§1.5 Interpreting, estimating, and using the derivative

Introduction

Units of the derivative function

Toward more accurate derivative estimates

Suppose that you are driving a car on the highway using cruise control so that you drive at a constant speed. The time, $T$ (measured in hours), it takes to drive 50 miles depends on the constant rate, $s$ (measured in miles per hour), at which you are driving. Since

, it follows that

, and thus we know that $T(s) = \frac{50}{s}$.

For instance, observe that $T(75) = \frac{50}{75} = \frac{2}{3}$, so it takes $\frac{2}{3}$ of an hour to drive $50$ miles at $75$ miles per hour, and the point $(75, \frac{2}{3})$ lies on the graph of $T(s)$.

Suppose that the function $y = P(t)$ measures the population of a city (in thousands) at the start of year $t$ (where $t = 0$ corresponds to the year 2020). What does

mean in context?

Consider the function $T(s) = \frac{50}{s}$ from Preview Activity 1.5, whose output tells us the amount of time (in hours) it takes to drive 50 miles at $s$ miles per hour. What is the meaning of

?

Suppose that $V(m)$ measures the value of a car (in dollars) after the car has been driven $m$ miles. Explain the meaning of the statement

by writing two sentences that address the instantaneous rate of change and the predicted change in the function, respectively.

Explicitly, your first sentence might have structure like

where the final blank should be completed using the units on the derivative. Your second sentence might have form

Suppose that $W(h)$ measures the amount of water (in liters) in a tank that is filled with water that is $h$ meters deep. Explain the meaning of the statement

by writing two sentences in the structure described above.

Suppose that $S(t)$ measures the temperature of a can of soda (in degrees Celsius) in a refrigerator at time $t$ in minutes. Explain the meaning of the statement

by writing two sentences in the structure described above.

Suppose that $C(s)$ measures the rate at which a person burns calories (in calories per hour) when riding a bike at a speed of $s$ kilometers per hour. Explain the meaning of the statement

by writing two sentences in the structure described above.

Suppose that $y = f(x)$ is a function for which three values are known: $f(1) = 2.5$, $f(2) = 3.25$, and $f(3) = 3.625$. Estimate $f'(2)$.

Use a central difference to estimate the instantaneous rate of change of the potato's temperature at $t= 20$. Include units on your answer.

Use a central difference to estimate the instantaneous rate of change of the potato's temperature at $t= 40$. Include units on your answer.

Without doing any calculation, which do you expect to be greater: $F'(50)$ or $F'(60)$? Why?

Suppose we know that $F(46) = 192.5$ and $F'(46) = 1.39$. What are the respective units on these two quantities? What do you expect the temperature of the potato to be when $t = 47$? when $t = 48$? Why?

Write a couple of careful sentences that describe the behavior of the temperature of the potato on the time interval $[10,40]$, as well as the behavior of the instantaneous rate of change of the potato's temperature on the same time interval.

Data provided by the car company tells us that $f(80) = 0.015$, $f(90) = 0.02$, and $f(100) = 0.027$. Use this information to estimate the instantaneous rate of change of fuel consumption with respect to speed at $s = 90$. Be as accurate as possible, use proper notation, and include units on your answer.

By writing a complete sentence, interpret the meaning (in the context of fuel consumption) of “$f(80) = 0.015$.”

Write at least one complete sentence that interprets the meaning of the value of $f'(90)$ that you estimated in (a).