Mathematics Questions

1. (8 points) Below is the graph of a function, y = f(x), on the domain [−1, 7]. (Note that this graph has a vertical tangent line at (0, 0).)

x

−1 0 1 2 3 4 5 6 7

y

−1

0

1

2

3

y = f(x)

(a) Please fill in the following table by writing a “Y” or “N” into each relevant box. Note: We don’t consider endpoints to be critical points, so we’ve filled that part of the table in for you.

critical point local max local min global max global min inflection point

x = −1 N

x = 0

x = 1

x = 2

x = 3

x = 4

x = 5

x = 6

x = 7 N

(b) In the interval 1 < x < 7, at approximately which value of x is f ′(x) minimized?

Answer: f ′(x) is minimized when x ≈ .

(You are not required to explain your answer.)

2. (10 points) Blue Angel is a company that produces soft drinks. They created a new energy drink called “The Beast’s Awakening” and have asked you to design the cans of this drink. The can must be cylindrical and have a volume of 1000 cm3, but the radius and height of the can are up to you to decide!

The top and bottom of the can are made from a thicker aluminum sheet than the sides. The top and bottom sheets cost $0.002/cm2 and the side sheet costs $0.001/cm2. What should be the radius and height of the can in order to minimize the costs of the materials needed to produce it? Show all of your reasoning in the box below. Please round your answers to the nearest 0.01.

The can should have a radius of and a height of .

3. (6 points) Graphed below is a confectionary company’s marginal cost and marginal revenue from producing and selling their signature chocolate eggs. Due to inventory constraints, they can produce at most 10,000 eggs. They have a fixed cost of $800.

y ($ per egg)

q (thousands of eggs)

MR(q)

MC(q)

0 1 2 3 4 5 6 7 8 9 10 0

1

2

3

4

5

6

How many chocolate eggs should the company produce in order to maximize their profits? What will their total profit be then? Please justify your answer. Show all of your reasoning in the box below.

The company should produce chocolate eggs.

Their profit will then be .

4. (8 points) You’re a happy alum of the University of Toronto who would like to endow scholarships for future students. Every year, a scholarship of 10,000 CAD will be given out in your name. The first scholarship will be given today. To thank you for your generosity, the University will cover the cost of the first two scholarships. However, you must give them your donation now (not two years from now), in one single payment that will fund the scholarship for the rest of time (i.e. forever).

(a) Assume the annual interest rate is 3.5%, compounded annually. How much should your donation be? Show your reasoning; use only a scientific calculator.

The donation should be at least CAD.

(b) What happens if the annual interest rate turns out to be lower than planned for? If the annual interest rate is only 3.1% (compounded annually), how many scholarships can be given out before the funds run out, with the initial donation calculated in (a)? Your count should include the initial two scholarships funded by the university. Show your reasoning; use only a scientific calculator.

The scholarship fund will run out after scholarships have been given.

5. (4 points) Jonah has just purchased a brand new pickup truck! The truck dealership has offered him two possible payment plans.

© Payment plan A: make one payment of $35,000 today, and the truck is yours! © Payment plan B: make payments of $1200 per month, starting today, every month for the

next three years (for a total of 36 payments).

Jonah has a bank account with an interest rate of 1.5% per month, compounded continuously. Which payment plan would you recommend he take? Fill in the appropriate bubble above and explain your reasoning below.

6. (4 points) The temperature T of a cooling cup of tea has been measured, in ◦C, at intervals of 30 seconds in the table below.

Time (s) 0 30 60 90 120 150 180

Temperature (◦C) 95 86 77 65 59 56 53

Using a Riemann sum, give an upper bound and a lower bound for the average temperature of the cup of tea during this 3 minute period. Explain your reasoning in the box below.

The average temperature is at least and at most .

7. (9 points) (a) Please sketch the graph of f(x) = (x − 1)2 on [−1, 3] using the grid below. Label at least 3 points on the graph.

y

x −1 0 1 2 3

−1

0

1

2

3

4

Your goal in this problem is to find an upper bound for∫ 3 0

(x − 1)2 dx

using Riemann sums. What will your strategy be?

© Use a left-hand sum.

© Use a right-hand sum.

© Break [0, 3] into two intervals and use a left-hand sum on one of them and a right-hand sum on the other.

(b) Divide [0, 3] into six subintervals of equal length, then draw the six corresponding rectangles you

will use to find an upper bound for

∫ 3 0

(x − 1)2 dx. Compute the upper bound and show your work.

My upper bound is .

(c) Using Wolfram Alpha, Excel or the Geogebra link provided in the L1 slides, what new upper bound would you obtain if you divide [0, 3] into 90 subintervals of equal length, using the same strategy you identified in (a)? (Please round to the nearest thousandth.)

Answer: My new upper bound is .

(d) What is the exact value of

∫ 3 0

(x−1)2 dx? Show your reasoning, using the Fundamental Theorem of Calculus.

My exact answer is .

8. (8 points) You’re a pharmaceutical chemist at Astra Zeneca working on the material properties of pill design. If you administer a certain pill to a patient at time t = 0, you find that the concentration of the drug in the patient’s bloodstream changes at a rate of

r(t) = e−0.25t − ae−0.1t

where a is a constant, r(t) is measured in ng/L per hour and t is measured in hours. This model accounts for both the drug entering the bloodstream due to the pill, as well as the drug leaving the bloodstream as the body metabolizes it. At t = 0, the patient had no drug in their bloodstream.

(a) In plain English, interpret the meaning of

∫ 4 0

r(t) dt.

(b) Which of the following best represents a simplification of

∫ 4 0

r(t) dt?

© −4 ( e−1 − 1

) + 10a(e−0.4 − 1)

© −4 ( e−1 − 1

) + 10a(e−0.4 − 1) + C, where C is an arbitrary constant.

© −4e−0.25t + 10ae−0.1t

© −4e−0.25t + 10ae−0.1t + C, where C is an arbitrary constant. © None of the above

(c) Find the value of

∫ ∞ 0

r(t) dt in terms of a.

∫ ∞ 0

r(t) dt =

(d) If the body eventually metabolizes all of the drug in the blood stream, what must the value of a be? Explain briefly.

a =

9. (7 points) Let f(x) be the function graphed below, with domain [−3, 3].

y

x

y = f(x)

−3 −2 −1 1 2 3

−2

−1

1

2

3

Let F (x) be a continuous function, such that F (x) is an antiderivative of f(x), satisfying F (0) = 0.

(a) Please fill in the table of values below:

x −3 −2 −1 0 1 2 3

F(x)

(b) Sketch the graph of F (x) below.

y

x −3 −2 −1 1 2 3

−2

−1

1

2

3

10. (1 point) The integral

∫ b a f(x + c) dx is equal to which of the following integrals? Select all correct

answers. You are not required to explain.

© ∫ b a f(x) dx ©

∫ b+c a+c

f(x + c) dx © ∫ b+c a+c

f(x) dx © ∫ b a f(u) du © None of these.

11. (1 point) The graph below represents the cost function for producing tablet computers. In which interval is the average cost minimized? Sketch on the graph to locate the point where the average cost is minimized, then select the corresponding interval below.

y

q10 20 30 40

y = C(q)

© 0 < q ≤ 10 © 10 < q ≤ 20 © 20 < q ≤ 30 © 30 < q ≤ 40 © More information is required to answer.

12. (4 points) You’re on a desert island and you find a treasure map that reads, “Starting at the rock with the skull carved on its side, I walk northward for two hours. Every ten minutes, I measure my speed in miles per minute. . . ” and there’s a formula:

12∑ k=1

10k

1 + (10k)2 10

Physically, explain what this sum is approximating. (Use the language of definite integrals, velocity, distance travelled, and so forth.)