미분 전체 문제

For the following functions, use ${f}^{''}(x)=\lim_{h\to 0}\frac{{f}^{'}(x+h)-{f}^{'}(x)}{h}$ to find ${f}^{''}(x).$

$f(x)=2-3x$

For the following functions, use ${f}^{''}(x)=\lim_{h\to 0}\frac{{f}^{'}(x+h)-{f}^{'}(x)}{h}$ to find ${f}^{''}(x).$

$f(x)=4{x}^{2}$

For the following functions, use ${f}^{''}(x)=\lim_{h\to 0}\frac{{f}^{'}(x+h)-{f}^{'}(x)}{h}$ to find ${f}^{''}(x).$

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

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

[T] $f(x)=-\frac{5}{x}$

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

[T] $f(x)=3{x}^{2}+2x+4.$

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

[T] $f(x)=\sqrt{x}+3x$

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

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

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

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

$P(x)$ denotes the population of a city at time $x$ in years.

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

$R(x)$ denotes the total cost (in thousands of dollars) of manufacturing $x$ clock radios.

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

$B(x)$ denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in $x$ years since $1990.$

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

Sketch the graph of a function $y=f(x)$ with all of the following properties:

a. ${f}^{'}(x)>0$ for $-2\le x<1$ b. ${f}^{'}(2)=0$ c. ${f}^{'}(x)>0$ for $x>2$ d. $f(2)=2$ and $f(0)=1$ e. $\lim_{x\to -\infty }f(x)=0$ and $\lim_{x\to \infty }f(x)=\infty$ f. ${f}^{'}(1)$ does not exist.

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 ${T}^{'}(x).$ b. If we know that ${T}^{'}(1000)=-0.1,$ explain the physical meaning.

Suppose the total profit of a company is $y=P(x)$ thousand dollars when $x$ units of an item are sold.

a. What does $\frac{P(b)-P(a)}{b-a}$ for $0<a<b$ measure, and what are the units? b. What does ${P}^{'}(x)$ measure, and what are the units? c. Suppose that ${P}^{'}(30)=5,$ what is the approximate change in profit if the number of items sold increases from $30\,\text{to}\,31?$

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 $50,000$ citizens.

a. Describe what ${N}^{'}(t)$ 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?

What is the physical meaning of ${h}^{'}(t)?$ What are the units?

[T] Construct a table of values for ${h}^{'}(t)$ and graph both $h(t)$ and ${h}^{'}(t)$ 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.)