Download Boxes - Calculus for the Social Sciences - Exam Key and more Exams Calculus in PDF only on Docsity!
Simon Fraser University
Department of Mathematics
Burnaby Campus
MATH 157 -3, 1107
Final Examination
December 9
th , 2010, 8:30 – 11:
PROVIDE THIS DATA AS IT APPEARS ON WebCT!
Last Name (please print): ______________ KEY ____________________
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
8 a-c /
8 d (^) /
Total /
(left blank intentionally)
- Find the following limits if they exist. [3 marks each = 12 marks]
a)
2
6 6 6
lim lim lim 6 12 x (^) 6 x (^) 6 x
x x^ x x x x ^ ^
^
b) (^) 0 0
sin8 sin 8 lim 2 lim 2 1 2 x (^) 4 x 8
x x
(^) x x
c)
7 6 5 6 5
1 39 1 38 37 1 38 37
1 1 ...^1 ... 1 7
lim lim lim x (^) 1 x (^) 1 ... 1 x ... 1 39
x x^ x^ x x x
(^) x (^) x x x x x
^ ^ ^
d)
2 3 4 5
3 5
lim x 5 4 3 2
x x x x x
x x x
5 4 3 2
5 4 2
lim 5 4 3 2
x
x x x x x
x x x
- Find the following derivatives. [4 marks each = 20 marks]
a) ( ) sin (^) , ( )
x f x e f x Do not simplify!
( ) cos 2
x x f x e e x
b)
2 100 2
100 ,
x d y y x dx
Do not simplify!
100
99 100
2 98 99 99 100 2
100 ln
100 99 ln100 ln100 100 ln
100
100 100
100 100 100 100
x
x x
x x x x
y x
dy x x dx
d y x x x x dx
c)
ln(sin ) ( ) , cos 4
x g x g x
Evaluate exactly!
2
2
2
2
cos ln(sin ) sin sin ( ) cos
cos 4 ln(sin ) sin 1 1 1 (^4 4) ln( ) sin 4 2 2 2
4 1 cos 4 2
ln( ) 2 2 ln( ) 2 2 2 2
x x x x g x x
g
^
- Trigonometric Functions: [2+4 marks= 6 marks]
a) Evaluate the expression
csc sin
^
. (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 .
We need a slope and a point.
Point:
, ,tan ,
f
^ ^
^ ^ ^ ^ ^
Slope:
2
2 2
( ) sec ( )
sec
f x x
f
^
Tangent line equation:
y x y 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
kt
Q t y e
We know y 0 (^) 5.3and
1 1.02 5.3 5.
k e.
Solving for k we get
1
ln1.
k e
k
Finally, (^)
ln1.02 ln1. ( ) 5.3 5.3 5.3 1.
t^ t^ t Q t e e
.
b) Find the length of time to the nearest integer required for the world
population to triple in size.
We need to solve 3 5.3 5.3 1.02
t for t.
ln 3 ln 1.02 ln 1.
ln 3
ln 1.
t
t
t t
t
Therefore, the population will triple in size in approximately 55 years.
- 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.
f '( ) x 0 (^)
2 2 1 x 0 x 1. So, the critical numbers are x 1.
Therefore, f is decreasing on (^) , (^1) (^) 1, (^) , and increasing on (^) 1,1 (^) .
b) Determine the intervals of concave up and concave down.
f ( )^ x 0 (^)
2 4 x x 3 0 x 0, 3.
Therefore, f is concave up on (^) ^ 3,0^ ^ 3,^ , and concave down on
,^3 ^ 0,^3 .
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.
T
T
F
F
T
T
- 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. [8 marks]
a) Determine the elasticity of demand function E ( p )at price p.
0.03^ ^12
d d x p dp dp
dx
dp
dx
dp
and
x p
x p
Then, (^)
p dx p p E p x dp p p
b) Solve E ( p ) 1 for p.
E p
p
p
p p
p
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.
F
T
T
T
- 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 ) 48 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]
Given 3
dp
dt
dollars/week.
Differentiating the given equation
p x implicitly w.r.t. t we get
d d p x dt dt
dp dx x dt dt
dx dp
dt dt x
^
When x 6, p 66 , and 3
dp
dt
we have
6, 66
x p^6
dx
dt (^)
Therefore, the supply is decreasing at the rate of 0.5 thousand tires/week.
(or the supply is decreasing at the rate of 500 tires/week)
- 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.
2/
12 1 1 0. 2
r monthly
b) Find the monthly payment.
300 1
r monthly P m i
t n
R
^
c) Find the total interest charges.
interest $3051.96 300 $500,000 $415,588.
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?
After 18 years, there are 7 years remaining, i.e. n 7 12 84.
84 1 (1.004532...) 3051.96 $212,818. 0.004532...
P
