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Unformatted text preview: 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 zezdm? (b) In the mgr—plane, the curve of the parametric equations a:(t) = 4cost
y(t) = 2sint 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 an“ m 1/1 + c,1 with c1 = 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 does w: i converge?
p n. 1 n? (f) 15 fog” sin mix = 0 true or false? (g) Is the following true or false? 5 1 d ——( —1)*15——4*1+1—§
0(w71)2$_ x 0m “4 Math 118 Final Exam, Spring 2005 Page 3 of 14 Name: [10] 2‘ Evaluate the following integrals.
(a)
x
f (1 +£2)3/2 dz Math 118 Final Exam, Spring 2005 Page 4 of 14 Name: [4] 3. Sketch the cardioid T = 1 + cos 6, and write down the fermula. for its area. specifying the limits of
integration. his not neccessary for you to simplify or Evaluate the integral. [10} 4. Convergence of series. Determine if the given series converges: (a) °° 1
Z n(1n 71)2 71:2 . Math 118 Final Exam, Spring 2005 Page 5 of 14 Name: (b) Math 118 Final Exam, Spring 2005 Page 6 of 14 Name: [5] 5. Convergence of power series. Determine the radius of convergence and interval of convergence
of
i (—2) ’1 1 En
n=1 3 n (Be sure to test the endpoints of the interval.) Math 118 Final Exam, Spring 2005 Page 7 of 14 Name: [12] 6. Taylor and Maclaurin series applications. (a) Give the Manlaurin series for the functions 1 1
and 1+: x/l+f3. Math 118 Final Exam, Spring 2005 Page 8 0f 14 Name: (b) Find the Macleurin series for I 1
cit.
fa \/1+t3 (0) Assuming that we have determined the radius of convergence ef the Meclaurin series in b)
to be equal to 1 write an inﬁnite series which can be used to approximate the value of 1/2 1
/ dt.
1] V1 + 153 Math 118 Final Exam, Spring 2005 Page 9 of 14 Name: [12] 7. Taylor series and Taylor polynomial approximations. (3,) Give the Binomial series formula including the interval of convergence. (b) Find the ﬁrst 5“ order Taylor polynomial of the Maciaurin series generated by the function f(&.“) = 1/1+$. Math 118 Final Exam, Spring 2005 Page 10 of 14 Name: (0) Write out an expression for the 5th order Taylor polynomial evaluated at 0.01 for the Maelan—
rin series in part in). Do not compute its value. (d) If you use the 5‘h order Taylor polynomial evaluated at 0.01 to approximate the value of
V1.01 give an upper bound for the error. Do not simplify your upper bound expression. {(3) Given the function f = sine use the Taylor Remainder Formula to compute an upper
bound for the error if we use the 5th order Taylor polynomial of the Maelaurin 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 foliowing differential equations for y in terms of m:
(a) d
3'
ya = (342 + 1)2
(b) d I
_y _ = 1/3
da", + ﬂy 6 a Math 118 Final Exam, Spring 2005 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 k; z 30 kg/sec. You may approximate the gravitational acceleration to be 10m/32. (a) Set up a differential equation which, when solved, will give us the velocity o(t) of the rock
at time it. Do not solve this diiferential equation. (b) Suppose the general solution to the differential equation in part a) is 110:) : 25 w glad—W5” 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 0f 14 Name: (c) Find an expression for the distance 55(6) between the rock and the helicopter at time 15. Your
expression for a:(t) should not contain any parameters. Math 118 Final Exam, Spring 2805 Page 14 of 14 Name: —'—~H——————n——u——..—_..__—___—_wnm_________ [Blank page] ...
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This note was uploaded on 04/09/2009 for the course MATH 118 taught by Professor Zhou during the Spring '08 term at Waterloo.
 Spring '08
 ZHOU

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