91Ó°ÊÓ

Mileage for an old car: The gas mileage \(M\) that you get on your car depends on its age \(t\) in years. a. Explain the meaning of \(\frac{d M}{d t}\) in practical terms. b. As your car ages and performance degrades, do you expect \(\frac{d M}{d t}\) to be positive or negative?

A better investment: You open an account by investing \(\$ 250\) with a financial institution that advertises an APR of \(5.75 \%\), with continuous compounding. a. Find an exponential formula for the balance in your account as a function of time. In your answer, give both the standard form and the alternative form for an exponential function. b. What account balance would you expect 5 years after your initial investment? Answer this question using both of the forms you found in part a. Which do you think gives a more accurate answer? Why?

The acceleration due to gravity: From the time of Galileo, physicists have known that near the surface of the Earth, gravity imparts a constant acceleration of 32 feet per second per second. Explain how this shows that if air resistance is ignored, velocity for a falling object is a linear function of time.

Water in a tank: Water is leaking out of a tank. The amount of water in the tank \(t\) minutes after it springs a leak is given by \(W(t)\) gallons. a. Explain what \(\frac{d W}{d t}\) means in practical terms. b. As water leaks out of the tank, is \(\frac{d W}{d t}\) positive or negative? c. For the first 10 minutes, water is leaking from the tank at a rate of 5 gallons per minute. What do you conclude about the nature of the function W during this period? d. After about 10 minutes, the hole in the tank suddenly gets larger, and water begins to leak out of the tank at 12 gallons per minute. i. Make a graph of W versus t. Be sure to incorporate linearity where it is appropriate. ii. Make a graph of dW dt versus t.