Chapter 3

Find \(d y / d x\). \(y=\frac{1}{\sqrt[3]{x^{2} \sin x}}\)

Evaluate the indicated derivative. $$ f^{\prime}(3) \text { if } f(x)=\left(\frac{x^{2}+1}{x+2}\right)^{3} $$

If \(s=\frac{1}{2} t^{4}-5 t^{3}+12 t^{2}\), find the velocity of the moving object when its acceleration is zero.

Show that the curves \(y=\sqrt{2} \sin x\) and \(y=\sqrt{2} \cos x\) intersect at right angles at a certain point with \(0

Find \(D_{x} y\) using the rules of this section. $$ y=\left(x^{2}+17\right)\left(x^{3}-3 x+1\right) $$

The radius of a circular oil spill is growing at a constant rate of 2 kilometers per day. At what rate is the area of the spill growing 3 days after it began?

Find \(D_{x} y\). $$ y=\tan \left(\cos ^{-1} x\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} e^{-1 / x^{2}} $$

Find \(d y / d x\). \(y=\sqrt[4]{1+\sin 5 x}\)

If \(s=\frac{1}{10}\left(t^{4}-14 t^{3}+60 t^{2}\right)\), find the velocity of the moving object when its acceleration is zero.

Evaluate the indicated derivative. $$ G^{\prime}(1) \text { if } G(t)=\left(t^{2}+9\right)^{3}\left(t^{2}-2\right)^{4} $$

. At time \(t\) seconds, the center of a bobbing cork is \(3 \sin 2 t\) centimeters above (or below) water level. What is the velocity of the cork at \(t=0, \pi / 2, \pi ?\)

Find \(D_{x} y\) using the rules of this section. $$ y=\left(x^{4}+2 x\right)\left(x^{3}+2 x^{2}+1\right) $$

The radius of a spherical balloon is increasing at the rate of \(0.25\) inch per second. If the radius is 0 at time \(t=0\), find the rate of change in the volume at time \(t=3\).

It can be shown that if \(\left|d^{2} y / d x^{2}\right| \leq M\) on a closed interval with \(c\) and \(c+\Delta x\) as end points, then $$ |\Delta y-d y| \leq \frac{1}{2} M(\Delta x)^{2} $$ Find, using differentials, the change in \(y=3 x^{2}-2 x+11\) when \(x\) increases from 2 to \(2.001\) and then give a bound for the error that you have made by using differentials.

Find \(D_{x} y\). $$ y=\sec ^{-1}\left(x^{3}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ y^{\prime} \text { if } y=e^{2 \ln x} $$

Find \(d y / d x\). \(y=\sqrt[4]{1+\cos \left(x^{2}+2 x\right)}\)

Two objects move along a coordinate line. At the end of \(t\) seconds their directed distances from the origin, in feet, are given by \(s_{1}=4 t-3 t^{2}\) and \(s_{2}=t^{2}-2 t\), respectively. (a) When do they have the same velocity? (b) When do they have the same speed? (c) When do they have the same position?

Evaluate the indicated derivative. $$ F^{\prime}(1) \text { if } F(t)=\sin \left(t^{2}+3 t+1\right) $$

$$ \begin{array}{l} \text { Use the definition of the derivative to show that }\\\ D_{x}\left(\sin x^{2}\right)=2 x \cos x^{2} \end{array} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\left(5 x^{2}-7\right)\left(3 x^{2}-2 x+1\right) $$

Use a graphing calculator or a CAS Draw the graph of \(y=f(x)=x^{3}-2 x^{2}+1 .\) Then find the slope of the tangent line at (a) \(-1\) (b) 0 (c) \(\overline{1}\) (d) \(3.2\)

Suppose that \(f\) is a function satisfying \(f(1)=10\), and \(f^{\prime}(1.02)=12 .\) Use this information to approximate \(f(1.02) .\)

Find \(D_{x} y\). $$ y=\left(\sec ^{-1} x\right)^{3} $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ y^{\prime} \text { if } y=e^{x / \ln x} $$

Find \(d y / d x\). \(y=\sqrt{\tan ^{2} x+\sin ^{2} x}\)

The positions of two objects, \(P_{1}\) and \(P_{2}\), on a coordinate line at the end of \(t\) seconds are given by \(s_{1}=3 t^{3}-12 t^{2}+\) \(18 t+5\) and \(s_{2}=-t^{3}+9 t^{2}-12 t\), respectively. When do the two objects have the same velocity?

Evaluate the indicated derivative. $$ g^{\prime}\left(\frac{1}{2}\right) \text { if } g(s)=\cos \pi s \sin ^{2} \pi s $$

$$ \begin{array}{l} \text { . Use the definition of the derivative to show that }\\\ D_{x}(\sin 5 x)=5 \cos 5 x \end{array} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\left(3 x^{2}+2 x\right)\left(x^{4}-3 x+1\right) $$

Use a graphing calculator or a CAS Draw the graph of \(y=f(x)=\sin x \sin ^{2} 2 x .\) Then find the slope of the tangent line at (a) \(\pi / 3\) (b) \(2.8\) (c) \(\pi\) (d) \(4.2\)

Suppose \(f\) is a function satisfying \(f(3)=8\) and \(f^{\prime}(3.05)=\frac{1}{4} .\) Use this information to approximate \(f(3.05) .\)

Find \(D_{x} y\). $$ y=\left(1+\sin ^{-1} x\right)^{3} $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} x^{3} e^{x} $$

If \(s^{2} t+t^{3}=1\), find \(d s / d t\) and \(d t / d s\).

Apply the Chain Rule more than once to find the indicated derivative. \(D_{x}\left[\sin ^{4}\left(x^{2}+3 x\right)\right]\)

Let \(f(x)=x \sin x\). (a) Draw the graphs of \(f(x)\) and \(f^{\prime}(x)\) on \([\pi, 6 \pi]\). (b) How many solutions does \(f(x)=0\) have on \([\pi, 6 \pi] ?\) How many solutions does \(f^{\prime}(x)=0\) have on this interval? (c) What is wrong with the following conjecture? If \(f\) and \(f^{\prime}\) are both continuous and differentiable on \([a, b]\), if \(f(a)=f(b)=0\), and if \(f(x)=0\) has exactly \(n\) solutions on \([a, b]\), then \(f^{\prime}(x)=0\) has exactly \(n-1\) solutions on \([a, b] .\) (d) Determine the maximum value of \(\left|f(x)-f^{\prime}(x)\right|\) on \([\pi, 6 \pi]\)

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{1}{3 x^{2}+1} $$

Use a graphing calculator or a CAS If a point moves along a line so that its distance \(s\) (in feet) from 0 is given by \(s=t+t \cos ^{2} t\) at time \(t\) seconds, find its instantaneous velocity at \(t=3\).

Find \(D_{x} y\). $$ y=\sin ^{-1}\left(\frac{1}{x^{2}+4}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} e^{x^{3} \ln x} $$

If \(y=\sin \left(x^{2}\right)+2 x^{3}\), find \(d x / d y\).

An object thrown directly upward from ground level with an initial velocity of 48 feet per second is \(s=48 t-16 t^{2}\) feet high at the end of \(t\) seconds. (a) What is the maximum height attained? (b) How fast is the object moving, and in which direction, at the end of 1 second? (c) How long does it take to return to its original position?

Apply the Chain Rule more than once to find the indicated derivative. \(D_{t}\left[\cos ^{5}(4 t-19)\right]\)

$$ \begin{array}{l} \text { . Let } f(x)=\cos ^{3} x-1.25 \cos ^{2} x+0.225 . \text { Find } f^{\prime}\left(x_{0}\right) \text { at }\\\ \text { that point } x_{0} \text { in }[\pi / 2, \pi] \text { where } f\left(x_{0}\right)=0 \text { . } \end{array} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{2}{5 x^{2}-1} $$

Use a graphing calculator or a CAS If a point moves along a line so that its distance \(s\) (in meters) from 0 is given by \(s=(t+1)^{3} /(t+2)\) at time \(t\) minutes, find its instantaneous velocity at \(t=1.6 .\)

A tank has the shape of a cylinder with hemispherical ends. If the cylindrical part is 100 centimeters long and has an outside diameter of 20 centimeters, about how much paint is required to coat the outside of the tank to a thickness of 1 millimeter?

Find \(D_{x} y\). $$ y=\tan ^{-1}\left(\ln x^{2}\right) $$