Page #1 of 6
Name: _____________________________
Student Number:____________________
MATH 1A03, 1ZA3
Spring Session, 2018
Test #2, Version 1
C. McLean
TERM TEST
EVENING CLASSES
DURATION OF TEST: 90 min (1.5 hrs.)
MCMASTER UNIVERSITY TERM TEST
Wednesday, May 30, 2018
THIS TEST INCLUDES 6 PAGES AND 18 QUESTIONS. YOU ARE
RESPONSIBLE FOR ENSURING YOUR COPY OF THE TEST IS COMPLETE.
BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.
Instructions:
1. NO calculator is allowed to be used on this test.
2. Make sure your name and student number at the top of each page.
3. For all questions, mark the answer in pencil on the OMR answer sheets according to the
OMR instructions on page 2. Only solutions on the scan card will be graded.
4. Each correct answer is worth one mark.
5. A blank answer is an automatic zero for any question, even if the correct solution is circled
on the question itself.
6. Incorrect or multiple answers are also worth zero marks. No negative marks or part marks will
be assigned.
7. Scrap paper for rough work has been provided. All rough work, and this question sheet must
be handed in with the test, but any solutions written either here or on the rough paper will
NOT be graded.
8. Good Luck!
_________________________________________________
PLEASE READ THE OMR INSTRUCTIONS ON PAGE #2
Continued on page #2
Page #2 of 6
Name: _____________________________
Student Number:____________________
OMR EXAMINATION INSTRUCTIONS
NOTE: IT IS YOUR RESPONSIBILITY TO ENSURE THAT THE ANSWER SHEET IS
PROPERLY COMPLETED: YOUR EXAMINATION RESULT DEPENDS ON PROPER ATTENTION TO THESE INSTRUCTIONS.
The scanner which reads the sheets senses the bubble-shaded areas by their non-reflection of light. A
heavy mark must be made, completely filling the circular bubble, with an HB pencil. Marks made with a
pen or a felt–tip marker will NOT be sensed. Erasures must be thorough or the scanner may still sense a
mark. Do NOT use correction fluid on the sheets. Do NOT put any unnecessary marks or writing on the
sheet.
1. On side 1 (red side) of the form, in the top box, in pencil, print your student number (NOTE: 9
digits), name, course name, section number, instructor name and date in the spaces provided. Then
you MUST sign in the space marked SIGNATURE.
2. In the second box, with a pencil, mark your student number, exam version number and course section number in the space provided and fill in the corresponding bubble numbers underneath.
3. To indicate your answers, mark only ONE choice from the alternatives (1, 2, 3, 4, 5, or A, B, C,
D, E) provided for each question. The question number is to the left of the bubbles. Make sure that
the number of the question on the scan sheet is the same as the question number on the test paper.
4. Pay particular attention to the Marking Directions on the form.
5. Begin answering the questions using the first set of bubbles, marked “1”.
SAMPLE OMR CARD ONLY: DO NOT USE
Continued on page #3
Page #3 of 6
Name: _____________________________
Student Number:____________________
1. A function, f (x), which is continuous for all real values is found to have a positive first derivative on
the intervals (1,3) and (5, 10) and negative on (−∞, 1), (3,5) and (10, ∞). Which of the following must
always be a true statement?
a) x = 10 is the location of a local maximum of f (x).
b) x = 3 is a critical number but is the location of neither
a local maximum nor minimum of of f (x).
c) f (x) has negative concavity at x = 5.
d) f (x) has an inflection point at x = 3.
e) x = 5 is the location of a local maximum of f (x).
__________________________
2. Evaluate the limit lim ln( x) .
x →∞ x 2
a) 0
c) ∞
b) –1
d) e
e) Cannot be evaluated
___________________________
40
3. Evaluate ∑ (6i 2 + 1)i
i =1
a)
1
(40)(41) ( 3(40)(41) + 1)
2
d)
b)
(40)41
( (40)(41)(81) + 40 )
2
c) (40)(41)(81) +
(40)41
2
(40)41
1
( (40)(41)(81) − 120 ) e) (39)(40) ( 3(39)(40) + 1) − 1
2
2
___________________________
4. Find the point on the curve y = x2 closest to the point (16, 1/2).
a) (2,4) b) (4,16) c) (–1,1) d) (5,4) e) (1,1)
_____________________________
5. A confused snail travels along a straight ruler at velocity given by: v(=
t ) 2t 3 − t 2 where t is time
measured in seconds and v is measured in cm/s. If the snail passes the 2cm mark after 1 second of motion, what was its initial position, in cm, at t = 0?
a) 1/4
b) 7/6 c) 11/6
d) 3/2
e) 1/3
___________________________
6. Given f '( x) =+
( x 1)15 ( x − 3)7 ( x − 5)8 which ONE of the following answers represents the ONLY intervals where f (x) is increasing?
a) (–1,3) (3,5)
b) (–1,3) only
c) (–1,3) (5,∞)
d) (–∞,–1) (3,5) (5,∞)
e) (–∞,–1) (3,5)
___________________________
Continued on page #4
Page #4 of 6
Name: _____________________________
Student Number:____________________
7. Which of the following are true statements about g ( x) =
e−2 x + 2
3 − e− x
I) It has a horizontal asymptote at y = 2/3
II) It has a horizontal asymptote at y = 0
III) It has a horizontal asymptote at y = –1
a) I only
b) II only
c) I and III
d) I and II
e) II and III
_____________________________
8. At what value of x does g(x) = x2 – 2x – 3, satisfy the conclusion of the Mean Value Theorem for the
interval [0,2]?
a) 1/4
b) 1
c) 1/5
d) 3/2
e) No such points exist
___________________________
9. Which of the following is the graph of e( −1/ x ) ?
2
a)
b)
d)
c)
e)
______________________________
10. Given the function: f ( x) =
7 − sin 2 ( x) + cos3 ( x) which of the following statements is true?
a) The function is odd (only).
b) The function is even (only). c) The function is neither even nor odd.
d) The function is both even and odd.
e) The function is constant.
_____________________________
Continued on page #5
Page #5 of 6
Name: _____________________________
Student Number:____________________
11. Evaluate the limit lim (1 + 2 x )1/ x .
x →0
a) 1/2
b) 2
c) ∞
d) e
e) e2
___________________________
12. If we know h”(x) = x2(x – 3)3(x – 5), how many inflection points does the graph of y = h(x) have?
a) 1
b) 2
c) 0
d) 3
e) 6
___________________________
13. Find the general antiderivative of: f ( x) =
a)
3
2 x
c)
3
1
1
+ + 7 sec2 ( x) .
3
x
x
+ ln | 3 x | +7 csc2 ( x) + C
b) 2 x + ln | 3 x | +7 tan −1 ( x) + C
1
1
1
+ ln x + 7 tan( x) + C d) 2 x + ln | x | +7 tan( x) + C e) 2 x + x ln | x | +7 x tan( x) + C
3
3
2 x3 3
___________________________
3
14. Given the graph to the right is the function f (x), which of the
following represents the graph of an antiderivative of this
function?
a)
b)
c)
c)
d)
___________________________
Continued on page #6
Page #6 of 6
Name: _______________________________
Student Number:____________________
100
100
50
50
i =1
i = 51
i =1
i =1
15. Given ∑ ai = 5 , ∑ ai = 3 , ∑ bi = −2 , evaluate ∑ ( ai − 2bi )
a) Insufficient Information
b) 5
c) 9
d) 4 + a50
______________________________
e) 6
16. Evaluate the limit lim ( sin( x) )ln( x ) .
x → 0+
a) 0
b) 1
c) ∞
d) e
e) Cannot be evaluated
___________________________
17. Given h(x) is differentiable for all x, and h(x) has three critical points, what is the minimum number
of possible inflection points?
a) 0
b) 1
c) 2
d) 3
e) 4
___________________________
18. A student creates an expression in Maple they call f, which represents a graph with a root on the
interval [0,3]. Which of the following statements will return the decimal value of that root?
a) evalf (f, 0<x<3); b) fsolve(f=0, x=0..3); c) rootof(f,[0,3]); d) fsolve(f, 0<x<3); e) evalf(f,x=0..3);
______________________________
THE END
Math 1A03/1ZA3 Test #2 Formula Sheet
d
1
arctan( x) =
dx
1 + x2
cos 2 (q ) =
1
(1 + cos(2q ))
2
d
arcsin( x) =
dx
sin 2 (q ) =
1
(1 - cos(2q ))
2
cos 2 (q ) + sin 2 (q ) = 1
n
å1= n
i =1
n
åi=
i =1
n(n + 1)
2
1
1 - x2
sin(q ) cos(q ) =
1
sin(2q )
2
1 + tan 2 (q ) = sec2 (q )
n
å i2 =
i =1
n(n + 1)(2n + 1)
6
n
æ n(n + 1) ö
å i3 = çè 2 ÷ø
i =1
2