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Simon Fraser University Department of Mathematics Burnaby Campus
MATH 157 -3, 1107 Final Examination December 9 th, 2010, 8:30 – 11:
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Last Name (please print): _________________________________________
First Name (please print): _________________________________________
SFU Student Number: _________________________________________
SFU email ID: [email protected]
Instructor: P. Menz
Instructions:
- DO NOT OPEN THIS BOOKLET UNTIL TOLD TO DO SO.
- Fill in the above box.
- This exam contains 14 pages with a total of 10 questions. Once the exam begins please check to make sure your exam is complete.
- SHOW ALL YOUR WORK!
- If you run out of space in a problem, use the space on the back of the cover page and clearly indicate where the solution continues.
- Only scientific, non-programmable calculators with no differentiation and integration capabilities are allowed.
- No book, paper, or device, other than the usual writing instruments, this booklet and an acceptable calculator, shall be within reach of a student during the examination.
- During the examination, speaking to, communicating with, or deliberately exposing written papers to the view of other examinees is forbidden.
Do not write in this table!
Question Marks 1 / 2 / 3 + / 4 / (^5) / 6 / 7 / 8 a-c / 8 d / (^9) / 10 /
Total /
(left blank intentionally)
- Find the following limits if they exist. [3 marks each = 12 marks]
a)
2 6
lim x 6
x x
b) 0
sin lim x 4
x x
c)
7 1 39
lim x 1
x x
d)
2 3 4 5 3 5
lim x 5 4 3 2
x x x x x x x x
- Find the following derivatives. [4 marks each = 20 marks]
a) f ( ) x sin e x , f ( ) x Do not simplify!
b)
2 100 100 , 2 y x x^ d y dx
Do not simplify!
c)
ln(sin ) ( ) , cos 4
x g x g x
Evaluate exactly!
- Trigonometric Functions: [2+4 marks= 6 marks]
a) Evaluate the expression
csc sin 1 1
^
. (textbook exercise 6.3 #20)
b) Find the equation of the tangent line to the graph of the function
f ( ) x tan( ) x at the point
x .
- The world population at the beginning of 1990 was 5.3 billion. Assume that the population continues to grow exponentially at its present rate of approximately 2%/year. (textbook exercise 5.5 #12 and #13) [5 marks] a) Find the function Q(t) that expresses the world population (in billions) as a function of time t (in years), with t 0 corresponding to the beginning of 1990.
b) Find the length of time to the nearest integer required for the world population to triple in size.
- You are given the function f, and its first and second derivatives: (^2)
x f x x
2
2 2
x f x x
2
2 3
x x f x x
. (textbook exercise 7.2 #72) [9 marks]
a) Determine the intervals of increase and decrease.
b) Determine the intervals of concave up and concave down.
c) Answer T (true) or F (false) in the boxes provided about the function f.
f is an odd function.
lim ( ) 0
x f x
1
lim ( ) x f^ x
and 1
lim ( ) x f^ x
f has a relative maximum at x 0.
f has an absolute minimum at x 1.
f has an inflection point at x 0.
- The demand equation for a product is x 0.03 p 12 where p is the price in
dollars per unit with 0 p 300 and x is the quantity in thousands of units demanded. [4 + 1 + 4 + 4 marks = 13 marks] a) Determine the elasticity of demand function E ( p )at price p.
b) Solve E ( p ) 1 for p.
c) Answer T (true) or F (false) in the boxes provided about the demand.
The demand is inelastic if 0 p 200.
The demand is inelastic if 200 p 300.
For p 50 an increase in the unit price will cause the revenue to
increase.
For p 50 a decrease in the unit price will cause the revenue to
decrease.
- Suppose the quantity x of Super Grip radial tires made available each week in the marketplace is related to the unit-selling price by the equation (^1 ) 2
p x , where x is measured in units of a thousand and p is in dollars.
How fast is the weekly supply of Supper Grip radial tires being introduced into the marketplace when x=6, p=66, and the price per tire is decreasing at the rate of $3/week? (textbook exercise 4.4 #54) [8 marks]
- The Smith family amortizes a loan of $500,000 for a new house by obtaining a 25 year mortgage with monthly payments at the nominal rate of 5.5 % compounded semiannually. [10 marks] a) Calculate the nominal rate compounded monthly.
b) Find the monthly payment.
c) Find the total interest charges.
d) After 18 years the Smith family decides to pay off the loan. How much money is needed to pay off the balance of the loan?
