}$, $a\gt 0$, at an arbitrary point meets the axes at $A$ (on $Ox$) and $B$ (on $Oy$). Prove that $\overline{OA}+\overline{OB}$ is constant.
The tangent at a point $P$ of the curve $y = \bigl(x-2\bigr)^{2}\bigl(x+2\bigr)^{3}$ meets $Ox$ at $T$ and $Oy$ at $T_{1}$; the normal at $P$ meets $Ox$ at $N$ and $Oy$ at $N_{1}$. Prove that $\overline{T_{1}N}:\overline{TN_{1}} = y'$.
Find the tangents at an arbitrary point $\bigl(x_{0},y_{0}\bigr)$ to 1° $x^{2}+y^{2} = a^{2}$; 2° $y^{2} = 2px$; 3° $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}} = 1$; 4° $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}} = 1$.
The tangent to $y = \dfrac{a^{2}}{2x}$ at an arbitrary point meets the axes at $A$ and $B$. 1° Show that the area of $\triangle OAB$ is constant and equal to $a^{2}$. 2° Find the distance from $O$ to the tangent.
Let $A_{1}$ and $A_{2}$ be points on the axis of the parabola $y^{2} = 2px$ at equal distances from the focus, and let $d_{1},d_{2}$ be their distances to a tangent of the parabola. Show that $d_{1}^{2}-d_{2}^{2}$ is constant.
Find the equations of the tangent and normal to the curve $x(x-y)+1=0$ at an arbitrary point, and determine the limiting position of the tangent as the point of tangency recedes to infinity.
The tangent to the curve $ay = x^{3}$ at a point $A$ meets the curve again at a point $B$. Find the locus of the midpoints of the segment $\overline{AB}$.
Given the curve $y=\dfrac{x^{2}}{x^{2}-1}$, write the equation of the tangent at an arbitrary point, then find the coordinates of the point at which this tangent meets the curve again.
A tangent is drawn at an arbitrary point of the curve $y = x+\dfrac{1}{x^{2}}$. Determine the coordinates of the point $M$ at which this tangent meets the curve again.
Show that every tangent to the curve $y = \dfrac{1}{2}\sqrt{x-4x^{2}}$ meets the $y$-axis at a point $T_{1}$ that is equidistant from the origin $O$ and the point of tangency $P$.
The tangent to the curve $x^{3}-ay^{2}=0$ at an arbitrary point $T$ meets the curve again at $S$. Show that the $x$-axis divides the segment $\overline{ST}$ in the ratio $8:1$.
Given the curve $x^{2}-xy+y^{2}-1=0$. 1° Show that there exist two distinct points on the curve where the tangent is parallel to a given line. Can an arbitrary line be used? 2° At which points of the curve must normals be drawn so that they pass through the origin?
Show that the tangents to the curves $y=e^{x}$, $y=e^{-x}$, and $y=\tfrac12(e^{x}+e^{-x})$ drawn at points with the same abscissa $x=a$ have a common point.
Three curves are given by $C_{1}\colon y=f(x)$, $C_{2}\colon y=\varphi(x)$, $C_{3}\colon y=\dfrac{f(x)+\lambda\varphi(x)}{1+\lambda}$, where $\lambda$ is a real parameter and $f,\varphi$ are differentiable. Show that the tangents to these three curves drawn at points with the same abscissa $x=a$ meet at a single point.
The normal to the curve $y^{2}=c\,e^{x/a}+4a(x+a)$ at a point $M$ meets the $Ox$ axis at the point $S$. Prove that the midpoint of the segment $\overline{MS}$ lies on the parabola $y^{2}=ax$.
For the curve $x=t^{2},\ y=t^{3}$, find the pairs of points at which the tangent line at one point coincides with the normal line at the other — and show that this happens at the parameter values $t=\pm\dfrac{\sqrt2}{3}$ (together with their partners).
Determine the line that is simultaneously a tangent at one point and a normal at another point of the curve $x=1+\dfrac{1}{t^{2}},\ y=t+\dfrac{1}{t^{2}}$.
Find the equations of the tangent and normal to each curve at the given parameter value. 1° $x=\ln(\cos t)+1,\ y=\tan t+\cot t,\ t=\dfrac\pi4$. 2° $x=\sin t,\ y=a^{t},\ t=0$.
The tangent to the cycloid $x=a(t-\sin t),\ y=a(1-\cos t)$ drawn at an arbitrary point $M$ intersects the coordinate axes at points $S$ (on $Oy$) and $T$ (on $Ox$). Form the ratio $\dfrac{\overline{SM}^{\,2}}{\overline{ST}^{\,2}}$ as a function of $t$, and find its limit as $t\to0$.
A cycloid is generated by rolling a circle of radius $a$ along a straight line. Prove that the normal to the cycloid at any point $M$ passes through the current contact point $Q$ of the rolling circle with the line.
Prove that the segment on the $Ox$ axis cut between the tangent and the normal to the curve $x=2a(\ln\sin t-\sin^{2}t),\ y=a\sin2t$ at an arbitrary point $M$ has length $2a$.
Consider the epicycloid $x=a\!\left[(1+\lambda)\cos t-\lambda\cos\mu t\right],\ y=a\!\left[(1+\lambda)\sin t-\lambda\sin\mu t\right]$, $\mu=1+\tfrac1\lambda$, $\lambda>0$, and the circle $x^{2}+y^{2}=a^{2}$. Prove that the normal to the epicycloid at the point with parameter $t$ intersects the circle at the point $Q=(a\cos t,\,a\sin t)$.
For the family of curves $\rho^{m}=k^{m}\cos(m\theta)$, prove: 1° the angle $\varphi$ between the tangent and the radius vector satisfies $\varphi=\dfrac{(2n+1)\pi}{2}+m\theta$; 2° the angle $\psi$ between the polar axis and the normal satisfies $\psi=(m+1)\theta+n\pi$ ($n\in\mathbb Z$).
Prove that the tangent to the curve $x=a(1-\cos t),\ y=a(1-\sin t)$ at an arbitrary point $M$ forms with the coordinate axes a triangle of constant perimeter.
Let $A,B,C$ be three points on the folium of Descartes ($a=1$), $x=\dfrac{3t}{1+t^{3}},\ y=\dfrac{3t^{2}}{1+t^{3}}$, that lie on a single line. The tangent at each of $A,B,C$ meets the curve again at a point $S_{1},S_{2},S_{3}$ respectively. Prove that $S_{1},S_{2},S_{3}$ are collinear.
Every radius vector of the curve $\rho=a\sin^{3}\dfrac{\varphi}{3}$ intersects it at three points (besides the pole). Prove that the tangents at those three points form an equilateral triangle.
By examining the curves $y=x$ and $y=a^{x}$, prove that the sequence $x_1=a,\ x_2=a^{a},\ x_3=a^{a^{a}},\ldots,\ x_n=a^{x_{n-1}}$ converges to a definite limit as $n\to\infty$, provided $e^{-e}\lt a\le e^{1/e}$.
Find the angle at which the curves $\Gamma_1\colon x=a\cos\varphi,\ y=a\sin\varphi$ and $\Gamma_2\colon x=\dfrac{at^2}{1+t^2},\ y=\dfrac{at\sqrt3}{1+t^2}$ intersect.
If the curve $y=\dfrac{f(x)}{f'(x)}\tan\lambda$ intersects the $Ox$ axis at several points, show that it always makes the same angle $\lambda$ with $Ox$ at each one.
Find the tangent of the angle between the polar axis and the tangent to each curve at the pole ($\rho=0$): 1° $\rho=\sin^{3}\varphi$; 2° $\rho=\sin3\varphi$.
Find the point where the tangent to the curve $\dfrac{a}{x^2}+\dfrac{y}{b^2}=1$ meets the $x$-axis, as a function of the abscissa $x_0$ of the point of tangency.
Show that for the catenary $y=a\cosh\dfrac{x}{a}$, the ordinate at any point is the geometric mean of the constant $a$ and the length of the normal at that point, i.e.\ that $y_0^2=a\cdot N$.
Find the length of the tangent, the length of the normal, the subtangent, and the subnormal of the curve $x=a(2\cos t-\cos 2t),\ y=a(2\sin t-\sin 2t)$.
Show that $4\rho^2=3p^2+4a^2$ for the curve $x=\dfrac a2(3\cos t+\cos3t),\ y=\dfrac a2(3\sin t+\sin3t)$, where $\rho$ is the distance from the origin to the point and $p$ is the distance from the origin to the tangent line there.
Show that the curves $\rho_1=f(\varphi)$ and $\rho_2=a+f(\varphi)$ have equal polar subnormals at points lying on the same radius vector (i.e.\ at points with the same value of $\varphi$).
On the parabola $y^2=4kx$ take $A(k\alpha^2,2k\alpha)$ and $B(k\beta^2,2k\beta)$, $\alpha\neq\beta$. 1° Find the tangent and normal at $A$. 2° Let $S$ be the intersection of the tangents at $A,B$ and $M$ the intersection of the normals at $A,B$. Find the locus of $M$ as $S$ moves along a line parallel to the axis of the parabola.
Through the origin, a line is drawn parallel to the tangent at a point $T$ of a curve. Find the locus of the intersection of this line with the vertical through $T$, for 1° $y^2=2px$ and 2° $x^2+y^2=a^2$.
Find the locus of the feet of the perpendiculars dropped from the origin onto the tangents of the curve $x=a(\cos t+t\sin t),\ y=a(\sin t-t\cos t)$ (the involute of a circle of radius $a$).
Apply the Maclaurin formula to $f(x)=\sin x$, estimate the remainder, and determine how many terms are needed so that the error is less than $10^{-3}$ for $x\in[0,\tfrac\pi4]$.
Prove that $\cos x = 1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots+(-1)^{n-1}\dfrac{x^{2n-2}}{(2n-2)!}+(-1)^n\dfrac{x^{2n}}{(2n)!}\cos\theta x$, $0\lt \theta\lt 1$.
Approximate $f(x)=(2x^2+3x)e^{-x}$ by its degree-4 Taylor (Maclaurin) polynomial about $x=0$ and estimate the error for $-\tfrac1{10}\lt x\lt \tfrac1{10}$.
Approximate the positive branch of $y^2(1-x^2)=x^2$ by its degree-3 Taylor (Maclaurin) polynomial about $x=0$, and estimate the error for $|x|\lt 10^{-1}$.
Compute the coefficients $a_k$ of the polynomial $P_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ using its successive derivatives at $x=0$. Find the degree-$n$ Maclaurin polynomial for $P_n(x)$.
Let $T_1(x)$ and $T_2(x)$ be the degree-$n$ Taylor polynomials of $f(x)$ and $\varphi(x)$ respectively, expanded about $x=a$. Prove that the degree-$n$ Taylor polynomial $T(x)$ of the product $y=f(x)\varphi(x)$ is $T(x)=T_1(x)T_2(x)-P(x)$, where $P(x)$ consists of all terms in the product $T_1(x)T_2(x)$ whose degree (in $x-a$) exceeds $n$.
Given $y=x^2\sqrt{1-x}$: 1° write the degree-$n$ Maclaurin polynomial and the remainder; 2° for $n=4$, find the error bound for $|x|\lt \tfrac1{10}$; 3° verify by multiplying the series for $f(x)=x^2$ and $\varphi(x)=\sqrt{1-x}$.
If $f(x)$ is a polynomial of degree $k$ and $\varphi(x)$ is an arbitrary (sufficiently differentiable) function, can the degree-$n$ Maclaurin polynomial of $f(x)\varphi(x)$ be obtained by multiplying $f(x)$ by the degree-$(n-k)$ Maclaurin polynomial of $\varphi(x)$?
Show that $y=\sqrt{x^2+1}-\ln\dfrac{\sqrt{1+x^2}-1}{x}$ (the log's argument taken positive for $x>0$) can be written $y=ax+b+\dfrac cx+o\!\left(\dfrac1x\right)$ as $x\to+\infty$, and determine $a,b,c$.