미분법 전체 문제

For the following exercises, find $\frac{dy}{dx}$ for each function.

Let $y={[f(x)]}^{2}$ and suppose that ${f}^{'}(1)=4$ and $\frac{dy}{dx}=10$ for $x=1.$ Find $f(1).$

For the following exercises, find $\frac{dy}{dx}$ for each function.

Let $y={(f(x)+5{x}^{2})}^{4}$ and suppose that $f(-1)=-4$ and $\frac{dy}{dx}=3$ when $x=-1.$ Find ${f}^{'}(-1)$

For the following exercises, find $\frac{dy}{dx}$ for each function.

Let $y={(f(u)+3x)}^{2}$ and $u={x}^{3}-2x.$ If $f(4)=6$ and $\frac{dy}{dx}=18$ when $x=2,$ find ${f}^{'}(4).$

[T] Find an equation of the tangent line to $y=-\sin \left(\frac{x}{2}\right)$ at the origin. Use a calculator to graph the function and the tangent line together.

[T] Find an equation of the tangent line to $y={\left(3x+\frac{1}{x}\right)}^{2}$ at the point $(1,16).$ Use a calculator to graph the function and the tangent line together.

Find the $x$-coordinates at which the tangent line to $y={\left(x-\frac{6}{x}\right)}^{8}$ is horizontal.

[T] Find an equation of the line that is normal to $g(\theta )={\sin }^{2}(\pi \theta )$ at the point $\left(\frac{1}{4},\frac{1}{2}\right).$ Use a calculator to graph the function and the normal line together.

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=f(g(x));a=0$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=g(f(x));a=0$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)={({x}^{4}+g(x))}^{-2};a=1$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

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

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=f(x+f(x));a=1$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)={(1+g(x))}^{3};a=2$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=f(g(\sin x));a=0$

[T] The position function of a freight train is given by $s(t)=100{(t+1)}^{-2},$ with $s$ in meters and $t$ in seconds. At time $t=6$ s, find the train’s

a. velocity and b. acceleration. c. Using a. and b. is the train speeding up or slowing down?

[T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where $t$ is measured in seconds and $s$ is in inches:

$s(t)=-3\cos \left(\pi t+\frac{\pi }{4}\right).$

a. Determine the position of the spring at $t=1.5$ s. b. Find the velocity of the spring at $t=1.5$ s.

[T] The total cost to produce $x$ boxes of Thin Mint Girl Scout cookies is $C$ dollars, where $C=0.0001{x}^{3}-0.02{x}^{2}+3x+300.$ In $t$ weeks production is estimated to be $x=1600+100t$ boxes.

a. Find the marginal cost ${C}^{'}(x).$ b. Use Leibniz’s notation for the chain rule, $\frac{dC}{dt}=\frac{dC}{dx}\cdot \frac{dx}{dt},$ to find the rate with respect to time $t$ that the cost is changing. c. Use b. to determine how fast costs are increasing when $t=2$ weeks. Include units with the answer.

[T] The formula for the area of a circle is $A=\pi {r}^{2},$ where $r$ is the radius of the circle. Suppose a circle is expanding, meaning that both the area $A$ and the radius $r$ (in inches) are expanding.

a. Suppose $r=2-\frac{100}{{(t+7)}^{2}}$ where $t$ is time in seconds. Use the chain rule $\frac{dA}{dt}=\frac{dA}{dr}\cdot \frac{dr}{dt}$ to find the rate at which the area is expanding. b. Use a. to find the rate at which the area is expanding at $t=4$ s.

[T] The formula for the volume of a sphere is $S=\frac{4}{3}\pi {r}^{3},$ where $r$ (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

a. Suppose $r=\frac{1}{{(t+1)}^{2}}-\frac{1}{12}$ where $t$ is time in minutes. Use the chain rule $\frac{dS}{dt}=\frac{dS}{dr}\cdot \frac{dr}{dt}$ to find the rate at which the snowball is melting. b. Use a. to find the rate at which the volume is changing at $t=1$ min.

[T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function $T(x)=94-10\cos \left[\frac{\pi }{12}(x-2)\right],$ where $x$ is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.