91Ó°ÊÓ

Sketching a graph with given concavity: a. Sketch a graph that is always decreasing but starts out concave down and then changes to concave up. There should be a point of inflection in your picture. Mark and label it. b. Sketch a graph that is always decreasing but starts out concave up and then changes to concave down. There should be a point of inflection in your picture. Mark and label it.

A rental: A rental car agency charges \(\$ 49.00\) per day and 25 cents per mile. a. Calculate the rental charge if you rent a car for 2 days and drive 100 miles. b. Use a formula to express the cost of renting a car as a function of the number of days you keep it and the number of miles you drive. Identify the function and each variable you use, and state the units. c. It is about 250 miles from Dallas to Austin. Use functional notation to express the cost to rent a car in Dallas, drive it to Austin, and return it in Dallas 1 week later. Use the formula from part b to calculate the cost.

A population of deer: When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number \(N\) of deer present at time \(t\) (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function $$ N=\frac{12.36}{0.03+0.55^{t}} $$ a. How many deer were initially on the reserve? b. Calculate \(N(10)\) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10 th to the 15 th year?

Falling with a parachute: If an average-size man jumps from an airplane with a properly opening parachute, his downward velocity \(v=v(t)\), in feet per second, \(t\) seconds into the fall is given by the following table. $$ \begin{array}{|c|c|} \hline \begin{array}{c} t=\text { Seconds } \\ \text { into the fall } \end{array} & v=\text { Velocity } \\ \hline 0 & 0 \\ \hline 1 & 16 \\ \hline 2 & 19.2 \\ \hline 3 & 19.84 \\ \hline 4 & 19.97 \\ \hline \end{array} $$ a. Explain why you expect \(v\) to have a limiting value and what this limiting value represents physically. b. Estimate the terminal velocity of the parachutist.

A car that gets 32 miles per gallon: The cost \(C\) of operating a certain car that gets 32 miles per gallon is a function of the price \(g\), in dollars per gallon, of gasoline and the distance \(d\), in miles, that you drive. The formula for \(C=C(g, d)\) is \(C=g d / 32\) dollars. a. Use functional notation to express the cost of operation if gasoline costs 98 cents per gallon and you drive 230 miles. Calculate the cost. b. Calculate \(C(1.03,172)\) and explain the meaning of the number you have calculated.

How much can I borrow? The function in Example \(1.2\) can be rearranged to show the amount of money \(P=P(M, r, t)\), in dollars, that you can afford to borrow at a monthly interest rate of \(r\) (as a decimal) if you are able to make \(t\) monthly payments of \(M\) dollars: $$ P=M \times \frac{1}{r} \times\left(1-\frac{1}{(1+r)^{r}}\right) . $$ Suppose you can afford to pay \(\$ 350\) per month for 4 years. a. How much money can you afford to borrow for the purchase of a car if the prevailing monthly interest rate is \(0.75 \%\) ? (That is \(9 \%\) APR.) Express the answer in functional notation, and then calculate it. b. Suppose your car dealer can arrange a special monthly interest rate of \(0.25 \%\) (or \(3 \%\) APR). How much can you afford to borrow now? c. Even at \(3 \%\) APR you find yourself looking at a car you can't afford, and you consider extending the period during which you are willing to make payments to 5 years. How much can you afford to borrow under these conditions?

Brightness of stars: The apparent magnitude \(m\) of a star is a measure of its apparent brightness as the star is viewed from Earth. Larger magnitudes correspond to dimmer stars, and magnitudes can be negative, indicating a very bright star. For example, the brightest star in the night sky is Sirius, which has an apparent magnitude of \(-1.45\). Stars with apparent magnitude greater than about 6 are not visible to the naked eye. The magnitude scale is not linear in that a star that is double the magnitude of another does not appear to be twice as dim. Rather, the relation goes as follows: If one star has an apparent magnitude of \(m_{1}\) and another has an apparent magnitude of \(m_{2}\), then the first star is \(t\) times as bright as the second, where \(t\) is given by $$ t=2.512^{m_{2}-m_{1}} . $$ The North Star, Polaris, has an apparent magnitude of \(2.04\). How much brighter than Polaris does Sirius appear?

Renting motel rooms: You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for \(\$ 85.00\), two for \(\$ 83.00\) each, and in general the group rate per room is found by taking \(\$ 2\) off the base of \(\$ 85\) for each extra room rented. a. How much money do you charge per room if a group rents 3 rooms? What is the total amount of money you take in? b. Use a formula to give the rate you charge for each room if you rent \(n\) rooms to an organization. c. Find a formula for a function \(R=R(n)\) that gives the total revenue from renting \(n\) rooms to a convention host. d. Use functional notation to show the total revenue from renting a block of 9 rooms to a group. Calculate the value.

Arterial blood flow: Medical evidence shows that a small change in the radius of an artery can indicate a large change in blood flow. For example, if one artery has a radius only \(5 \%\) larger than another, the blood flow rate is \(1.22\) times as large. Further information is given in the table below. $$ \begin{array}{|c|c|} \hline \text { Increase in radius } & \begin{array}{c} \text { Times greater blood } \\ \text { flow rate } \end{array} \\ \hline 5 \% & 1.22 \\ \hline 10 \% & 1.46 \\ \hline 15 \% & 1.75 \\ \hline 20 \% & 2.07 \\ \hline \end{array} $$ a. Use the average rate of change to estimate how many times greater the blood flow rate is in an artery that has a radius \(12 \%\) larger than another. b. Explain why if the radius is increased by \(12 \%\) and then we increase the radius of the new artery by \(12 \%\) again, the total increase in the radius is \(25.44 \%\). c. Use parts a and \(b\) to answer the following question: How many times greater is the blood flow rate in an artery that is \(25.44 \%\) larger in radius than another? d. Answer the question in part c using the average rate of change.

Widget production: The following table shows, for a certain manufacturing plant, the number \(W\) of widgets, in thousands, produced in a day as a function of \(n\), the number of full-time workers. $$ \begin{array}{|c|c|} \hline n=\begin{array}{c} \text { Number of } \\ \text { workers } \end{array} & \begin{array}{c} W=\text { Thousands of } \\ \text { widgets produced } \end{array} \\ \hline 10 & 25.0 \\ \hline 20 & 37.5 \\ \hline 30 & 43.8 \\ \hline 40 & 46.9 \\ \hline 50 & 48.4 \\ \hline \end{array} $$ a. Make a table showing, for each of the 10 -worker intervals, the average rate of change in \(W\) per worker. b. Describe the general trend in the average rate of change. Explain in practical terms what this means. c. Use the average rate of change to estimate how many widgets will be produced if there are 55 full-time workers. d. Use your answer to part \(b\) to determine whether your estimate in part \(\mathrm{c}\) is likely to be too high or too low.