§1.5 Interpreting, estimating, and using the derivative (continued)

Summary

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. - Use a central difference with $h = 0.01$ to estimate the value of $F'(10)$. - What are the units on the value of $F'(10)$ that you computed in (a)? What is the practical meaning of the value of $F'(10)$? - Which do you expect to be greater: $F'(10)$ or $F'(20)$? Why? - Write a sentence that describes the behavior of the function $y = F'(t)$ on the time interval $0 \le t \le 30$. How do you think its graph will look? Why?

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 $T = f(q)$. - What does it mean to say $f(50) = 0.75$? Write a complete sentence to explain, using correct units. - A person's sensitivity, $s$, to the drug is defined by the function $s(q) = f'(q)$. What are the units of sensitivity? - Suppose that $f'(50) = -0.02$. Write a complete sentence to explain the meaning of this value. Include in your response the information given in (a).

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 $t = 2$. - When is the ball's velocity greatest? - Determine the value of $v'(1)$. Justify your thinking. - 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 at time $t = 1$? - What is the physical meaning of the function $v'(t)$?

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 $V = h(m)$. - 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? - 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 meaning of your result, including units on your answer. - Which value do you expect to be greater: $h'(30000)$ or $h'(80000)$? Why? - Write a sentence to describe the long-term behavior of the function $V = h(m)$, plus another sentence to describe the long-term behavior of $h'(m)$. Provide your discussion in practical terms regarding the value of the car and the rate at which that value is changing.