Interpreting, estimating, and using the derivative

A potato is placed in an oven whose temperature is 350 degrees Fahrenheit, and the potato's temperature $F$ (in degrees Fahrenheit) is recorded in the following table. Time $t$ is measured in minutes.

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

2. Use a central difference to estimate the instantaneous rate of change o…

Researchers at a major car company have found a function that relates gasoline consumption to speed for a particular model of car. In particular, they have determined that the consumption $C$, in liters per kilometer, at a given speed $s$, is given by a function $C = f(s)$, where $s$ is the car's speed in kilometers per hour.

1. 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 esti…

A cup of coffee has its temperature $F$ (in degrees Fahrenheit) at time $t$ given by the function $F(t) = 75 + 110 e^{-0.05t}$, where time is measured in minutes.

1. Use a central difference with $h = 0.01$ to estimate the value of $F'(10)$.

2. What are the units on the value of $F'(10)$ that you computed in (a)? What is the practical meaning of the value of $F'(10)$?

3. Which do you expect to be greater: $F'(10)$ or $F'(20)$? Why?

4. Write a sentence that describes the behavior of the…

The temperature change $T$ (in Fahrenheit degrees), in a patient, that is generated by a dose $q$ (in milliliters), of a drug, is given by the function $T = f(q)$.

1. What does it mean to say $f(50) = 0.75$? Write a complete sentence to explain, using correct units.

2. A person's sensitivity, $s$, to the drug is defined by the function $s(q) = f'(q)$. What are the units of sensitivity?

3. Suppose that $f'(50) = -0.02$. Write a complete sentence to explain the mean…

The velocity of a ball that has been tossed vertically in the air is given by $v(t) = 16 - 32t$, where $v$ is measured in feet per second, and $t$ is measured in seconds. The ball is in the air from $t = 0$ until $t = 2$.

1. When is the ball's velocity greatest?

2. Determine the value of $v'(1)$. Justify your thinking.

3. What are the units on the value of $v'(1)$? What does this value and the corresponding units tell you about the behavior of the ball…

The value, $V$, of a particular automobile (in dollars) depends on the number of miles, $m$, the car has been driven, according to the function $V = h(m)$.

1. Suppose that $h(40000) = 15500$ and $h(55000) = 13200$. What is the average rate of change of $h$ on the interval $[40000,55000]$, and what are the units on this value?

2. In addition to the information given in (a), say that $h(70000) = 11100$. Determine the best possible estimate of $h'(55000)$ and write one sentence to explain the mea…