Chapter 2

Fill in the blanks: (a) If \(f^{\prime \prime}\) is positive on an interval, then \(f^{\prime}\) is _________ on that interval, and \(f\) is ___________ on that interval. (b) If \(f^{\prime \prime}\) is negative on an interval, then \(f^{\prime}\) is __________ on that interval, and \(f\) is __________ on that interval.

(a) Estimate \(f^{\prime}(2)\) using the values of \(f\) in the table. (b) For what values of \(x\) does \(f^{\prime}(x)\) appear to be positive? Negative? $$\begin{array}{c|c|c|c|c|c|c|c} \hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline f(x) & 10 & 18 & 24 & 21 & 20 & 18 & 15 \\ \hline \end{array}$$

The cost, \(C\) (in dollars), to produce \(g\) gallons of a chemical can be expressed as \(C=f(g) .\) Using units, explain the meaning of the following statements in terms of the chemical: (a) \(\quad f(200)=1300\) (b) \(\quad f^{\prime}(200)=6\)

The distance, \(s,\) a car has traveled on a trip is shown in the table as a function of the time, \(t,\) since the trip started. Find the average velocity between \(t=2\) and \(t=5\) $$\begin{array}{c|c|c|c|c|c|c}\hline t \text { (hours) } & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline s(\mathrm{km}) & 0 & 45 & 135 & 220 & 300 & 400 \\\\\hline\end{array}$$

The table shows values of \(f(x)=x^{3}\) near \(x=2\) (to three decimal places). Use it to estimate \(f^{\prime}(2)\).$$\begin{array}{c|ccccc}\hline x & 1.998 & 1.999 & 2.000 & 2.001 & 2.002 \\\\\hline x^{3} & 7.976 & 7.988 & 8.000 & 8.012 & 8.024 \\\\\hline\end{array}$$.

The table gives the position of a particle moving along the \(x\) -axis as a function of time in seconds, where \(x\) is in meters. What is the average velocity of the particle from \(t=0\) to \(t=4 ?\) $$\begin{array}{c|c|c|c|c|c}\hline t & 0 & 2 & 4 & 6 & 8 \\\\\hline x(t) & -2 & 4 & -6 & -18 & -14 \\\\\hline\end{array}$$

Find approximate values for \(f^{\prime}(x)\) at each of the \(x\) values given in the following table. $$\begin{array}{c|c|c|c|c|c} \hline x & 0 & 5 & 10 & 15 & 20 \\ \hline f(x) & 100 & 70 & 55 & 46 & 40 \\ \hline \end{array}$$

The time for a chemical reaction, \(T\) (in minutes), is a function of the amount of catalyst present, \(a\) (in milliliters), so \(T=f(a).\) (a) If \(f(5)=18,\) what are the units of \(5 ?\) What are the units of \(18 ?\) What does this statement tell us about the reaction? (b) If \(f^{\prime}(5)=-3,\) what are the units of \(5 ?\) What are the units of \(-3 ?\) What does this statement tell us?

By choosing small values for \(h\), estimate the instantaneous rate of change of the function \(f(x)=x^{3}\) with respect to \(x\) at \(x=1\).

The table gives the position of a particle moving along the \(x\) -axis as a function of time in seconds, where \(x\) is in angstroms. What is the average velocity of the particle from \(t=2\) to \(t=8 ?\) $$\begin{array}{c|c|c|c|c|c} \hline t & 0 & 2 & 4 & 6 & 8 \\\\\hline x(t) & 0 & 14 & -6 & -18 & -4 \\\\\hline\end{array}$$

Values of \(f(x)\) are in the table. Where in the interval \(-12 \leq x \leq 9\) does \(f^{\prime}(x)\) appear to be the greatest? Least? $$\begin{array}{c|c|c|c|c|c|c|c|c} \hline x & -12 & -9 & -6 & -3 & 0 & 3 & 6 & 9 \\ \hline f(x) & 1.02 & 1.05 & 1.12 & 1.14 & 1.15 & 1.14 & 1.12 & 1.06 \\ \hline \end{array}$$

The temperature, \(T\), in degrees Fahrenheit, of a cold yam placed in a hot oven is given by \(T=f(t),\) where \(t\) is the time in minutes since the yam was put in the oven. (a) What is the sign of \(f^{\prime}(t) ?\) Why? (b) What are the units of \(f^{\prime}(20) ?\) What is the practical meaning of the statement \(f^{\prime}(20)=2 ?\)

The income that a company receives from selling an item is called the revenue. Production decisions are based, in part, on how revenue changes if the quantity sold changes; that is, on the rate of change of revenue with respect to quantity sold. Suppose a company's revenue, in dollars, is given by \(R(q)=100 q-10 q^{2},\) where \(q\) is the quantity sold in kilograms. (a) Calculate the average rate of change of \(R\) with respect to \(q\) over the intervals \(1 \leq q \leq 2\) and \(2 \leq q \leq 3\). (b) By choosing small values for \(h,\) estimate the instantaneous rate of change of revenue with respect to change in quantity at \(q=2\) kilograms.

The temperature, \(H\), in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by \(H=f(t)\) where \(t\) is in minutes since the coffee was put on the counter. (a) Is \(f^{\prime}(t)\) positive or negative? Give a reason for your answer. (b) What are the units of \(f^{\prime}(20) ?\) What is its practical meaning in terms of the temperature of the coffee?

(a) Make a table of values rounded to two decimal places for the function \(f(x)=e^{x}\) for \(x=\) \(1,1.5,2,2.5,\) and \(3 .\) Then use the table to answer parts (b) and (c). (b) Find the average rate of change of \(f(x)\) between \(x=1\) and \(x=3\) (c) Use average rates of change to approximate the instantaneous rate of change of \(f(x)\) at \(x=2\).

decide if the function is differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=x \cdot|x|$$

Graph the functions described in parts (a)-(d). (a) First and second derivatives everywhere positive. (b) Second derivative everywhere negative; first derivative everywhere positive. (c) Second derivative everywhere positive; first derivative everywhere negative. (d) First and second derivatives everywhere negative.

The cost, \(C\) (in dollars), to produce \(q\) quarts of ice cream is \(C=f(q) .\) In each of the following statements, what are the units of the two numbers? In words, what does each statement tell us? (a) \(\quad f(200)=600\) (b) \(f^{\prime}(200)=2\)

(a) Make a table of values, rounded to two decimal places, for \(f(x)=\log x\) (that is, log base 10 ) with \(x=1,1.5,2,2.5,3 .\) Then use this table to answer parts (b) and (c). (b) Find the average rate of change of \(f(x)\) between \(x=1\) and \(x=3\) (c) Use average rates of change to approximate the instantaneous rate of change of \(f(x)\) at \(x=2\).

decide if the function is differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=\left\\{\begin{array}{ll} -2 x & \text { for } x<0 \\ x^{2} & \text { for } x \geq 0 \end{array}\right.$$

Sketch the graph of a function whose first derivative is everywhere negative and whose second derivative is positive for some \(x\)-values and negative for other \(x\)-values.

An economist is interested in how the price of a certain item affects its sales. At a price of \(\$ p,\) a quantity, \(q,\) of the item is sold. If \(q=f(p),\) explain the meaning of each of the following statements: (a) \(\quad f(150)=2000\) (b) \(\quad f^{\prime}(150)=-25\)

decide if the function is differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=\left\\{\begin{array}{ll} (x+1)^{2} & \text { for } x<0 \\ 2 x+1 & \text { for } x \geq 0 \end{array}\right.$$

Sketch the graph of the height of a particle against time if velocity is positive and acceleration is negative.

At time \(t\) in seconds, a particle's distance \(s(t),\) in \(\mathrm{mi}-\) crometers \((\mu \mathrm{m}),\) from a point is given by \(s(t)=e^{t}-1\) What is the average velocity of the particle from \(t=2\) to \(t=4 ?\)

Let \(S(t)\) be the amount of water, measured in acre-feet, \(?\) that is stored in a reservoir in week \(t.\) (a) What are the units of \(S^{\prime}(t) ?\) (b) What is the practical meaning of \(S^{\prime}(t)>0 ?\) What circumstances might cause this situation?

Graph \(f(x)=\sin x,\) and use the graph to decide whether the derivative of \(f(x)\) at \(x=3 \pi\) is positive or negative.

Decide if the functions in are differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=(x+|x|)^{2}+1$$

At time \(t\) in seconds, a particle's distance \(s(t),\) in centimeters, from a point is given by \(s(t)=4+3 \sin t .\) What is the average velocity of the particle from \(t=\pi / 3\) to \(t=7 \pi / 3 ?\)

The wind speed, \(W,\) in meters per second, at a distance \(x\) km from the center of a hurricane is given by \(W=h(x).\) (a) Give the the units of \(d W / d x.\) (b) For a certain hurricane, \(h^{\prime}(15)>0 .\) What does this tell you about the hurricane?

Decide if the functions in are differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=\left\\{\begin{array}{ll} x \sin (1 / x)+x & \text { for } x \neq 0 \\ 0 & \text { for } x=0 \end{array}\right.$$

In a time of \(t\) seconds, a particle moves a distance of \(s\) meters from its starting point, where \(s=3 t^{2}\) (a) Find the average velocity between \(t=1\) and \(t=\) \(1+h\) if: (i) \(\quad h=0.1\) (ii) \(h=0.01,\) (iii) \(h=0.001\) (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time \(t=1\)

Suppose \(C(r)\) is the total cost of paying off a car loan borrowed at an annual interest rate of \(r \% .\) What are the units of \(C^{\prime}(r) ?\) What is the practical meaning of \(C^{\prime}(r) ?\) What is its sign?

Estimate \(f^{\prime}(2)\) for \(f(x)=3^{x} .\) Explain your reasoning.

Decide if the functions in are differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=\left\\{\begin{array}{ll} x^{2} \sin (1 / x) & \text { for } x \neq 0 \\ 0 & \text { for } x=0 \end{array}\right.$$

In a time of \(t\) seconds, a particle moves a distance of \(s\) meters from its starting point, where \(s=4 t^{3}\) (a) Find the average velocity between \(t=0\) and \(t=h\) if: (i) \(\quad h=0.1\) (ii) \(\quad h=0.01,\) (iii) \(\quad h=0.001\) (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time \(t=0\)

Let \(f(x)\) be the elevation in feet of the Mississippi River \(x\) miles from its source. What are the units of \(f^{\prime}(x) ?\) What can you say about the sign of \(f^{\prime}(x) ?\)

In each of the following cases, sketch the graph of a continuous function \(f(x)\) with the given properties. (a) \(f^{\prime \prime}(x)>0\) for \(x<2\) and for \(x>2\) and \(f^{\prime}(2)\) is undefined. (b) \(f^{\prime \prime}(x)>0\) for \(x<2\) and \(f^{\prime \prime}(x)<0\) for \(x>2\) and \(f^{\prime}(2)\) is undefined.

In a time of \(t\) seconds, a particle moves a distance of \(s\) meters from its starting point, where \(s=\sin (2 t)\) (a) Find the average velocity between \(t=1\) and \(t=\) \(1+h\) if: (i) \(\quad h=0.1\) (ii) \(\quad h=0.01,\) (iii) \(\quad h=0.001\) (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time \(t=1\)

Give the units and sign of the derivative. \(h^{\prime}(t),\) where \(h(t)\) is the altitude of a parachutist, in feet, \(t\) seconds after he jumps out of a plane.

A car is driven at a constant speed. Sketch a graph of the distance the car has traveled as a function of time.

Give the units and sign of the derivative. \(f^{\prime}(t),\) where \(f(t)\) is the temperature of a room, in degrees Celsius, \(t\) minutes after a heater is turned on.

The acceleration due to gravity, \(g,\) varies with height above the surface of the earth, in a certain way. If you go down below the surface of the earth, \(g\) varies in a different way. It can be shown that \(g\) is given by $$ g=\left\\{\begin{array}{l} \frac{G M r}{R^{3}} \text { for } r

A car is driven at an increasing speed. Sketch a graph of the distance the car has traveled as a function of time.

Give the units and sign of the derivative. \(P^{\prime}(r),\) where \(P(r)\) is the monthly payment on a car loan, in dollars, if the annual interest rate is \(r \%.\)

An electric charge, \(Q,\) in a circuit is given as a function of time, \(t,\) by $$ Q=\left\\{\begin{array}{ll} C & \text { for } t \leq 0 \\ C e^{-t / \hbar C} & \text { for } t>0 \end{array}\right. $$ where \(C\) and \(R\) are positive constants. The electric current, \(I\), is the rate of change of charge, so 1 $$ I=\frac{d Q}{d t} $$ (a) Is the charge, \(Q,\) a continuous function of time? (b) Do you think the current, \(I\), is defined for all times, t? [Hint: To graph this function, take, for example, \(C=1 \text { and } R=1 .]\)

The position of a particle moving along the \(x\) -axis is given by \(s(t)=5 t^{2}+3 .\) Use difference quotients to find the velocity \(v(t)\) and acceleration \(a(t)\)

A car starts at a high speed, and its speed then decreases slowly. Sketch a graph of the distance the car has traveled as a function of time.

Sketch the graph of \(f(x),\) and use this graph to sketch the graph of \(f^{\prime}(x)\). $$f(x)=5 x$$

Give the units and sign of the derivative. \(T^{\prime}(v),\) where \(T(v)\) is the time, in minutes, that it takes to drive from Tucson to Phoenix at a constant speed of \(v\) miles per hour.