50
1.5. INTERPRETING, ESTIMATING, AND USING THE DERIVATIVE
for every one unit of input. Thus, we expect f (8) to be approximately 2 units greater
than f (7). The value is approximate because we don’t know that the rate of change
stays the same as x changes.
Exercises
1. A cup of coffee has its temperature F (in degrees Fahrenheit) at time t given by the
function F(t) = 75 + 110e −0.05t , where time is measured in minutes.
(a) Use a central difference with h = 0.01 to estimate the value of F ′ (10).
(b) 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)?
(c) Which do you expect to be greater: F ′ (10) or F ′ (20)? Why?
(d) Write a sentence that describes the behavior of the function y = F ′ (t) on the
time interval 0 ≤ t ≤ 30. How do you think its graph will look? Why?
2. 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).
(a) What does it mean to say f (50) = 0.75? Write a complete sentence to explain,
using correct units.
(b) A person’s sensitivity, s, to the drug is defined by the function s(q) = f ′ (q).
What are the units of sensitivity?
(c) 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).
3. 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.
(a) When is the ball’s velocity greatest?
(b) Determine the value of v ′ (1). Justify your thinking.
(c) 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?
(d) What is the physical meaning of the function v ′ (t)?
4. 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).
(a) 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?
1.5. INTERPRETING, ESTIMATING, AND USING THE DERIVATIVE
for every one unit of input. Thus, we expect f (8) to be approximately 2 units greater
than f (7). The value is approximate because we don’t know that the rate of change
stays the same as x changes.
Exercises
1. A cup of coffee has its temperature F (in degrees Fahrenheit) at time t given by the
function F(t) = 75 + 110e −0.05t , where time is measured in minutes.
(a) Use a central difference with h = 0.01 to estimate the value of F ′ (10).
(b) 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)?
(c) Which do you expect to be greater: F ′ (10) or F ′ (20)? Why?
(d) Write a sentence that describes the behavior of the function y = F ′ (t) on the
time interval 0 ≤ t ≤ 30. How do you think its graph will look? Why?
2. 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).
(a) What does it mean to say f (50) = 0.75? Write a complete sentence to explain,
using correct units.
(b) A person’s sensitivity, s, to the drug is defined by the function s(q) = f ′ (q).
What are the units of sensitivity?
(c) 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).
3. 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.
(a) When is the ball’s velocity greatest?
(b) Determine the value of v ′ (1). Justify your thinking.
(c) 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?
(d) What is the physical meaning of the function v ′ (t)?
4. 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).
(a) 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?