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Unformatted text preview: Name (Print): UW Student Id. Number: University of Waterloo
Final Examination ' Math 118
Calculus 2 for Engineering Instructor: See table below
Date: Wednesday, August 3, 2005
Term: 1055 Number of exam pages: 14
(including cover page) Section: See table below
Time: 12:30 — 3:00 p.m.
Duration of exam: 2.5 hours Exam type: Closed book Additional material allowed: No calculators or additional material allowed Circle your instructor’s name and section number Instructor R. André R. André J. West B. Richmond Instructions . Write your name and ID number on this
page. . Questions are to be answered in the space
provided. Give REASONS for your answers
and show all your work required to obtain
your answers. If the space is insufﬁcient,
use the back and indicate where your work
continues. ' . There are 14 pages in this examination.
Check that you have all the pages. Write
your name at the top of each page. . No calculators or additional aids allowed. Section 001
002
003
004 Question Mark 1 /10 Math 118 Final Exam, Spring 2005 Page 2 of 14 Name: [7] 1. General short answer questions. (a) What technique would you use to evaluate f xexdm? (b) In the my—plane, the curve of the parametric equations x(t) = 400st
y(t) = 2 sint can be described by which of the following: (Circle the correct answer) (i) a circle (ii) an ellipse (iii) a spiral (iv) a parabola. (c) Consider the recursive sequence deﬁned by cn+1 = V1 + c" with cl = 1. If the sequence is and monotone, then it converges. (d) Assuming the sequence in(c) converges, ﬁnd its limit. (e) For what values of 19 does 22:1 nip converge? (f) Is f3" sin mdw = 0 true or false? (g) Is the following true or false? 5
1
d 2— —1—15=—43l 1:
/0(.’E—1)2m (27 ) '0 + Math 118 Final Exam, Spring 2005 Page 3 of 14 Name: [10] 2. Evaluate the following integrals.
(a)
:1: 1v+1‘
/$2_4da: Math 118 Final Exam, Spring 2005 Page 4 of 14 Name: [4] 3. Sketch the cardioid 7" = 1 + cos 0, and write down the formula for its area specifying the limits of
integration. his not neccessary for you to simplify or evaluate the integral. [10] 4. Convergence of series. Determine if the given series converges: (a) °° 1
Z n(ln n)2 n=2 Math 118 Final Exam, Spring 2005 Page 5 of 14 Name: (b) Math 118 Final Exam, Spring 2005 Page 6 of 14 Name: [5] 5. Convergence of power series. Determine the radius of convergence and interval of convergence
of ..
00 n
——2 1
E (—3 ) —:c".
n=l Tl, (Be sure to test the endpoints of the interval.) Math 118 Final Exam, Spring 2005 Page 7 of 14 Name: [12] 6. Taylor and Maclaurin series applications. (a) Give the Maclaurin series for the functions 1 1
and " 1+t \/1+t3' Math 118 Final Exam, Spring 2005 Page 8 of 14 Name: (b) Find the Maclaurin series for (0) Assuming that we have determined the radius of convergence of the Maclaurin series in b)
to be equal to 1 write an inﬁnite series which can be used to approximate the value of 1/2 1
/ dt.
0 V 1 + t3 Math 118 Final Exam, Spring 2005‘ Page 9 of 14 Name: [12] 7. Taylor series and Taylor polynomial approximations. (a) Give the Binomial series formula including the interval of convergence. (b) Find the ﬁrst 5th order Taylor polynomial of the Maclaurin series generated by the function f(a:) 2 V1 +23. Math 118 Final Exam, Spring 2005 Page 10 of 14 Name: (c) Write out an expression for the 5th order Taylor polynomial evaluated at 0.01 for the Maclau—
rin series in part b). Do not compute its value. (d) If you use the 5th order Taylor polynomial evaluated at 0.01 to approximate the value of
\/ 1.01 give an upper bound for the error. Do not simplify your upper bound expression. (e) Given the function f (11:) = sina: use the Taylor Remainder Formula to compute an upper
bound for the error if we use the 5th order Taylor polynomial of the Maclaurin series generated
by sinx to approximate the value of sin 0.1. Do not simplify your upper bound expression. Math 118 Final Exam, Spring 2005 Page 11 of 14 Name: [8] 8. Differential equations. Solve the following differential equations for y in terms of 3:: (a)
dy ___= 2 2
ydx (11 +1) Math 118 Final Exam, Spring 2005 l Page 12 of 14 Name: [12] 9. Newtonian mechanics. A rock whose mass is 75 kg is dropped from a helicopter hovering
2000 m above the ground and falls toward the ground under the inﬂuence of gravity. Assume that
the force due to air resistance is proportional to the velocity of the rock, with the proportionality
constant 191 = 30 kg/sec. You may approximate the gravitational acceleration to be 10m/52. (a) Set up a differential equation which, when solved, will give us the velocity v(t) of the rock
at time t. Do not solve this differential equation. (b) Suppose the general solution to the differential equation in part a) is v(t) = 25 — %e(”2/5)‘ Where M is any nonzero number. Find the velocity at 1 second after the rock is dropped.
Do not simplify your answer. Math 118 Final Exam, Spring 2005 Page 13 of 14 Name: (c) Find an expression for the distance $(t) between the rock and the helicopter at time t. Your
expression for a;(t) should not contain any parameters. ' Math 118 Final Exam, Spring 2005 Page 14 of 14 Name: [Blank page] ...
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 Spring '08
 ZHOU
 Math, Calculus

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