MAT 1308 A
Introduction to Calculus
Final Exam
December 2008
Instructor: Dr. Steve Desjardins
Duration: 3 hours
Name:
Student Number:
Instructions:
- Print your name and student number on this page.
- Verify that your copy of the exam has all 13 pages.
- You must answer all questions. There are 12 questions worth a total of
50 marks.
- Write your answers in the spaces below the questions. You may use the
backs of the pages if necessary.
- No Notes or Books.
- Basic scientific calculators only – graphing and/or programmable calculators and calculators with differentiation or integration capabilities are NOT permitted.
MAT 1308 A F08 – Final Exam
Question 1 (3 marks) Use the definition of the derivative to find f 0 (x) if f (x) =
2
3
.
x+2
MAT 1308 A F08 – Final Exam


2x + 1 x < 0
Question 2 (4 marks) Consider the function f (x) = x2
0≤x≤2


4
x>2
Is it continuous at x = 0 and x = 2 ?
3
MAT 1308 A F08 – Final Exam
4
Question 3 (6 marks) Find the derivatives of the following functions. Simplify where
possible.
(a) f (x) = (3x + 2)4 (x2 + 4x)3
(b) g(x) = x2 e4x + 7x ln(x2 + 2)
(c) h(x) =
5x2 + 2x
x2 − x − 1
3
MAT 1308 A F08 – Final Exam
Question 4 (6 marks) Find the following integrals.
Z
(a) x (3x2 + 2)5/4 dx
Z
(b)
1
xe2x dx
0
Z
(c)
1
dx
4x + 3
5
MAT 1308 A F08 – Final Exam
Question 5 (6 marks) Sketch the graph of y = f (x) =
6
x
.
x+2
MAT 1308 A F08 – Final Exam
7
Question 6 (4 marks) A product has demand function D(p) = −0.2p2 + 70p + 43 . What
price will maximize revenue ? What is the elasticity of demand when p = $240 ? Should the
price of the product be increased or decreased ?
MAT 1308 A F08 – Final Exam
8
Question 7 (4 marks) An open-topped box with a square base is to have a volume of
4000 cm3 . What dimensions will require the least amount of materials ?
MAT 1308 A F08 – Final Exam
9
Question 8 (3 marks) The supply curve for a product is S(q) = 0.1q 2 + 10q . If 10 units
are sold, what is the Producer Surplus ?
MAT 1308 A F08 – Final Exam
Question 9 (3 marks) If x2 y 3 + 5xy 2 = 3x , what is
10
dy
?
dx
MAT 1308 A F08 – Final Exam
11
Question 10 (4 marks) Money is transferred into an account at the rate of 500 + 100t
dollars per year. The account earns 5% annual interest, compounded continuously. What
is the value of the account after 3 years ?
MAT 1308 A F08 – Final Exam
Question 11 (3 marks) If f (x) = x3 e2x + 3x4 ln x , what is f 000 (x) ?
12
MAT 1308 A F08 – Final Exam
13
4
Question 12 (4 marks) Find the absolute maximim and minimum of the function f (x) = x + on
x
the interval 1 ≤ x ≤ 8 .