McMaster University
Department of Mathematics and Statistics
Winter 2015
Test 2
VERSION 1
Calculus for Science I / Engineering Mathematics I
MATH 1A03 / 1ZA3
Thursday March 19, 2015
Time: 75 minutes
Examiner: Dr. Ihsan Topaloglu
Student name (last, first)
Student number (Mac ID)
Signature
INSTRUCTIONS
1. Please fill out your answers clearly in the computer card. The exam booklet will not be graded.
2. Each problem is worth 1 point.
3. This is a closed book exam.
4. Only the McMaster standard calculator Casio fx-991 is allowed. Other electronic devices such as smart phones,
tablets, computers are not permitted.
5. Your name and signature must be written in pen on both parts of the test paper and on the computer card.
6. The rest of the computer card must be filled out in pencil.
This exam comprises the cover page and seven pages of questions and two blank pages at the end.
MATH 1A03/1ZA3 – Winter 2015
Version 1
1.) Find f (x) if f 00 (x) = 12x2 + 6x
4, f (0) = 4, and f (1) = 1.
(a) 4x3 + 3x2
4
(b) x + x
3
Page 1
4x
2x2 + 4
(c) x4 + x3 + 2x2 + 4
(d) x4 + x3
4
(e) x + x
2x2
3
3x + 4
2
2x + 3x + 4
2.) Find the point on the curve y =
p
x that is closest to the point (3, 0).
(a) (0, 0)
p
5/2)
p
(c) (3/2, 3/2)
(b) (5/2,
(d) (4, 2)
p
(e) (2, 2)
3.) Find the derivative of the function g(x) =
(a) g 0 (x) =
(b)
(c)
(d)
(e)
p
x 1
p
g 0 (x) = x2 1
p
g 0 (x) = 2x x2 1 2
p
g 0 (x) = x2 1 2
p
g 0 (x) = 2x x2 1
Z
x2
5
p
u
1 du.
MATH 1A03/1ZA3 – Winter 2015
Version 1
Page 2
4.) Which of the following expressions determines the area under the parabola y = x2 between the
lines x = 2 and x = 2?
◆2
n ✓
4 X 4i
(a) lim
n!1 n
n
i=1
◆2
n ✓
4i
4X
2+
(b) lim
n!1 n
n
i=1
✓
◆
n
2
2 X 4i
(c) lim
n!1 n
n
i=1
◆2
n ✓
4X
i
2+
(d) lim
n!1 n
n
i=1
✓
◆2
n
1X
4i
(e) lim
1+
n!1 n
n
i=1
5.) Find the area bounded by the graph of the parabola y = x2 + 2, the line y = x + 4, and the lines
x = 2 and x = 3.
y
-4
-3
(a) 5/6
(b) 15/6
(c) 27/6
(d) 32/6
(e) 49/6
-2
-1
0
1
2
3
4
x
MATH 1A03/1ZA3 – Winter 2015
6.) If f (1) = 12 and
Z
4
Version 1
f 0 (x) dx = 17, what is the value of f (4)?
1
(a) 0
(b) 29
(c) 12
(d) 5
(e) 4
7.) Evaluate the integral
Z
1
3
x2 ex dx.
0
(a) e
(b) e
1
(c) e/3
1/3
(d) e/2
1/2
(e) e/4
1/4
8.) Find the general indefinite integral of
(a)
(b)
(c)
(d)
(e)
3p
3
x2 +
2
3p
3
x5 +
5
3p
3
x5 +
2
2p
3
x5 +
5
2p
3
x2 +
5
2p 3
x +C
3
2p 5
x +C
5
p
2
x5 + C
3
3p 5
x +C
5
3p 3
x +C
5
Z p
x3 +
p
3
x2 dx.
Page 3
MATH 1A03/1ZA3 – Winter 2015
Version 1
Page 4
p
9.) Consider the region enclosed by the functions y = x and y = x/2 (see the graph below). Which
of the integrals below describe the volume of the solid obtained by rotating this region about the
y-axis?
y
4
3
2
1
-4
(a) ⇡
(b) ⇡
(c) ⇡
(d) ⇡
(e) ⇡
Z
Z
Z
Z
Z
-3
-2
4
x
0
4
0
2
⇣
x
2y
-1
0
x2
dx
4
x ⌘2
2
y2
1
2
3
4
x
dx
2
dy
0
2
4y 2
y 4 dy
0
2
y2
2y
2
dy
0
p
10.) Consider the region enclosed by the functions y = x and y = x/2 (see the graph in Question
10). Which of the integrals below describe the volume of the solid obtained by rotating this region
about the line y = 2?
Z 4
x2
(a) ⇡
x
dx
4
0
Z 4⇣
p 2
x ⌘2
(b) ⇡
2
(2
x) dx
2
0
Z 4⇣
p
x ⌘2
(c) ⇡
2+
(2 + x)2 dx
2
0
Z 4
⇣
p
x ⌘2
(d) ⇡
(2 + x)2
2+
dx
2
0
Z 4
p
(e) ⇡
2 + ( x x/2)2 dx
0
MATH 1A03/1ZA3 – Winter 2015
Version 1
11.) Find the area of the largest rectangle that can be inscribed in a semicircle of radius
(Hint: Any point (x, y) that lies on the circle of radius r satisfies the equation x2 + y 2 = r 2 .)
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
Page 5
p
2.
MATH 1A03/1ZA3 – Winter 2015
12.) Evaluate the integral
Z
1
p
Version 1
Page 6
2x + 1 dx.
0
(a) 0
p
(b) 2 1/ 2
p
(c) 3
p
(d) 2 1/2
p
(e) 3 1/3
13.) If
Z
10
f (x) dx = 12 and
0
(a) 8
(b) 12
(c) 10
(d) 16
(e) 0
Z
8
f (x) dx =
0
4, find
Z
10
f (x) dx.
8
MATH 1A03/1ZA3 – Winter 2015
14.) Find
(a)
(b)
Z
ex sin x dx.
1 x
2 e (sin x
1 x
2 e (cos x
x
cos x) + C
sin x) + C
(c) e (sin x
cos x) + C
x
sin x) + C
(d) e (cos x
x
(e) e sin x + C
15.) Evaluate
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4
Z
1
x ex dx.
0
Version 1
Page 7
MATH 1A03/1ZA3 – Winter 2015
Version 1
This page is left blank intentionally.
Page 8
MATH 1A03/1ZA3 – Winter 2015
Version 1
This page is left blank intentionally.
Page 9