ConcepTest • Section 2.4 • Question 1

1. For Problems 1-3, the function C(r) is the total cost, in dollars, of paying off a car loan borrowed at an interest rate of r% per year. ConcepTest• Section 2.4 •Question 1 • Year/$ • $/Year • $/(%/Year) • (%/Year)/$

2. ANSWER ConcepTest• Section 2.4 •Answer 1 (c) COMMENT: Remind students that many times looking at the units will help them solve the problem.

3. For Problems 1-3, the function C(r) is the total cost of paying off a car loan borrowed at an interest rate of r% per year. What is the practical meaning of C’(5)? ConcepTest• Section 2.4 •Question 2 • The rate of change of the total cost of the car loan is C’(5) • If the interest rate increases by 1%, then the total cost of the loan increases by about C’(5). • If the interest rate increases by 1%, then the total cost of the loan increases by about C’(5) when the interest rate is 5%. • If the interest rate increases by 5%, then the total cost of the loan increases by about C’(5).

4. ANSWER ConcepTest• Section 2.4 •Answer 2 (c). C’(5) requires the interest rate to be 5%. COMMENT: This problem is designed to help students make connections between real-world problems and mathematical concepts.

5. For Problems 1-3, the function C(r) is the total cost of paying off a car loan borrowed at an interest rate of r% per year. What is the sign of C’(5)? (a) Positive (b) Negative (c) Not enough information ConcepTest• Section 2.4 •Question 3

6. ANSWER ConcepTest• Section 2.4 •Answer 3 (a). If the interest rate increases, then the total car loan cost also increases, thus C’(5) is positive. COMMENT: Students don’t always realize how to make connections between real-world problems and mathematical concepts.

7. For Problems 4-5, you invest $1000 at an annual interest rate of r%, compounded continuously. At the end of 10 years, you have a balance of B dollars, where B = g(r). What is the financial interpretation of g(5) = 1649? (a) When r is 5%, B is $1649. (b) When the interest rate is 5, you have $1649. (c) When the interest rate is 5%, there is $1649. (d) When the interest rate is 5%, then in 10 years you have a balance of $1649. ConcepTest• Section 2.4 •Question 4

8. ANSWER ConcepTest• Section 2.4 •Answer 4 (d) contains the most information. COMMENT: Try to get your students to see the subtle differences between the four choices.

9. For Problems 4-5, you invest $1000 at an annual interest rate of r%, compounded continuously. At the end of 10 years, you have a balance of B dollars, where B = g(r). What is the financial interpretation of g’(5) = 165? (a) The balance in your account after 5 years is $165. (b) The balance grows at a rate of $165 per % when r = 5%. (c) If the interest rate increases from 5% to 6%, you would expect about $165 more in your account after 10 yrs. (d) If the interest rate increases from 5% to 6% you would expect about $1814 in your account after 10 yrs. ConcepTest• Section 2.4 •Question 5

10. ANSWER ConcepTest• Section 2.4 •Answer 5 (c) and (d) are equivalent. (d) uses the information from the previous problem. COMMENT: Students should start thinking of rate as an incremental change.

11. Let N = f(t) be the total number of cans of cola Sean has consumed by age t in years. Interpret the following in practical terms, paying close attention to units. (a) f (14) = 400 (b) f -1 (50) = 6 (c) f ‘(12) = 50 (d) (f -1)’(450) = 1/70 ConcepTest• Section 2.4 •Question 6

12. ANSWER ConcepTest• Section 2.4 •Answer 6 • By age 14, Sean had consumed 400 cans of cola. • Sean had consumed 50 cans of cola by age 6. • At age 12 Sean was consuming approximately 50 cans of cola per year. • At the time Sean had consumed 450 cans of cola, it took approximately one year to consume 70 additional cans. COMMENT: Students have a difficult time with derivatives of inverses.

13. For Problems 7-9, let A = f(t) be the depth of tread, in centimeters, on a radial tire as a function of the time elapsed t, in months, since the purchase of the tire. Interpret the following in practical terms, paying close attention to units. (a) f (6) = 0.5 (b) f -1 (0.31) = 15 (c) f ‘(12) = – 0.015 (d) (f -1)’(0.4) = – 60 ConcepTest• Section 2.4 •Question 7

14. ANSWER ConcepTest• Section 2.4 •Answer 7 • After 6 months, the depth of tread was 0.5 cm. • When the depth of tread was 0.31 cm, the tire had been used for 15 months. • After 12 months, the depth of tread was decreasing by 0.015 cm per month. • When the depth of tread was 0.4 cm, it took 60 months to reduce the tread by 1 cm (or it took one month to reduce the depth of tread by 1/60 cm). COMMENT: For a challenge have students construct a similar word problem.

15. For Problems 7-9, let A = f(t) be the depth of tread, in centimeters, on a radial tire as a function of the time elapsed t, in months, since the purchase of the tire. What is the sign of f ‘(t)? Explain why. ConcepTest• Section 2.4 •Question 8

16. ANSWER ConcepTest• Section 2.4 •Answer 8 The sign of f ’(t) is negative, because tread wears with use, so its depth is always decreasing. COMMENT: Follow-up Question. What would f ’(t) > 0 mean?

17. For Problems 7-9, let A = f(t) be the depth of tread, in centimeters, on a radial tire as a function of the time elapsed t, in months, since the purchase of the tire. What is the sign of (f -1)‘(A)? Explain why. ConcepTest• Section 2.4 •Question 9

18. ANSWER ConcepTest• Section 2.4 •Answer 9 A = f(t) is a decreasing function, so its inverse will also be decreasing, and (f -1)’(A) will be negative. COMMENT: This reasoning might be pointed out graphically.