48
1.5. INTERPRETING, ESTIMATING, AND USING THE DERIVATIVE
Activity 1.13.
A company manufactures rope, and the total cost of producing r feet of rope is C(r)
dollars.
(a) What does it mean to say that C(2000) = 800?
(b) What are the units of C ′ (r)?
(c) Suppose that C(2000) = 800 and C ′ (2000) = 0.35. Estimate C(2100), and
justify your estimate by writing at least one sentence that explains your thinking.
(d) Which of the following statements do you think is true, and why?
• C ′ (2000) < C ′ (3000)
• C ′ (2000) = C ′ (3000)
• C ′ (2000) > C ′ (3000)
(e) Suppose someone claims that C ′ (5000) = −0.1. What would the practical
meaning of this derivative value tell you about the approximate cost of the next
foot of rope? Is this possible? Why or why not?
⊳
Activity 1.14.
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.
(a) 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.
(b) By writing a complete sentence, interpret the meaning (in the context of fuel
consumption) of “ f (80) = 0.015.”
(c) Write at least one complete sentence that interprets the meaning of the value
of f ′ (90) that you estimated in (a).
⊳
In Section 1.4, we learned how use to the graph of a given function f to plot the
graph of its derivative, f ′ . It is important to remember that when we do so, not only
does the scale on the vertical axis often have to change to accurately represent f ′ , but
the units on that axis also differ. For example, suppose that P(t) = 400 − 330e −0.03t tells
us the temperature in degrees Fahrenheit of a potato in an oven at time t in minutes. In
Figure 1.24, we sketch the graph of P on the left and the graph of P ′ on the right.
1.5. INTERPRETING, ESTIMATING, AND USING THE DERIVATIVE
Activity 1.13.
A company manufactures rope, and the total cost of producing r feet of rope is C(r)
dollars.
(a) What does it mean to say that C(2000) = 800?
(b) What are the units of C ′ (r)?
(c) Suppose that C(2000) = 800 and C ′ (2000) = 0.35. Estimate C(2100), and
justify your estimate by writing at least one sentence that explains your thinking.
(d) Which of the following statements do you think is true, and why?
• C ′ (2000) < C ′ (3000)
• C ′ (2000) = C ′ (3000)
• C ′ (2000) > C ′ (3000)
(e) Suppose someone claims that C ′ (5000) = −0.1. What would the practical
meaning of this derivative value tell you about the approximate cost of the next
foot of rope? Is this possible? Why or why not?
⊳
Activity 1.14.
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.
(a) 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.
(b) By writing a complete sentence, interpret the meaning (in the context of fuel
consumption) of “ f (80) = 0.015.”
(c) Write at least one complete sentence that interprets the meaning of the value
of f ′ (90) that you estimated in (a).
⊳
In Section 1.4, we learned how use to the graph of a given function f to plot the
graph of its derivative, f ′ . It is important to remember that when we do so, not only
does the scale on the vertical axis often have to change to accurately represent f ′ , but
the units on that axis also differ. For example, suppose that P(t) = 400 − 330e −0.03t tells
us the temperature in degrees Fahrenheit of a potato in an oven at time t in minutes. In
Figure 1.24, we sketch the graph of P on the left and the graph of P ′ on the right.