Chapter 3: DerivativesContent check — Calculus Vol. 1, Chapter 3: Derivatives

Find the slope of the line tangent to the graph of $f\left(x\right)=\sqrt{x}$ at $x=4.$

For $f\left(x\right)=x^2,$ use a table to estimate $f^{\prime }(3)$ using Equation 3.6.

For $f\left(x\right)=$ $x^2+$ $3x+$ $2,$ find $f^{\prime }\left(1\right).$

A rock is dropped from a height of $64$ feet. Its height above ground at time $t$ seconds later is given by $s\left(t\right)=$ $\mathrm{-16}{t}^2+$ $64,$ $0\le t\le 2.$ Find its instantaneous velocity $1$ second after it is dropped, using Equation 3.5.

A coffee shop determines that the daily profit on scones obtained by charging $s$ dollars per scone is $P\left(s\right)=$ $\mathrm{-20}{s}^2+$ $150s-$ $10.$ The coffee shop currently charges $\text{\$}3.25$ per scone. Find $P^{\prime }(3.25),$ the rate of change of profit when the price is $\text{\$}3.25$ and decide whether or not the coffee shop should consider raising or lowering its prices on scones.

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f\left(x\right).$

$f\left(x\right)=$ $4x+$ $7;$ $x_1=$ $2,$ $x_2=$ $5$

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f\left(x\right).$

$f\left(x\right)=$ $x^2+$ $2x+$ $1;$ $x_1=$ $3,$ $x_2=$ $3.5$

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f\left(x\right).$

$f\left(x\right)=$ $\frac{4}{3x-1};$ $x_1=$ $1,$ $x_2=$ $3$

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f\left(x\right).$

$f\left(x\right)=$ $\sqrt{x};$ $x_1=$ $1,$ $x_2=$ $16$

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f\left(x\right).$

$f\left(x\right)=$ $x^{1\text{/}3}+$ $1;$ $x_1=$ $0,$ $x_2=$ $8$

For the following functions,

$f\left(x\right)=$ $3-$ $4x,$ $a=$ $2$

For the following functions,

$f\left(x\right)=$ $x^2+$ $x,$ $a=$ $1$

For the following functions,

$f\left(x\right)=\frac{7}{x},a=3$

For the following functions,

$f\left(x\right)=$ $2-$ $3x^2,$ $a=$ $\mathrm{-2}$

For the following functions,

$f\left(x\right)=$ $\frac{2}{x+3},$ $a=$ $\mathrm{-4}$

For the following functions $y=f\left(x\right),$ find $f^{\prime }\left(a\right)$ using Equation 3.5.

$f\left(x\right)=$ $5x+$ $4,$ $a=$ $\mathrm{-1}$

For the following functions $y=f\left(x\right),$ find $f^{\prime }\left(a\right)$ using Equation 3.5.

$f\left(x\right)=$ $x^2+$ $9x,$ $a=$ $2$

For the following functions $y=f\left(x\right),$ find $f^{\prime }\left(a\right)$ using Equation 3.5.

$f\left(x\right)=\sqrt{x},a=4$

For the following functions $y=f\left(x\right),$ find $f^{\prime }\left(a\right)$ using Equation 3.5.

$f\left(x\right)=\frac{1}{x},a=2$

For the following functions $y=f\left(x\right),$ find $f^{\prime }\left(a\right)$ using Equation 3.5.

$f\left(x\right)=$ $\frac{1}{x^3},$ $a=$ $1$

For the following exercises, given the function $y=f\left(x\right),$

[T] $f\left(x\right)=$ $x^2+$ $3x+$ $4,$ $P(1,8)$ (Round to $6$ decimal places.)

x	Slope $m_{PQ}$	x	Slope $m_{PQ}$
1.1	(i)	0.9	(vii)
1.01	(ii)	0.99	(viii)
1.001	(iii)	0.999	(ix)
1.0001	(iv)	0.9999	(x)
1.00001	(v)	0.99999	(xi)
1.000001	(vi)	0.999999	(xii)

For the following exercises, given the function $y=f\left(x\right),$

[T] $f\left(x\right)=$ $10e^{0.5x},$ $P(0,10)$ (Round to $4$ decimal places.)

x	Slope $m_{PQ}$
$\mathrm{-0.1}$	(i)
$\mathrm{-0.01}$	(ii)
$\mathrm{-0.001}$	(iii)
$\mathrm{-0.0001}$	(iv)
$\mathrm{-0.00001}$	(v)
−0.000001	(vi)

[T] For the following position functions $y=s\left(t\right),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find

$s\left(t\right)=\frac{1}{3}t+5$

[T] For the following position functions $y=s\left(t\right),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find

$s\left(t\right)=2t^3+3$

[T] For the following position functions $y=s\left(t\right),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find

Use the following graph to evaluate a. $f^{\prime }\left(1\right)$ and b. $f^{\prime }\left(6\right).$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

$f\left(x\right)=$ $x^{1\text{/}3},$ $x=$ $0$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

$$f\left(x\right)=\{\begin{array}{c}1,x<1\\ x,x\ge 1\end{array},x=1$$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

[T] The position in feet of a race car along a straight track after $t$ seconds is modeled by the function $s\left(t\right)=$ $8t^2-$ $\frac{1}{16}{t}^3.$

a. Find the average velocity of the vehicle over the following time intervals to four decimal places:

i. [4, 4.1]

ii. [4, 4.01]

iii. [4, 4.001]

iv. [4, 4.0001]

b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at $t=4$ seconds.

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by $s=f(t)$ and $s=g\left(t\right),$ where $s$ is measured in feet and $t$ is measured in seconds.

a. Which vehicle has traveled farther at $t=2$ seconds?

b. What is the approximate velocity of each vehicle at $t=3$ seconds?

c. Which vehicle is traveling faster at $t=4$ seconds?

d. What is true about the positions of the vehicles at $t=4$ seconds?

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

[T] For the function $f\left(x\right)=$ $x^3-$ $2x^2-$ $11x+$ $12,$ do the following.

a. Use a graphing calculator to graph f in an appropriate viewing window.

b. Use the ZOOM feature on the calculator to approximate the two values of $x=a$ for which $m_{\text{tan}}=$ $f^{\prime }\left(a\right)=$ $0.$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

Suppose that $N\left(x\right)$ computes the number of gallons of gas used by a vehicle traveling $x$ miles. Suppose the vehicle gets $30$ mpg.

a. Find a mathematical expression for $N\left(x\right).$

b. What is $N(100\text{)?}$ Explain the physical meaning.

c. What is $N^{\prime }(100)?$ Explain the physical meaning.

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.

[T] For the function $f\left(x\right)=$ $\frac{x^2}{x^2+1},$ do the following.

a. Use a graphing calculator to graph $f$ in an appropriate viewing window.

b. Use the $\text{nDeriv}$ function on a graphing calculator to find $f^{\prime }\left(\mathrm{-4}\right),$ $f^{\prime }(\mathrm{-2}),$ $f^{\prime }(2),$ and $f^{\prime }(4).$

Find the derivative of $f\left(x\right)=x^2.$

Sketch the graph of $f\left(x\right)=x^2-4.$ On what interval is the graph of $f^{\prime }\left(x\right)$ above the $x$-axis?

Find values of $a$ and $b$ that make $$f\left(x\right)=\{\begin{array}{c}ax+b\text{if}x<3\\ x^2\text{if}x\ge 3\end{array}$$ both continuous and differentiable at $3.$

Find $f^{″}(x)$ for $f\left(x\right)=x^2.$

For $s\left(t\right)=t^3,$ find $a(t).$

For the following exercises, use the definition of a derivative to find $f^{\prime }\left(x\right).$

$f\left(x\right)=2-3x$

For the following exercises, use the definition of a derivative to find $f^{\prime }\left(x\right).$

$f\left(x\right)=4x^2$

For the following exercises, use the definition of a derivative to find $f^{\prime }\left(x\right).$

$f\left(x\right)=\sqrt{2x}$

For the following exercises, use the definition of a derivative to find $f^{\prime }\left(x\right).$

$f\left(x\right)=\frac{9}{x}$

For the following exercises, use the definition of a derivative to find $f^{\prime }\left(x\right).$

$f\left(x\right)=\frac{1}{\sqrt{x}}$

For the following exercises, use the graph of $y=f(x)$ to sketch the graph of its derivative $f^{\prime }\left(x\right).$

For the following exercises, use the graph of $y=f(x)$ to sketch the graph of its derivative $f^{\prime }\left(x\right).$

For the following exercises, the given limit represents the derivative of a function $y=f\left(x\right)$ at $x=a.$ Find $f\left(x\right)$ and $a.$

$\underset{h\to 0}{\text{lim}}\frac{\left[3{\left(2+h\right)}^2+2\right]-14}{h}$

For the following exercises, the given limit represents the derivative of a function $y=f\left(x\right)$ at $x=a.$ Find $f\left(x\right)$ and $a.$

$\underset{h\to 0}{\text{lim}}\frac{{\left(2+h\right)}^4-16}{h}$

For the following exercises, the given limit represents the derivative of a function $y=f\left(x\right)$ at $x=a.$ Find $f\left(x\right)$ and $a.$

$\underset{h\to 0}{\text{lim}}\frac{e^h-1}{h}$

For the following functions,

$$f(x)=\{\begin{array}{l}3,x<1\\ 3x,x\ge 1\end{array}$$

For the following functions,

$$f(x)=\{\begin{array}{l}2x,x\le 1\\ \frac{2}{x},x>1\end{array}$$

For the following graphs,

For the following functions, use $f^{″}\left(x\right)=$ $\underset{h\to 0}{\text{lim}}\frac{f^{\prime }\left(x+h\right)-f^{\prime }(x)}{h}$ to find $f^{″}(x).$

$f(x)=2-3x$

For the following functions, use $f^{″}\left(x\right)=$ $\underset{h\to 0}{\text{lim}}\frac{f^{\prime }\left(x+h\right)-f^{\prime }(x)}{h}$ to find $f^{″}(x).$

$f(x)=x+\frac{1}{x}$

For the following exercises, use a calculator to graph $f\left(x\right).$ Determine the function $f^{\prime }\left(x\right),$ then use a calculator to graph $f^{\prime }\left(x\right).$

[T] $f\left(x\right)=$ $3x^2+$ $2x+$ $4.$

For the following exercises, use a calculator to graph $f\left(x\right).$ Determine the function $f^{\prime }\left(x\right),$ then use a calculator to graph $f^{\prime }\left(x\right).$

[T] $f\left(x\right)=\frac{1}{\sqrt{2x}}$

For the following exercises, use a calculator to graph $f\left(x\right).$ Determine the function $f^{\prime }\left(x\right),$ then use a calculator to graph $f^{\prime }\left(x\right).$

[T] $f\left(x\right)=x^3+1$

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

$C(x)$ denotes the total amount of money (in thousands of dollars) spent on concessions by $x$ customers at an amusement park.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

$g(x)$ denotes the grade (in percentage points) received on a test, given $x$ hours of studying.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

$p\left(x\right)$ denotes atmospheric pressure in Torrs at an altitude of $x$ feet.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

Suppose temperature $T$ in degrees Fahrenheit at a height $x$ in feet above the ground is given by $y=T(x).$

a. Give a physical interpretation, with units, of $T^{\prime }(x).$

b. If we know that $T^{\prime }\left(1000\right)=$ $\mathrm{-0.1},$ explain the physical meaning.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

The graph in the following figure models the number of people $N(t)$ who have come down with the flu $t$ weeks after its initial outbreak in a town with a population of $\mathrm{50,000}$ citizens.

a. Describe what $N^{\prime }(t)$ represents and how it behaves as $t$ increases.

b. What does the derivative tell us about how this town is affected by the flu outbreak?

For the following exercises, use the following table, which shows the height $h$ of the Saturn $\text{V}$ rocket for the Apollo $11$ mission $t$ seconds after launch.

Time (seconds)	Height (meters)
$0$	$0$
$1$	$2$
$2$	$4$
$3$	$13$
$4$	$25$
$5$	$32$

[T] Construct a table of values for $h^{\prime }\left(t\right)$ and graph both $h\left(t\right)$ and $h^{\prime }\left(t\right)$ on the same graph. (Hint: for interior points, estimate both the left limit and right limit and average them. An interior point of an interval I is an element of I which is not an endpoint of I.)

For the following exercises, use the following table, which shows the height $h$ of the Saturn $\text{V}$ rocket for the Apollo $11$ mission $t$ seconds after launch.

Time (seconds)	Height (meters)
$0$	$0$
$1$	$2$
$2$	$4$
$3$	$13$
$4$	$25$
$5$	$32$

[T] The best quadratic fit to the data is given by $G\left(t\right)=$ $1.429t^2+$ $0.0857t-$ $0.1429,$ where $G$ is the height of the rocket (in meters) and $t$ is the time elapsed since takeoff. From this equation, determine $G^{\prime }\left(t\right).$ Graph $G\left(t\right)$ with the given data and, on a separate coordinate plane, graph $G^{\prime }\left(t\right).$

For the following exercises, use the following table, which shows the height $h$ of the Saturn $\text{V}$ rocket for the Apollo $11$ mission $t$ seconds after launch.

Time (seconds)	Height (meters)
$0$	$0$
$1$	$2$
$2$	$4$
$3$	$13$
$4$	$25$
$5$	$32$

Using the best linear, quadratic, and cubic fits to the data, determine what $H^{″}\left(t\right),$ $G^{″}\left(t\right)\text{and}{F}^{″}\left(t\right)$ are. What are the physical meanings of $H^{″}\left(t\right),$ $G^{″}\left(t\right)\text{and}{F}^{″}\left(t\right),$ and what are their units?

Find the derivative of $g\left(x\right)=\mathrm{-3}.$

Find $\frac{d}{dx}\left(x^4\right).$

Find the derivative of $f\left(x\right)=x^7.$

Find the derivative of $f\left(x\right)=$ $2x^3-$ $6x^2+$ $3.$

Find an equation of the line tangent to the graph of $f\left(x\right)=3x^2-11$ at $x=2.$ Use the point-slope form.

Use the product rule to obtain the derivative of $j\left(x\right)=$ $2x^5(4x^2+x).$

Find the derivative of $h\left(x\right)=$ $\frac{3x+1}{4x-3}.$

Find the derivative of $g\left(x\right)=\frac{1}{x^7}$ using the extended power rule.

Find $\frac{d}{dx}\left(3f\left(x\right)-2g\left(x\right)\right).$

Find the values of $x$ for which the graph of $f\left(x\right)=$ $4x^2-$ $3x+$ $2$ has a tangent line parallel to the line $y=2x+3.$

For the following exercises, find $f^{\prime }(x)$ for each function.

$f\left(x\right)=$ $5x^3-$ $x+$ $1$

For the following exercises, find $f^{\prime }(x)$ for each function.

$f\left(x\right)=$ $8x^4+$ $9x^2-$ $1$

For the following exercises, find $f^{\prime }(x)$ for each function.

$f\left(x\right)=$ $3x(18x^4+\frac{13}{x+1})$

For the following exercises, find $f^{\prime }(x)$ for each function.

$f\left(x\right)=$ $x^2(\frac{2}{x^2}+\frac{5}{x^3})$

For the following exercises, find $f^{\prime }(x)$ for each function.

$f\left(x\right)=$ $\frac{4x^3-2x+1}{x^2}$

For the following exercises, find $f^{\prime }(x)$ for each function.

$f\left(x\right)=$ $\frac{x+9}{x^2-7x+1}$

For the following exercises, find an equation of the tangent line $T\left(x\right)$ to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.

[T] $y=\frac{2}{x^2}+1$ at $\left(1,3\right)$

For the following exercises, find an equation of the tangent line $T\left(x\right)$ to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.

[T] $y=\frac{2}{x}-\frac{3}{x^2}$ at $\left(1,\mathrm{-1}\right)$

For the following exercises, assume that $f(x)$ and $g\left(x\right)$ are both differentiable functions for all $x.$ Find the derivative of each of the functions $h\left(x\right).$

$h\left(x\right)=x^{3}f(x)$

For the following exercises, assume that $f(x)$ and $g\left(x\right)$ are both differentiable functions for all $x.$ Find the derivative of each of the functions $h\left(x\right).$

$h\left(x\right)=$ $\frac{3f\left(x\right)}{g\left(x\right)+2}$

For the following exercises, assume that $f(x)$ and $g\left(x\right)$ are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

$x$	$1$	$2$	$3$	$4$
$f(x)$	$3$	$5$	$\mathrm{-2}$	$0$
$g(x)$	$2$	$3$	$\mathrm{-4}$	$6$
$f^{\prime }(x)$	$\mathrm{-1}$	$7$	$8$	$\mathrm{-3}$
$g^{\prime }(x)$	$4$	$1$	$2$	$9$

Find $h^{\prime }\left(2\right)$ if $h\left(x\right)=$ $\frac{f(x)}{g(x)}.$

For the following exercises, assume that $f(x)$ and $g\left(x\right)$ are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

$x$	$1$	$2$	$3$	$4$
$f(x)$	$3$	$5$	$\mathrm{-2}$	$0$
$g(x)$	$2$	$3$	$\mathrm{-4}$	$6$
$f^{\prime }(x)$	$\mathrm{-1}$	$7$	$8$	$\mathrm{-3}$
$g^{\prime }(x)$	$4$	$1$	$2$	$9$

Find $h^{\prime }\left(4\right)$ if $h\left(x\right)=$ $\frac{1}{x}+$ $\frac{g(x)}{f(x)}.$

For the following exercises, use the following figure to find the indicated derivatives, if they exist.

Let $h\left(x\right)=$ $f\left(x\right)g\left(x\right).$ Find

a. $h^{\prime }\left(1\right),$

b. $h^{\prime }\left(3\right),$ and

c. $h^{\prime }\left(4\right).$

For the following exercises,

[T] $f\left(x\right)=$ $2x^3+$ $3x-$ $x^2,$ $a=$ $2$

For the following exercises,

[T] $f\left(x\right)=$ $x^2-$ $x^{12}+$ $3x+$ $2,$ $a=$ $0$

For the following exercises,

Find an equation of the tangent line to the graph of $f\left(x\right)=$ $2x^3+$ $4x^2-$ $5x-$ $3$ at $x=\mathrm{-1}.$

For the following exercises,

Find an equation of the tangent line to the graph of $f\left(x\right)=$ $(3x-x^2)(3-x-x^2)$ at $x=1.$

For the following exercises,

Find an equation of the line passing through the point $P(3,3)$ and tangent to the graph of $f\left(x\right)=\frac{6}{x-1}.$

For the following exercises,

Find a quadratic polynomial such that $f\left(1\right)=$ $5,$ $f^{\prime }\left(1\right)=$ $3$ and $f^{″}\left(1\right)=\mathrm{-6}.$

For the following exercises,

[T] A herring swimming along a straight line has traveled $s\left(t\right)=\frac{t^2}{t^2+2}$ feet in $t$ seconds.

Determine the velocity of the herring when it has traveled 3 seconds.

For the following exercises,

[T] The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C\left(t\right)=$ $\frac{2t^2+t}{t^3+50},$ where $C$ is measured in milligrams per liter of blood.

a. Find the rate of change of $C\left(t\right).$

b. Determine the rate of change for $t=8,12,24,$ and $36.$

c. Briefly describe what seems to be occurring as the number of hours increases.

For the following exercises,

[T] According to Newton’s law of universal gravitation, the force $F$ between two bodies of constant mass $m_1$ and $m_2$ is given by the formula $F=\frac{G{m}_1{m}_2}{d^2},$ where $G$ is the gravitational constant and $d$ is the distance between the bodies.

a. Suppose that $G,m_1,\text{and}{m}_2$ are constants. Find the rate of change of force $F$ with respect to distance $d.$

b. Find the rate of change of force $F$ with gravitational constant $G=$ $6.67\times {10}^{\mathrm{-11}}$ ${\text{Nm}}^2\text{/}{\text{kg}}^2,$ on two bodies 10 meters apart, each with a mass of 1000 kilograms.

Given $f\left(10\right)=\mathrm{-5}$ and $f^{\prime }\left(10\right)=6,$ estimate $f\left(10.1\right).$

A particle moves along a coordinate axis. Its position at time $t$ is given by $s\left(t\right)=$ $t^2-$ $5t+$ $1.$ Is the particle moving from right to left or from left to right at time $t=3?$

The current population of a mosquito colony is known to be 3,000; that is, $P\left(0\right)=\mathrm{3,000}.$ If $P^{\prime }\left(0\right)=100,$ estimate the size of the population in 3 days, where $t$ is measured in days.