3.2 The derivative as a function  (Page 5/10)

$\underset{h\to 0}{\text{lim}}\frac{{\left(1+h\right)}^{2\text{/}3}-1}{h}$

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

$\underset{h\to 0}{\text{lim}}\frac{\text{cos}\left(\pi +h\right)+1}{h}$

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

$\underset{h\to 0}{\text{lim}}\frac{[2{\left(3+h\right)}^2-\left(3+h\right)]-15}{h}$

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

$f(x)=\{\begin{array}{l}2\sqrt{x},0\le x\le 1\\ 3x-1,x>1\end{array}$

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

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

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

Use the graph to evaluate a. $f^{\prime }\left(\mathrm{-0.5}\right),$ b. $f^{\prime }\left(0\right),$ c. $f^{\prime }\left(1\right),$ d. $f^{\prime }\left(2\right),$ and e. $f^{\prime }\left(3\right),$ if it exists.

$f(x)=2-3x$

$f(x)=4x^2$

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

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

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

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

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

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

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

$P(x)$ 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 $1990.$

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

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

1. $f^{\prime }\left(x\right)>0$ for $\mathrm{-2}\le x<1$
2. $f^{\prime }\left(2\right)=0$
3. $f^{\prime }\left(x\right)>0$ for $x>2$
4. $f\left(2\right)=2$ and $f\left(0\right)=1$
5. $\underset{x\to \text{−}\infty }{\text{lim}}f\left(x\right)=0$ and $\underset{x\to \infty }{\text{lim}}f\left(x\right)=\infty$
6. $f^{\prime }\left(1\right)$ does not exist.

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

1. Give a physical interpretation, with units, of $T^{\prime }(x).$
2. If we know that $T^{\prime }\left(1000\right)=\mathrm{-0.1},$ 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.

1. What does $\frac{P\left(b\right)-P(a)}{b-a}$ for $0<a<b$ measure, and what are the units?
2. What does $P^{\prime }(x)$ measure, and what are the units?
3. Suppose that $P^{\prime }\left(30\right)=5,$ 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 $\mathrm{50,000}$ citizens.

1. Describe what $N^{\prime }(t)$ represents and how it behaves as $t$ increases.
2. What does the derivative tell us about how this town is affected by the flu outbreak?
   

What is the physical meaning of $h^{\prime }\left(t\right)?$ What are the units?

[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.)

[T] The best linear fit to the data is given by $H\left(t\right)=7.229t-4.905,$ where $H$ is the height of the rocket (in meters) and $t$ is the time elapsed since takeoff. From this equation, determine $H^{\prime }\left(t\right).$ Graph $H\left(t\right)$ with the given data and, on a separate coordinate plane, graph $H^{\prime }\left(t\right).$

[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 $G^{\prime }\left(t\right).$

[T] The best cubic fit to the data is given by $F\left(t\right)=0.2037t^3+2.956t^2-2.705t+0.4683,$ where $F$ is the height of the rocket (in m) and $t$ is the time elapsed since take off. From this equation, determine $F^{\prime }\left(t\right).$ Graph $F\left(t\right)$ with the given data and, on a separate coordinate plane, graph $F^{\prime }\left(t\right).$ Does the linear, quadratic, or cubic function fit the data best?

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