Chapter 3

\(\lim _{h \rightarrow 0} \frac{(3+h)^{2}+2(3+h)-15}{h}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\left(x^{4}-1\right)\left(x^{2}+1\right) $$

The rate of change of electric charge with respect to time is called current. Suppose that \(\frac{1}{3} t^{3}+t\) coulombs of charge flow through a wire in \(t\) seconds. Find the current in amperes (coulombs per second) after 3 seconds. When will a 20 -ampere fuse in the line blow?

In Problems 21-28, find the indicated derivative. $$ \frac{d y}{d t}, \text { where } y=\left[\sin t \tan \left(t^{2}+1\right)\right] $$

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

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.

Find the indicated derivative. \(D_{x} e^{\sqrt{x+2}}\)

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

\(\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\left(x^{2}+17\right)\left(x^{3}-3 x+1\right) $$

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?

In Problems 29-32, evaluate the indicated derivative. $$ f^{\prime}(3) \text { if } f(x)=\left(\frac{x^{2}+1}{x+2}\right)^{3} $$

A snowball melts at a rate proportional to its surface area. (a) Show that its radius shrinks at a constant rate. (b) If it melts to \(\frac{8}{27}\) its original volume in one hour, how long will it take to melt completely?

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 the indicated derivative. \(D_{x} e^{-1 / x^{2}}\)

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

\(\lim _{x \rightarrow 3} \frac{x^{3}+x-30}{x-3}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\left(x^{4}+2 x\right)\left(x^{3}+2 x^{2}+1\right) $$

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 \(t=3\). GG Use a graphing calculator or a CAS to do Problems 31-34.

In Problems 29-32, evaluate the indicated derivative. $$ G^{\prime}(1) \text { if } G(t)=\left(t^{2}+9\right)^{3}\left(t^{2}-2\right)^{4} $$

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?

Find the indicated derivative. \(y^{\prime}\) if \(y=e^{2 \ln x}\)

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

\(\lim _{t \rightarrow x} \frac{t^{2}-x^{2}}{t-x}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\left(5 x^{2}-7\right)\left(3 x^{2}-2 x+1\right) $$

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) 1 (d) \(3.2\)

In Problems 29-32, evaluate the indicated derivative. $$ F^{\prime}(1) \text { if } F(t)=\sin \left(t^{2}+3 t+1\right) $$

Use the definition of the derivative to show that \(D_{x}(\sin 5 x)=5 \cos 5 x .\)

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?

Find the indicated derivative. \(y^{\prime}\) if \(y=e^{x / \ln x}\)

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

\(\lim _{p \rightarrow x} \frac{p^{3}-x^{3}}{p-x}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\left(3 x^{2}+2 x\right)\left(x^{4}-3 x+1\right) $$

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) \(4.2\)

In Problems 29-32, evaluate the indicated derivative. $$ g^{\prime}\left(\frac{1}{2}\right) \text { if } g(s)=\cos \pi s \sin ^{2} \pi s $$

Use the definition of the derivative to show that \(D_{x}(\sin 5 x)=5 \cos 5 x .\)

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

Find the indicated derivative. \(D_{x} x^{3} e^{x}\)

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{1}{3 x^{2}+1} $$

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 \(t=3\).

In Problems 33-40, 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 \(f(x)=\cos ^{3} x-1.25 \cos ^{2} x+0.225\). Find \(f^{\prime}\left(x_{0}\right)\) at that point \(x_{0}\) in \([\pi / 2, \pi]\) where \(f\left(x_{0}\right)=0\).

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?

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

Find the indicated derivative. \(D_{x} e^{x^{3} \ln x}\)

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{2}{5 x^{2}-1} $$

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 \(t=1.6\).

In Problems 33-40, apply the Chain Rule more than once to find the indicated derivative. $$ D_{t}\left[\cos ^{5}(4 t-19)\right] $$