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Sample Tests and Exams
This part contains actual tests and mid-year exams for Arts Sci 1D6,
given in Fall 2012 and Fall 2013.
Exams and tests might contain multiple choice, true/false and/or
standard question-and-answer problems. You will find examples of
all three types among the sample questions.
Information on how tests and exams are marked is given in the
Solutions to Sample Tests and Exams section in this booklet.
109
110
Arts & Science 1D6 Test #1
Day Class
Test #1
Duration of test: 60 minutes
McMaster University
30 October, 2012
Dr. Matt Valeriote
Last Name:
Initials:
Student No.:
Your TA’s Name:
This test has 8 pages and 8 questions and is printed
on BOTH sides of the paper. Pages 7 and 8 contain no questions
and can be used for scratch work.
You are responsible for ensuring that your copy of the paper is complete.
Bring any discrepancies to the attention of the invigilator.
Attempt all questions and write your answers in the space provided.
Marks are indicated next to each question; the total number of marks is 40.
Any Casio fx991 calculator is allowed. Other aids are not permitted.
Use pen to write your test. If you use a pencil, your test will not
be accepted for regrading (if needed).
Good Luck!
Score
Question
Points
Score
1
3
2
3
3
6
4
6
5
6
6
6
7
4
8
6
Total
40
continued . . .
2
Multiple Choice Questions.
Indicate your answers to questions 1 and 2 by circling only ONE of the letters
⎧
⎪
⎪
⎪
⎨
−1
3x
1. [3 ] If f (x) = ⎪
2
⎪
⎪
⎩
x+2
if
if
if
if
x ≤ −1
−1 < x < 1
x=1
x>1
then f is discontinuous at
(A)
-1 only
(B)
1 only
(C)
2. [3 ] If f (3) = 3, and f (3) = −5 then the value of
-1 and 1 only
f (x)
x
(D)
no point of
the domain
at x = 3 is
(A)
-6
(B)
-3
(C)
-2
(D)
-5/3
(E)
1
(F)
2
(G)
3
(H)
6
continued . . .
A&S 1D6 Test # 1, 30 October, 2012
Student #
Initials
Page 3
Questions 3–8: you must show work to receive full credit.
3. [6 ] Let f (x) = ln(2 + ln x).
(a) Find the domain of f .
(b) Find f −1 .
(c) Find the domain of f −1 .
(d) Does f have any horizontal asymptotes?
continued . . .
4
4. [6 ] Evaluate the following limits, if they exist. Justify your answers.
(a) lim−
x→0
1
1
−
|x| x
(b) lim 2 arctan(2x)
x→∞
5. [6 ] Differentiate the following functions. You do not need to simplify your answers.
(a) h(x) = −2x3 +
(b) f (x) =
√
3
+ x − 2.
2
x
tan(3x)
ex
6
.
continued . . .
A&S 1D6 Test # 1, 30 October, 2012
6.
Student #
Initials
Page 5
For each of the following statements, determine whether it is true or false. To receive credit,
justify your answers. Answering true or false without a correct justification carries no credit.
(a) [3 ] If f is an odd function that has an inverse, then f −1 is also an odd function.
(b) [3 ] If f is a function with domain (−∞, ∞) and with f (−1) = 1 and f (1) = 3, then there
is some number c with f (c) = 2.
7. [4 ] Find the equation of the tangent line to the curve y = ln(x) − sin(πx) at x = 1.
continued . . .
6
8. [6 ] The following is the graph of the derivative f (x) of some continuous function f (x).
(a) On what interval(s) is f (x) decreasing? No justification required.
(b) On what interval(s) is f (x) concave down? No justification required.
continued . . .
Name
Student Number
Your TA’s Name:
Arts & Science 1D6
DAY CLASS
DECEMBER EXAM
DURATION OF EXAM: 2 Hours
MCMASTER UNIVERSITY
DR. MATT VALERIOTE
11 December, 2012
THIS EXAMINATION PAPER INCLUDES 12 PAGES AND 10 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING
ANY DISCREPANCIES TO THE ATTENTION OF YOUR INVIGILATOR.
Attempt all questions.
The total number of available points is 50.
Marks are indicated next to each question.
Use of a Casio fx991 calculator only is allowed.
Write your answers in the space provided.
You must show your work to get full credit.
Use the last two pages for rough work.
Good Luck.
Score
Question
Points
Score
Question
Points
Score
1–3
6
4
6
5
8
6
9
8
6
9
4
10
5
Total
50
7
6
Continued on Page 2 . . .
A&S 1D6 December Exam
Student #
Initials
Page
2
Multiple Choice Questions.
Indicate your answers to questions 1, 2, and 3 by circling only ONE of the letters. You do not need
to provide justifications for your answers to these three questions. Each of these questions is worth
2 marks. No partial credit will be given for these three questions.
1. [2 ] The absolute maximum value of the function f (x) =
2x3
+ 3x2 on the interval [−4, 1] is:
3
(A)
0
(B)
7/3
(C)
16/3
(D)
17/3
(E)
20/3
(F)
9
(G)
27
(H)
does not exist
Continued on Page 3. . .
A&S 1D6 December Exam
Student #
Initials
Page
3
2. [2 ] Which of the following three functions is/are odd?
(I) sin x + x cos x
(II) sin(2x)
(III) sin x sin(2x)
(A)
none
(B)
I only
(C)
II only
(D)
III only
(E)
I and II
(F)
I and III
(G)
II and III
(H)
all three
3. [2 ] Let g(x) = xf (x2 ), where f is a differentiable function with f (4) = 1 and f (4) = −1. Then
g (2) is equal to:
(A)
−7
(B)
−4
(C)
−3
(D)
0
(E)
1
(F)
4
(G)
7
(H)
9
Continued on Page 4. . .
A&S 1D6 December Exam
Student #
Initials
Page
Questions 4–10: you must show work to receive full credit.
4. [6 ] Evaluate the following limits, if they exist. Justify your answers.
(a) x→π
lim
2 tan x
x−π
sin2 (x)
x→∞
x
(b) lim
Continued on Page 5. . .
4
A&S 1D6 December Exam
Student #
Initials
Page
5
5. [8 ] (a) State the Intermediate Value Theorem.
(b) Prove that the equation sin(x) + x − 1 = 0 has a solution.
(c) Use Newton’s Method with initial approximation x1 = 0 to find x3 , the third approximation to the solution of the equation sin(x) + x − 1 = 0.
Continued on Page 6. . .
A&S 1D6 December Exam
Student #
6. Let f (x) = x(5/3) − 5x(2/3) ; then f (x) =
Initials
Page
6
5(x − 2)
10(x + 1)
√
and f (x) =
.
3
3 x
9x(4/3)
(a) [6 ] For the function f , find the domain, x−, y− intercepts, any symmetries, asymptotes,
critical numbers and the intervals of increase and decrease, all intervals where f is concave
up and concave down, and inflection points. Place your answers in the following table.
Use the next page for rough work. Note that f (x) can also be written as x(2/3) (x − 5).
ANSWERS:
domain of f :
x−intercept(s):
y−intercept(s):
symmetries:
horizontal asymptote(s):
vertical asymptote(s):
critical numbers (if any):
f is increasing on:
f is decreasing on:
inflection points (if any):
f is concave up on:
f is concave down on:
(b) [3 ] Sketch the graph of y = f (x).
y
8
4
−6
−4
−2
2
4
6
−4
−8
−12
Continued on Page 7 . . .
x
A&S 1D6 December Exam
Student #
Initials
Page
Space for rough work for question #6.
Continued on Page 8 . . .
7
A&S 1D6 December Exam
7. [6 ] Find
Student #
Initials
Page
8
dy
for each of the following.
dx
(a) xy = sin(y).
(b) y = arcsin(2x ).
8. [6 ] (a) Find the most general anti-derivative of the function g(x) = sec2 (x) + e3x + 2
(b) A particle moves in a straight line and has acceleration a(t) = π sin(πt) − t (m/sec2 ). If
the initial velocity of the particle (at time t = 0 seconds) is 2 m/sec, what is the particle’s
velocity at time t = 4?
Continued on Page 9 . . .
A&S 1D6 December Exam
Student #
Initials
Page
9
9. [4 ] For each part, indicate your answer by circling only ONE of TRUE or FALSE. To receive
credit for your solutions, you must justify your answers. Each part is worth 2 marks; partial
credit may be assigned.
(a)
3
−3
(x2 + x7 cos(x) + 5)dx =
TRUE
3
−3
(x2 + 5)dx.
FALSE
(b) If f (x) is a function such that the function |f (x)| is continuous, then f (x) is a continuous
function.
TRUE
FALSE
Continued on Page 10 . . .
A&S 1D6 December Exam
Student #
Initials
Page
2
1
10. [5 ] (a) Estimate the definite integral
dx by evaluating the Riemann sum for
0 1 + x2
1
, with n = 4 and by taking the sample points to be the right endpoints.
f (x) =
1 + x2
(b) Find the exact value of the definite integral
2
0
1
dx.
1 + x2
Continued on Page 11 . . .
10
Arts & Science 1D6 Test #1
Day Class
Test #1
Duration of test: 75 minutes
McMaster University
29 October, 2013
Dr. Matt Valeriote
Last Name:
First Name:
Student No.:
Your TA’s Name:
This test has 8 pages and 8 questions and is printed
on BOTH sides of the paper. Page 8 contains no questions
and can be used for rough work.
You are responsible for ensuring that your copy of the paper is complete.
Bring any discrepancies to the attention of the invigilator.
Attempt all questions and write your answers in the space provided.
Marks are indicated next to each question; the total number of marks is 40.
Any Casio fx991 calculator is allowed. Other aids are not permitted.
Use pen to write your test. If you use a pencil, your test will not
be accepted for regrading (if needed).
Good Luck!
Score
Question
Points
Score
1
2
2
2
3
6
4
5
5
6
6
7
7
6
8
6
Total
40
continued . . .
2
Multiple Choice Questions.
Indicate your answers to questions 1 and 2 by circling only ONE of the letters. Partial credit will
not be given for incorrect answers.
1. [2 ] Let
f (x) =
ex
1
.
+1
Which of the following statements are true?
(I) The domain of f (x) is (−∞, ∞).
(II) f (x) is an odd function.
(III) f (x) has an inverse.
(A)
none
(B)
I only
(C)
II only
(D)
III only
(E)
I and II
(F)
I and III
(G)
II and III
(H)
all three
2. [2 ] Suppose that F (x) = f (g(x)) and g(3) = 5, g (3) = 3, f (3) = 1, f (5) = 4. Find the value
of F (3).
(A)
3
(B)
4
(C)
7
(D)
9
(E)
12
(F)
15
(G)
17
(H)
20
continued . . .
A&S 1D6 Test # 1, 29 October, 2013
Student #
Initials
Page 3
Questions 3–8: you must show work to receive full credit.
3. [6 ] Evaluate the following limits, if they exist. Justify your work.
√
x2 + 4x + 1 − x .
(a) lim
x→∞
(b)
lim
x→(π/2)+
x sec(x).
continued . . .
4
4. [5 ] Find the value(s) of x for which the function
⎧
⎪
⎪
⎪
⎨
f (x) = ⎪
⎪
⎪
⎩
x−1
if x = 1
x3 − x
1/2
if x = 1
is not continuous. Justify your answer.
5. [6 ] Let f (x) = f (x) =
1
.
x+1
(a) Use the definition of the derivative to compute f (x).
(b) Find the equation of the tangent line to the curve y =
1
at x = 1.
x+1
continued . . .
A&S 1D6 Test # 1, 29 October, 2013
Student #
Initials
Page 5
6. [7 ] Calculate the derivatives of the following functions:
(a) f (x) =
1 − sin(x)
. Simplify your answer.
1 + sin(x)
√
√
(b) g(x) = ( x + 2)(2 x − 4). Simplify your answer.
(c) h(x) = arctan(ln(x2 )).
continued . . .
6
7.
For each of the following statements, determine whether it is true or false. To receive credit,
justify your answers. For statements that are false, provide an example that demonstrates
this. Answering true or false without a correct justification carries no credit.
(a) [2 ] If lim f (x) = 4, then the function f is continuous at 2 and f (2) = 4.
x→2
(b) [2 ] For any two functions f and g, (f ◦ g) = (g ◦ f ).
(c) [2 ] If u(x) and v(x) are odd functions, then the function w(x) = u(x)v(x) is an even
function.
continued . . .
A&S 1D6 Test # 1, 29 October, 2013
Student #
Initials
Page 7
8 (a) [2 ] Carefully state the Intermediate Value Theorem (IVT).
8 (b) [2 ] Show that the equation sin x = x2 − 1 has a solution with x ≥ 0.
8 (c) [2 ] How many solutions does the equation sin x = x2 − 1 have? Explain (You may use a graph
of the functions involved).
continued . . .
Name
Student Number
Your TA’s Name:
Arts & Science 1D6
DAY CLASS
DECEMBER EXAM
DURATION OF EXAM: 2 Hours
MCMASTER UNIVERSITY
DR. MATT VALERIOTE
6 December, 2013
THIS EXAMINATION PAPER INCLUDES 12 PAGES AND 11 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING
ANY DISCREPANCIES TO THE ATTENTION OF YOUR INVIGILATOR.
Attempt all questions.
The total number of available points is 50.
Marks are indicated next to each question.
Use of a Casio fx991 calculator only is allowed.
Write your answers in the space provided.
You must show your work to get full credit.
Use the last two pages for rough work.
Good Luck.
Score
Question
Points
Score
Question
Points
Score
1–3
6
4
4
5
6
6
4
7
6
8
4
9
6
10
8
11
6
Total
50
Page 1 of 12
A&S 1D6 December Exam
Student #
Initials
Page
2
Multiple Choice Questions.
Indicate your answers to questions 1, 2, and 3 by circling only ONE of the letters. You do not need
to provide justifications for your answers to these three questions. Each of these questions is worth
2 marks. No partial credit will be given for these three questions.
1. [2 ] Let f (x) = e10 . Then f (x) =
(A)
e10
(B)
10e9
(C)
10
(D)
0
2. [2 ] For which value (if any) of the constant k is the following function continuous everywhere?
f (x) =
(A)
−1
(B)
0
⎧
⎪
⎨
kx + 1
⎪
⎩
2
if x < −1
.
kx − 1 if x ≥ −1
(C)
1
(D)
no value
Page 2 of 12
A&S 1D6 December Exam
Student #
Initials
Page
3
3. [2 ] The domain of the function f (x) = arcsin(2x + 1) is equal to:
(A)
[−1, 1]
(B)
[− π2 , π2 ]
(C)
[−1, 0]
(D)
(−∞, ∞)
Questions 4–11: you must show work to receive full credit.
4. [4 ] Consider the curve defined by the equation y 2 + x3 + tan(y) = 8.
(a) Find
dy
.
dx
(b) Find the equation of the tangent line to the curve at the point (2, 0).
Page 3 of 12
A&S 1D6 December Exam
Student #
Initials
Page
4
5. [6 ] Compute the derivatives of the following functions.
(a) f (x) = cos(x2 + 1) + xe−x .
(b) g(x) =
x2 √
0
e t dt, x > 0.
6. [4 ] Find the function f (x) that satisfies f (x) = 6 + 12x, f (−1) = −1, and f (1) = 4.
Page 4 of 12
A&S 1D6 December Exam
Student #
Initials
Page
5
7. [6 ] Solve the following limits.
(a)
lim +
x→−3
(b) lim
x→0
x2
.
9 − x2
x
.
arctan(7x)
Page 5 of 12
A&S 1D6 December Exam
Student #
Initials
Page
6
8. [4 ] For each part, indicate your answer by circling only ONE of TRUE or FALSE. To receive
credit for your solutions, you must justify your answers. Partial credit may be assigned.
(a) The function f (x) = 2x + cos(x) has an inverse.
TRUE
FALSE
(b) If the function f (x) is continuous on the interval [a, b] and has an absolute minimum at
c in (a, b), then f (c) exists and equals 0.
TRUE
FALSE
Page 6 of 12
A&S 1D6 December Exam
Student #
Initials
Page
7
9. [6 ] Consider the function f (x) = 2x3 + 6x2 − 1.
(a) Prove that there is some c in (0, 4) such that f (c) = 0.
(b) Find the absolute maximum and minimum values of f on the interval [0, 4].
(c) Use Newton’s Method to find x2 , the second approximation to a root of f (x), starting
with x1 = 1.
Page 7 of 12
A&S 1D6 December Exam
Student #
Initials
Page
8
10. Given that
f (x) =
x2
,
(x + 2)2
f (x) =
4x
,
(x + 2)3
f (x) =
8 − 8x
,
(x + 2)4
(a) [5 ] Find the domain, x− and y− intercepts, all asymptotes, all local extreme values and
intervals of increase and decrease, all intervals where f is concave up and concave down,
and inflection points. Place your answers in the following table. Use the next page for
rough work.
ANSWERS:
domain of f :
x−intercept(s):
y−intercept(s):
horizontal asymptote(s):
vertical asymptote(s):
local extreme values (if any):
f is increasing on:
f is decreasing on:
inflection points (if any):
f is concave up on:
f is concave down on:
(b) [3 ] Sketch the graph of y = f (x) on the following grid.
Page 8 of 12
A&S 1D6 December Exam
Student #
Initials
Page
9
— Space for rough work for question #10 —
Page 9 of 12
A&S 1D6 December Exam
11. [6 ] Let f (x) =
Student #
Initials
Page
10
1
, x > 0.
x
11
(a) Estimate the definite integral
f (x)dx by evaluating the Riemann sum for f (x) with
1
n = 5, taking the sample points to be the right endpoints.
(b) Find an antiderivative for f (x).
(c) Use the Fundamental
Theorem of Calculus and your answer from part (b) to find the
exact value of
11
1
f (x)dx.
Page 10 of 12