Last name (Print):
First name (Print):
UW Student ID Number:
University of Waterloo
Final Examination
MATH 116
Calculus 1 for Engineering
Instructor: Matthew Douglas Johnston
Section: 001
Date: Monday, July 30, 2012
Time: 4:00 - 6:30 p.m.
Term: 1125
Duration of exam: 2 hours, 30 minutes
Number of exam pages: 10
(including cover page)
Exam type: Closed Book
Formulas
• sin 2α = 2 sin α cos α
• sin2 α = (1 − cos 2α)/2
• cos2 α = (1 + cos 2α)/2
• sin2 α + cos2 α = 1
• tan2 α + 1 = sec2 α
Instructions
1. Write your name and ID number at the top
of this page.
2. Answer the questions in the spaces provided,
using the backs of pages for overflow or
rough work.
FOR EXAMINERS’ USE ONLY
Page
Mark
2
/11
3. No graphing calculators allowed.
3
/8
4. Show all your work required to obtain your
answers.
4
/10
5
/8
6
/10
7
/9
8
/9
9
/10
Total
/75
MATH 116 Final Exam, Spring 2012
Page 2 of 10
Name:
1. Miscellaneous Topics
1
(a) Consider the functions f (x) = x2 − 1 and g(x) = √ .
x
[2]
(i) Find (f ◦ g)(x). What is its domain?
[2]
(ii) Find (g ◦ f )(x). What is its domain?
[3]
(iii) Set h(x) = (f ◦ g)(x) and find h−1 (x). What is its domain? (Hint: Be careful
to consider the domain of h(x) and how it relates to the range of h−1 (x).)
[2]
(b) Consider the Heaviside function given by
1,
H(x) =
0,
for x ≥ 0
for x < 0.
State the following as a piecewise defined function:
f (x) = −x + |x|H(x).
[2]
(c) Set-up the partial fraction decomposition of
f (x) =
x2 + 1
.
(2x − 3)(x − 1)2 (x2 + 4)
Do not attempt to evaluate it!
MATH 116 Final Exam, Spring 2012
Page 3 of 10
Name:
2. Short answer questions
(a) Answer the following definition questions:
d
f (x).
dx
[1]
(i) State the formal definition of the derivative
[1]
(ii) Suppose f (x) is continuous on an interval [a, b], f (a) < 0 and f (b) > 0. Name
the theorem which allows us to conclude that there is a c ∈ (a, b) such that
f (c) = 0.
[1]
(iii) Suppose f (x) is continuously differentiable in the interval (a, b). Name the
theorem which allows us to conclude that there is a c ∈ (a, b) such that f 0 (c)
equals the average rate of change over the interval.
(b) Evaluate the exact values of the following:
[1]
(i) arcsec(2)
(Draw the triangle for full marks)
[2]
(ii) sin(arctan(2/3))
[2]
(iii) p0 (0) given that p(t) =
(Draw the triangle for full marks)
1
, r(0) = 2, and r0 (0) = −4.
r(t)
MATH 116 Final Exam, Spring 2012
Page 4 of 10
Name:
3. Limits
Evaluate the following limits:
√
x4 − 1 + x
4x2 − 1
[3]
(a) lim
[4]
x1
1
(b) lim
x→∞
x
[3]
(c) lim
x→∞
cos2 (7t) − 1
t→0
t sin(4t)
sin(kt)
= k.)
t→0
t
(Hint: You may use the fact that lim
MATH 116 Final Exam, Spring 2012
Page 5 of 10
4. Derivatives
Find f 0 (x) =
dy
for the following:
dx
[3]
(a) f (x) = √
[3]
(b) ey =
1
1 + e2x
x
y
Z
[2]
(c) f (x) =
x
x2
√
cos( s) ds,
x>1
Name:
MATH 116 Final Exam, Spring 2012
Page 6 of 10
Name:
5. Approximation
Many problems in mathematics do not have explicit solutions; instead, we must rely
on approximations. Consider the following problems.
[4]
(a) Use linear approximation to estimate the value of arctan(1.02). (Hint: Note that
π
arctan(1) = .)
4
[4]
(b) Use Newton’s Method to estimate the value of arctan(1.02). It is sufficient to
compute the first iterate x1 (i.e. apply Newton’s Method once). (Hint: The exact
value of arctan(1.02) is the root of f (x) = tan(x) − 1.02 nearest x0 = π/4.)
Newton’s Formula: xn+1 = xn −
[2]
f (xn )
f 0 (xn )
(c) Set up the Riemann sum for evaluating the area below the curve f (x) = ln(x)
between the bounds x = 1 and x = 2 using the right end-point of each interval.
Do not attempt to evaluate the sum!
MATH 116 Final Exam, Spring 2012
Page 7 of 10
Name:
6. Curve Sketching
An important application of the tools developed in this class is curve sketching. Con2
sider the function f (x) = xe−x /2 .
[2]
(a) Find all vertical and horizontal asymptotes of f (x). (Hint: Consider writing the
exponent in the denominator and using L’Hopital’s rule.)
[1]
(b) Determine all roots of f (x).
[2]
(c) Given that f 0 (x) = (1 − x2 )e−x /2 , determine all critical points and the intervals
where f (x) is increasing and decreasing.
[2]
(d) Given that f 00 (x) = x(x2 − 3)e−x /2 , determine all points of inflection and the intervals where f (x) is concave up and concave down
[2]
(e) Based on the information from parts (a)-(d), sketch the
(x). Label all
√ graph of f−1/2
3
≈
1.7,
e
≈ 0.6 and
roots,
critical
points,
and
points
of
inflection.
(Hint:
√ −3/2
3e
≈ 0.4.)
2
2
y
2
1
x
-5
-4
-3
-2
-1
1
-1
-2
2
3
4
5
MATH 116 Final Exam, Spring 2012
Page 8 of 10
Name:
7. Integration
Evaluate the following integrals:
Z
[2]
(a)
cos(ax) dx,
Z
[3]
(b)
1
a 6= 0
√
2x 1 + x2 dx
0
8. Area between the curves
[4]
Determine the area of the region bound by the functions f (x) =
(Hint: f (x) lies above g(x) in the relevant interval.)
x2
and g(x) = x2 − 2x.
2
MATH 116 Final Exam, Spring 2012
Page 9 of 10
Name:
9. Optimization
[5]
After a long illustrious career teaching mathematics, Matthew decides to retire to a
peaceful life of recreational farming. He wants to fence in a rectangular plot of land
adjacent to a road (leaving the side facing the road unfenced). If he decides to plant 800
m2 worth of crops, what are the dimensions of the field which minimizes the amount of
fencing he needs to purchase?
Farm
x
Area = 800 m2
x
y
10. Related Rates
[5]
Suppose you are observing a cycling race around a circular track of radius 1 km. Suppose
the leader is travelling in the counterclockwise direction with constant velocity 30 km/h.
If you are watching the race from the western-most point on the track (pictured below),
how quickly is the leader moving away from you when he is at the northern-most point
on the track? (Hint: A constant counterclockwise velocity of 30 km/h around a circle
dθ
= −30 radians/h.)
of radius 1 km corresponds to
dt
y
(x2,y2)=(cos(θ),sin(θ))
-x
(y
)12 +
(x 2
2=
D
(x1,y1)=(-1,0)
)2
-2 y 1
dθ
=-30
dt
r=1
θ= π
2
x
MATH 116 Final Exam, Spring 2012
Page 10 of 10
Name:
THIS PAGE IS FOR ROUGH WORK