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2.3. THE PRODUCT AND QUOTIENT RULES
his land to grow corn, and that land had an average yield of 170 bushels per acre. At
the current time, he plans to increase his number of acres devoted to growing corn at a
rate of 600 acres/year, and he expects that right now his average yield is increasing
at a rate of 8 bushels per acre per year. Use this information to answer the following
questions.
(a) Say that the present year is t = 0, that A(t) denotes the number of acres the
farmer devotes to growing corn in year t, Y (t) represents the average yield in
year t (measured in bushels per acre), and C(t) is the total number of bushels
of corn the farmer produces. What is the formula for C(t) in terms of A(t) and
Y (t)? Why?
(b) What is the value of C(0)? What does it measure?
(c) Write an expression for C ′ (t) in terms of A(t), A ′ (t), Y (t), and Y ′ (t). Explain
your thinking.
(d) What is the value of C ′ (0)? What does it measure?
(e) Based on the given information and your work above, estimate the value of
C(1).
5. Let f (v) be the gas consumption (in liters/km) of a car going at velocity v (in km/hour).
In other words, f (v) tells you how many liters of gas the car uses to go one kilometer
if it is traveling at v kilometers per hour. In addition, suppose that f (80) = 0.05 and
f ′ (80) = 0.0004.
(a) Let g(v) be the distance the same car goes on one liter of gas at velocity v.
What is the relationship between f (v) and g(v)? Hence find g(80) and g ′ (80).
(b) Let h(v) be the gas consumption in liters per hour of a car going at velocity v.
In other words, h(v) tells you how many liters of gas the car uses in one hour if
it is going at velocity v. What is the algebraic relationship between h(v) and
f (v)? Hence find h(80) and h ′ (80).
(c) How would you explain the practical meaning of these function and derivative
values to a driver who knows no calculus? Include units on each of the function
and derivative values you discuss in your response.
2.3. THE PRODUCT AND QUOTIENT RULES
his land to grow corn, and that land had an average yield of 170 bushels per acre. At
the current time, he plans to increase his number of acres devoted to growing corn at a
rate of 600 acres/year, and he expects that right now his average yield is increasing
at a rate of 8 bushels per acre per year. Use this information to answer the following
questions.
(a) Say that the present year is t = 0, that A(t) denotes the number of acres the
farmer devotes to growing corn in year t, Y (t) represents the average yield in
year t (measured in bushels per acre), and C(t) is the total number of bushels
of corn the farmer produces. What is the formula for C(t) in terms of A(t) and
Y (t)? Why?
(b) What is the value of C(0)? What does it measure?
(c) Write an expression for C ′ (t) in terms of A(t), A ′ (t), Y (t), and Y ′ (t). Explain
your thinking.
(d) What is the value of C ′ (0)? What does it measure?
(e) Based on the given information and your work above, estimate the value of
C(1).
5. Let f (v) be the gas consumption (in liters/km) of a car going at velocity v (in km/hour).
In other words, f (v) tells you how many liters of gas the car uses to go one kilometer
if it is traveling at v kilometers per hour. In addition, suppose that f (80) = 0.05 and
f ′ (80) = 0.0004.
(a) Let g(v) be the distance the same car goes on one liter of gas at velocity v.
What is the relationship between f (v) and g(v)? Hence find g(80) and g ′ (80).
(b) Let h(v) be the gas consumption in liters per hour of a car going at velocity v.
In other words, h(v) tells you how many liters of gas the car uses in one hour if
it is going at velocity v. What is the algebraic relationship between h(v) and
f (v)? Hence find h(80) and h ′ (80).
(c) How would you explain the practical meaning of these function and derivative
values to a driver who knows no calculus? Include units on each of the function
and derivative values you discuss in your response.