M253M2

MATH 253 L01 - MIDTERM Winter 2012

Faculty of ScienceDepartment of Mathematics & Statistics

MATH 253 L01MIDTERM 2 WINTER 2013

March 15, 2012

COURSE INFORMATION AND STUDENT IDENTIFICATION

VERSION SECTION LAB I.D. LAST FIRSTNUMBER NUMBER NUMBER NUMBER NAME NAME

11

CIRCLE THE LAB SECTION WHERE YOU WOULD LIKE TO PICK UP YOUREXAM

If you do not circle a lab section, your exam WILL NOT BE RETURNED TO YOU

Lab 1 Lab 2 Lab 3 Lab 4 Lab 5Diane Fenton Diane Fenton Monir Rezai rad Monir Rezai rad Diane Fenton

M 12:00 M 13:00 M 14:00 T 12:00 T 13:00

Lab 6 Lab 7 Lab 8 Lab 9 Lab 10Juan Dong Rebecca Meissen Fataneh Esteki Diane Fenton Michael Cavers

T 14:00 W 10:00 W 15:00 R 8:00 R 12:00

EXAMINATION RULES

1. ENTER YOUR NAME, ID NUMBER, EXAM No., LAB No. and LECTURE No. on the EXAM BOOKLETand SCANTRON SHEET

2. No Calculators, electronic equipment, or other paper material than this examination and scantron sheet allowed.

3. Use the back of the previous page for rough drafts or calculations.

4. All enquiries and requests must be addressed to supervisors only.

5. Candidates are strictly cautioned against:

(a) speaking to other candidates or communicating with them under any circumstances whatsoever;

(b) bringing into the examination room any textbook, notebook or memoranda not authorized by the examiner;

(c) making use of calculators, cell phones or other portable computing machines not authorized by the instructor;

(d) leaving answer papers exposed to view;

(e) attempting to read other students examination papers.

The penalty for violation of these rules is suspension or expulsion or such other penalty as may be determined.

6. A candidate must report to a supervisor before leaving the examination room, of if they become ill or otherwise unable to completethe exam.

7. Answer books must be handed to the supervisor-in-charge promptly when the signal is given. Failure to comply with this regulationwill be cause for rejection of an answer paper.

Page 1


Part I. Follow the directions on the scantron sheet provided to indicate the correct answers to eachquestion. Each question is worth 5 points. Make sure to include your name, student number, andversion number on the scantron sheet.

1. Which of these is the value of the integral

31

1

(x 2)2dx ?A) 2

B) 0

C) 2

D *The integral diverges.

E) None of these.

2. Consider the integral 1

1

xex dx.

Which of these is a true statement regarding this integral?

A) The integral

1

1

xex dx diverges by comparison to

1

1

xdx.

B) The integral

1

1

xex dx diverges by comparison to

1

ex dx.

C) *The integral

1

1

xex dx converges by comparison to

1

ex dx.

D) The integral

1

1

xex dx converges by comparison to

1

1

xdx.

E) None of these.

Page 2


-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

3

Figure 1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

3

Figure 2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

3

Figure 3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

3

Figure 4

3. Which of the graphs above is the region in the xy plane bounded by 1 y 3 x2 and x 1?

A) Figure 1 B) Figure 2 C) *Figure 3 D) Figure 4 E) None of these.

4. Calculate the area between the two curves y = x3 + 6x and y = x3 + 6x2 from x = 0 to x = 2.

A) 36

B) 8

C) 4

D) *6

E) 4

Page 3


5. A solid object is formed by rotating the region bounded by

1 y x2

1 x 2about the line x = 1. In finding the volume of this solid using cylindrical shells, what are thedimensions (radius r, height h and width or thickness) of a typical cylindrical shell?

A) *r = x+ 1, h = x2 1, with width dx;

B) r = x, h = 1 x2, with width dx;

C) r = 1 +y, h = y, with width dy;

D) r = y, h =y + 1, with width dy;

E) None of these.

6. A solid object is formed by rotating the region bounded by

1 y x

1 x 2about the line y = 3. In finding the volume of this solid using the slicing method, which of the followingbest describes a typical slice of the object?

A) It is a disk with radius r = 3x and thickness dx.B) It is a washer with inner radius r = 3 y2, outer radius R = 2 and thickness dy.C) *It is a washer with inner radius r = 3x, outer radius R = 2, and thickness dx.D) It is a disk with radius r = 3 y2 and thickness dy.E) None of these.

Page 4


7. A surface is formed by rotating the segment of the curve x =

1 y2 between (1, 0) and (22,22

) aboutthe line x = 1. In finding the surface area of this object, which of the following best describes a typicalslice of the object?

A) It is a cylinder with radius r = |1 x| and height dx.

B) It is a cylinder with radius r = |1 y| and height ds =

1 +

(y1y2

)dy.

C) It is a cylinder with radius r = |11 y2| and height dy.D) It is a cylinder with radius r = |11 y2| and height ds = 1 + ( y

1y2

)dy.

E) *None of these.

8. A surface is formed by rotating the curve y = ex

0 x 1

about the y-axis. Which of these integrals computes the surface area of this object?

A) *2pi

10

x

1 + e2x dx;

B) 2pi

e1

y

1 +

(1

x

)2dy;

C) 2pi

10

ex

1 + e2x dx;

D) 2pi

e1

ln y dy;

E) None of these.

Page 5


9. Compute the arc length of the curve y = 23x3/2 on the interval 0 x 3.

A) 6

B) tan1 3 tan1 0C) *

14

3

D)21

2E) None of these.

10. Which of the following is an integral that determines the perimeter of the region enclosed by the curvesy = x2 1 and y = 1 x2?

A) 2pi

11

2

1 + 4x2 dx.

B) *2

11

1 + 4x2 dx.

C)

11

1 + 4x2 dx+

11

1 4x2 dx.

D)

11x

1 + 4x2 dx+

21x

1 4x2 dx.E) None of these.

Page 6


11. Which of the following is an integral that determines the volume of the solid generated by taking theregion in the plane enclosed by y = x3 and y =

x between x = 0 and x = 1 about the x-axis?

A)

10

2pix(x x3) dx

B) *

10

pi(x x6) dx

C)

10

pi(x x3)2 dx

D)

10

pi( 3y y2)2 dy

E)

10

pi(y4 3y2) dy

12. What does the integral

31

2pixe2x dx compute?

A) It computes the volume of the solid obtained by rotating the area bounded by 0 y

12e2x and

1 x 3 about the x-axis.B) *It computes the volume of the solid obtained by rotating the area bounded by 0 y e2x and1 x 3 about the y-axis.C) It computes the surface area of the surface obtained by rotating the curve y = xe2x from x = 1 tox = 3 about the y-axis.

D) It computes the arc length of the curve y =xe2x from x = 1 to x = 3.

E) None of these.

Page 7


13. Newtons Law of Cooling and Heating describes the tempertature, T of an object after s seconds whenthe object is placed in a room whose ambient temperature is a constant, A. Newtons Law can bedescribed by the differential equation

dT

ds= k(A T ),

where k > 0 is a proportionality constant. Which of the following statements is a consequence of thedifferential equation?

A) * If T > A, then the temperature of the object will DECREASE.

B) If T > A, then the temperature of the object will INCREASE.

C) If T > A, then the ambient temperature A will DECREASE.

D) If T > A, then the ambient temperature A will INCREASE.

E) None of these.

14. Which of these is a solution to the differential equation 4y = 25y?A) y(t) = sin(t)

B) y(t) = sin(2

5t)

C) *y(t) = cos(5

2t)

D) y(t) = cos(t)

E) None of these.

Page 8


Part II Answer Questions 15 - 16 in the space provided. Be sure to show all of your work, aspartially correct answers may be worth partial credit. Please put your answer in the answer box.Each question is worth 10 points.

15. Determine whether the integral0xex

2dx is convergent or divergent. Evaluate the integral if

it is convergent.

ANSWER:

Page 9


16. The region below the curve y = 1 x2 and above the x-axis is rotated about the line y = 1.Find the volume of the resulting solid. Please make sure your work includes a sketch of the regionand axis of revolution.

ANSWER:

Page 10


Formulas

cos2 x = 12

(1 + cos 2x)

sin2 x = 12

(1 cos 2x)

cos 2x = cos2 x sin2 x

sin 2x = 2 cos x sinx

cos ( + ) = cos cos sin sin

sin ( + ) = sin cos + sin cos

ddx

sinh1 x =1

x2 + 1

ddx

cosh1 x =1

x2 1

ddx

tanh1 x =1

1 x2

1

1 + x2dx = tan1 x+ C

1 + u2 du =u

1 + u2

2+

ln (u+

1 + u2)

2+ C

secu du = ln |secu+ tanu|+ C

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