math 103

UNIVERSITY OF REGINA DEPARTMENT OF MATHEMATICS AND STATISTICS

MATH 103–001, 991 Calculus Final Examination, Semester 2013-10

23 April 2013 Instructors: C.-H. Guo (001), J. Hengen (991)

Family name: First name:

Student ID:

READ THESE INSTRUCTIONS CAREFULLY

1. This examination has 13 questions. If you need extra space, you can write on the back of the page.

2. You have 3 hours to complete this examination.

3. This is a closed book examination, and no notes of any kind are allowed.

4. You may use a calculator, but it must be one that is approved for the course.

5. Cell phones, pagers, or any text storage or communication devices are not permitted. (Warning: Any access of UR Courses during the examination will be detected by the instructors and may lead to a zero mark on the final examination.)

6. Read each question carefully.

7. Where it is possible to check your work, do so.

8. Good luck!

DO NOT OPEN THIS BOOKLET UNTIL YOU HAVE BEEN

TOLD TO DO SO

1. (6 marks) Using the definition of the derivative as a limit, find f ′(1) if f(x) = 4x2 −3.

2. (5 marks) List all the values of x for which f(x) is continuous. Give reasons for your answer.

f(x) =

{ 2 − 3x if x ≤−1 x2 −x + 3 if x > −1

2

3. (6 marks) Evaluate the following limit, if it exists.

(a) lim x→2

x2 + 4

x− 2 .

(b) lim x→1

x2 + x− 2 x2 − 1

.

4. (6 marks) Find the vertical and horizontal asymptotes of the graph of the function

f(x) = 5x2

x2 − 3x− 4 .

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5. (16 marks) Find the following derivatives. (You do not need to simplify your answers.)

(a) f ′(x) if f(x) = (x− 1)(3x + 2)4.

(b) g′(t) if g(t) = et − 3t t2 + 5

.

(c) dy

dx if y = ex

2+3x + 3 ln x− ln 5.

(d) f ′(x) if f(x) = e−x(x2 + 2)3

(x2 + 1)4 , using logarithmic differentiation.

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6. (5 marks) Find the equation of the tangent line to the curve y = √

3x2 + 1 at x = −1.

7. (5 marks) If f ′(x) = 2 √ x−

3

x and f(1) = 2, find f(x).

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8. (9 marks) For the function f(x) = x3 − 6x2 + 9x− 2, determine where the function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative exterma and inflection points, and sketch the graph of the function.

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9. (9 marks) The ABC store has been selling radios at $6 a piece, and at this price, consumers have been buying 3000 radios per month. The store estimates that for each $1 increase in the price, 1000 fewer radios will be sold each month. These radios can be produced at a cost of $4 per radio. At what price should the store sell the radios to generate the greatest possible profit?

10. (5 marks) Suppose that you can invest your money at an annual interest rate of 8%, compounded quarterly. How much money should you invest today so that it will be worth $5000 in 6 years?

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11. (5 marks) A radioactive substance has a half-life of 1280 years. How long will it take for a 10-gram sample to be reduced to 2 grams?

12. (7 marks) Sketch the region bounded by the curve y = x2 −3x and the line y = x + 5, and then find the area of the region.

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13. (16 marks) Evaluate the following integrals. Simplify your answers.

(a) ∫ (

1

2x −

2

x2 +

3 √ x

+ ex )

dx

(b) ∫

ln (3x)

x dx

(c) ∫ 1 −1

3x + 6

(x2 + 4x + 5)2 dx

(d) ∫ 1 0

6t

t2 + 1 dt

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