Chapter 3

In Exercises 1 through 8 , find the slope of the tangent line to the graph at the point \(\left(x_{1}, y_{1}\right) .\) Make a table of values of \(x, y\), and \(m\) at various points on the graph, and include in the table all points where the graph has a horizontal tangent. Draw a sketch of the graph. $$ y=9-x^{2} $$

In Exercises 1 through 14, do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ \begin{gathered} f(x)=\left\\{\begin{aligned} x+2 & \text { if } x \leq-4 \\ -x-6 & \text { if } x>-4 \end{aligned}\right. \\ x_{1}=-4 \end{gathered} $$

In Exercises 1 through 26 , differentiate the given function by applying the theorems of this section. $$ f(x)=x^{3}-3 x^{2}+5 x-2 $$

In Exercises 1 through 20 , find the derivative of the given function. $$ F(x)=\left(x^{2}+4 x-5\right)^{3} $$

In Exercises 1 through 18 , find the derivative of the given function. $$ f(x)=(3 x+5)^{2 / 3} $$

In Exercises 1 through 16, find \(D_{x} y\) by implicit differentiation. $$ x^{2}+y^{2}=16 $$

If \(A\) in. \(^{2}\) is the area of a square and \(s\) in. is the length of a side of the square, find the average rate of change of \(A\) with respect to \(s\) as \(s\) changes from (a) \(4.00\) to \(4.60\); (b) \(4.00\) to 4.30; (c) \(4.00\) to 4.10. (d) What is the instantaneous rate of change of \(A\) with respect to \(s\) when \(s\) is \(4.00\) ?

A kite is flying at a height of \(40 \mathrm{ft}\). A boy is flying it so that it is moving horizontally at a rate of \(3 \mathrm{ft} / \mathrm{sec}\). If the string is taut, at what rate is the string being paid out when the length of the string released is \(50 \mathrm{ft}\) ?

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ f(x)=x^{5}-2 x^{3}+x $$

Find the slope of the tangent line to the graph at the point \(\left(x_{1}, y_{1}\right) .\) Make a table of values of \(x, y\), and \(m\) at various points on the graph, and include in the table all points where the graph has a horizontal tangent. Draw a sketch of the graph. $$ y=x^{2}-6 x+9 $$

Do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ f(x)=\left\\{\begin{array}{cc} 3-2 x & \text { if } x<2 \\ 3 x-7 & \text { if } x \geq 2 \\ & x_{1}=2 \end{array}\right. $$

Find the derivative of the given function. $$ f(x)=(10-5 x)^{4} $$

Find the derivative of the given function. $$ f(s)=\sqrt{2-3 s^{2}} $$

Find \(D_{x} y\) by implicit differentiation. $$ 2 x^{3} y+3 x y^{3}=5 $$

Suppose a right-circular cylinder has a constant height of \(10.00\) in. If \(V\) in. \({ }^{3}\) is the volume of the right-circular cylinder, and \(r\) in. is the radius of its base, find the average rate of change of \(V\) with respect to \(r\) as \(r\) changes from (a) \(5.00\) to \(5.40\); (b) \(5.00\) to 5.10; (c) \(5.00\) to 5.01. (d) Find the instantaneous rate of change of \(V\) with respect to \(r\) when \(r\) is \(5.00\).

A spherical balloon is being inflated so that its volume is increasing at the rate of \(5 \mathrm{ft}^{3} / \mathrm{min}\). At what rate is the diameter increasing when the diameter is \(12 \mathrm{ft}\) ?

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ F(x)=7 x^{3}-8 x^{2} $$

Find the slope of the tangent line to the graph at the point \(\left(x_{1}, y_{1}\right) .\) Make a table of values of \(x, y\), and \(m\) at various points on the graph, and include in the table all points where the graph has a horizontal tangent. Draw a sketch of the graph. $$ y=7-6 x-x^{2} $$

In Exercises 3 through 8 , a particle is moving along a horizontal line according to the given equation of motion, where \(s \mathrm{ft}\) is the directed distance of the particle from a point \(O\) at \(t \mathrm{sec}\). Find the instantaneous velocity \(v\left(t_{1}\right) \mathrm{ft} / \mathrm{sec}\) at \(t_{1} \mathrm{sec}\); and then find \(v\left(t_{1}\right)\) for the particular value of \(t_{1}\) given. $$ s=3 t^{2}+1 ; t_{1}=3 $$

Do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ f(x)=|x-3| ; x_{1}=3 $$

Differentiate the given function by applying the theorems of this section. $$ f(x)=\frac{1}{8} x^{8}-x^{4} $$

Find the derivative of the given function. $$ f(t)=\left(2 t^{4}-7 t^{3}+2 t-1\right)^{2} $$

Find the derivative of the given function. $$ g(x)=\sqrt{\frac{2 x-5}{3 x+1}} $$

Find \(D_{x} y\) by implicit differentiation. $$ x^{3}+y^{3}=8 x y $$

Let \(r\) be the reciprocal of a number \(n .\) Find the instantaneous rate of change of \(r\) with respect to \(n\) and the relative rate of change of \(r\) per unit change in \(n\) when \(n\) is (a) 4 and (b) 10 .

A spherical snowball is being made so that its volume is increasing at the rate of \(8 \mathrm{ft}^{3} / \mathrm{min}\). Find the rate at which the radius is increasing when the snowball is \(4 \mathrm{ft}\) in diameter.

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ g(s)=2 s^{4}-4 s^{3}+7 s-1 $$

A particle is moving along a horizontal line according to the given equation of motion, where \(s \mathrm{ft}\) is the directed distance of the particle from a point \(O\) at \(t \mathrm{sec}\). Find the instantaneous velocity \(v\left(t_{1}\right) \mathrm{ft} / \mathrm{sec}\) at \(t_{1} \mathrm{sec}\); and then find \(v\left(t_{1}\right)\) for the particular value of \(t_{1}\) given. $$ s=8-t^{2} ; t_{1}=5 $$

Do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ f(x)=1+|x+2| ; x_{1}=-2 $$

Find the derivative of the given function. $$ g(r)=\left(2 r^{4}+8 r^{2}+1\right)^{5} $$

Find the derivative of the given function. $$ h(t)=\frac{\sqrt{t-1}}{\sqrt{t+1}} $$

Find \(D_{x} y\) by implicit differentiation. $$ x^{2}=\frac{x+2 y}{x-2 y} $$

Let \(s\) be the principal square root of a number \(x\). Find the instantaneous rate of change of \(s\) with respect to \(x\) and the relative rate of change of \(s\) per unit change in \(x\) when \(x\) is (a) 9 and (b) 4 .

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ G(t)=t^{3}-t^{2}+t $$

Find the slope of the tangent line to the graph at the point \(\left(x_{1}, y_{1}\right) .\) Make a table of values of \(x, y\), and \(m\) at various points on the graph, and include in the table all points where the graph has a horizontal tangent. Draw a sketch of the graph. $$ y=x^{3}-3 x $$

A particle is moving along a horizontal line according to the given equation of motion, where \(s \mathrm{ft}\) is the directed distance of the particle from a point \(O\) at \(t \mathrm{sec}\). Find the instantaneous velocity \(v\left(t_{1}\right) \mathrm{ft} / \mathrm{sec}\) at \(t_{1} \mathrm{sec}\); and then find \(v\left(t_{1}\right)\) for the particular value of \(t_{1}\) given. $$ s=\sqrt{t+1} ; t_{1}=3 $$

Do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ \begin{gathered} f(x)= \begin{cases}-1 & \text { if } x<0 \\ x-1 & \text { if } x \geq 0\end{cases} \\ x_{1}=0 & \end{gathered} $$

Differentiate the given function by applying the theorems of this section. $$ F(t)=\frac{1}{4} t^{4}-\frac{1}{2} t^{2} $$

Find the derivative of the given function. $$ f(x)=(x+4)^{-2} $$

Find the derivative of the given function. $$ f(x)=4 x^{1 / 2}+5 x^{-1 / 2} $$

Find \(D_{x} y\) by implicit differentiation. $$ \frac{1}{x}+\frac{1}{y}=1 $$

If water is being drained from a swimming pool and \(V\) gal is the volume of water in the pool \(t\) min after the draining starts, where \(V=250(40-t)^{2}\), find (a) the average rate at which the water leaves the pool during the first \(5 \mathrm{~min}\), and (b) how fast the water is flowing out of the pool 5 min after the draining starts.

Sand is being dropped at the rate of \(10 \mathrm{ft}^{3} / \mathrm{min}\) onto a conical pile. If the height of the pile is always twice the base radius, at what rate is the height increasing when the pile is \(8 \mathrm{ft}\) high?

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ f(x)=\sqrt{x^{2}+1} $$

Find the slope of the tangent line to the graph at the point \(\left(x_{1}, y_{1}\right) .\) Make a table of values of \(x, y\), and \(m\) at various points on the graph, and include in the table all points where the graph has a horizontal tangent. Draw a sketch of the graph. $$ y=x^{3}-x^{2}-x+10 $$

Do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ f(x)= \begin{cases}x & \text { if } x \leq 0 \\ x^{2} & \text { if } x>0 \\\ x_{1}=0\end{cases} $$

Find the derivative of the given function. $$ H(z)=\left(z^{3}-3 z^{2}+1\right)^{-3} $$

Find the derivative of the given function. $$ g(y)=\left(y^{2}+3\right)^{1 / 3}\left(y^{3}-1\right)^{1 / 2} $$

Find \(D_{x} y\) by implicit differentiation. $$ \frac{x}{y}-4 y=x $$

The supply equation for a certain kind of pencil is \(x=3 p^{2}+2 p\) where \(p\) cents is the price per pencil when \(1000 x\) pencils are supplied. (a) Find the average rate of change of the supply per 1 cent change in the price when the price is increased from 10 cents to 11 cents. (b) Find the instantaneous (or marginal) rate of change of the supply per 1 cent change in the price when the price is 10 cents.