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Unformatted text preview: Math 201 —— Fall 2008—09
Calculus and Analytic Geometry III,—sections 21—23
Quiz 2, December 2 — Duration: 70 minutes GRADES (each problem is worth 12 points): YOUR NAME: YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 21 Section 22 Section 23
Recitation M 11 Recitation M 9 Recitation M 1
Ms. Itani Professor Makdisi Ms. Itani
INSTRUCTIONSi , p
1. Write yoiir NAME and AUB ID number, and circle your SECTION above. 2. 3. Solve the. problems inside the booklet. Explain your steps precisely and clearly to
ensure full credit. Partial solutions will receive partial credit. You may use the back of each page «for scratchwork OR for solutions. There are
three extra blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE
THE GRADER SHOULD CONTINUE READING. . Open book and notes. NO CALCULATORS ALLOWED. 'I‘urrr OFF and put away any cell phones.
GOOD LUCK! An overview of the exam problems. Each problem is worth 12 points.
Take a minute to look at all the questions, THEN
solve each problem on its corresponding page INSIDE the booklet. 1. Let f(:r,y,z) = e” — .732.
(2 pts) a) Find the gradient V f .
(5 pts) b) Let P(t) be a moving point with position 7"(t) and velocity 17(t). Assume we know that at time t = 0, we have 77(0) = (1,1,2) and 17(0) = (—1,3, 1). Find t D. (5 pts) 0) Let S = {(1r,y,z) e R3 ( f(:r,y,z) = —1} be the level set of f passing through the
point P0(2,0,1). Find the equation of the tangent plane to S at P0. 2. (5 pts) a) Sketch the curve r = sin2 0 in polar coordinates.
(2 pts) b) Write a parametrization of your curve in the form 77(9) = (22(6), 31(6)), for 0 S 0 S 27r.
(Yes, I want you to use 6 instead of t as the time variable.)
(5 pts) c) When 9 = 7r/4, ﬁnd the position vector F9=ﬂ4 and the velocity vector 17 [9: Use 17 to deduce the speed of the moving point at 6 = 7T—/4. 3. Let f(:r,y) = 1n(x2 + y). (4 pts) a) Sketch the domain of f and brieﬂy indicate why the domain is not closed. (4 pts) b) Find the directional derivative of f at the point P0(3, 1) in the direction of the vector
A = (3, 4). (Suggestion: do not simplify fractions; for example, keep 40/70 the way it is, and do
not reduce it to 4/7.) (4 pts) c) Approximately how much is f(3.03, 1.04)? This can be done either using (b) or by
a direct calculation. You may choose whichever method you prefer. 7r/4' 4. (3 pts) a) Use integration by parts twice to show that if n is a constant, then +0. I 2 . ——a:2 cos nx 2:2.sin nm 2 cos m: I
:r smnmdmz —— + —— + n n2 n3
(7 pts) b) Let f(:z:) be given by f(a:) :1 2’2, _Extend f periodically to ave period 27L Your job is to sketch the graph ~of f and to compute ONLY the coefﬁcien s bn ,
in the Fourier series f(m) = (20/2 +2211 an cos m: + 22:, bn sin nz. m. Ji‘
(2 pts) c) At what points as does the Fourier series of f not' converge to f What the
,‘ "D value of the Fourier series at those points? '1 5. We wish to ﬁnd the maximum and minimum of f(:c,y) = :r(y2 — 1) in the half disk D deﬁned
'1, 11. by D = {(m,y) E R2  m2 + y? 5 10,3; 2 0}. (Note that D is closed and bounded, so f attains a
maximum and minimum on D.) (3 pts) 3.) Show that f has exactly two critical points in the plane R2, but that only one of
these points belongs to D. (3 pts) b) Test the top part of the boundary of D (this means the semicircle m2 + y2 = 10
where y _>_ 0) for possible maxima and minima of f. Parametrize the boundary in the following
way: P(t) = (t, v 10 — 152). Remember to specify the range'of t for your parametrization. (3 pts) c) Test the bottom part of the boundary (this means the line segment) for possible
maxima and minima. (3 pts) d) Make a table of values that includes the points you found above, as well as the
corners, and deduce the maximum and minimum values of f, as well as the points where they are
attained. 6. (3 pts) a) State the ﬁrst three'terms in the Maclaurin series of (1 +u)1/3. Express your answer
in the form (1 + u)1/3 = (2—) bu + cu2 + 0(u3). (In other words, ﬁnd speciﬁc numbers a, b, and c.) (4 pts) b) Show that :r + y = 0(As), where A5 = 3/32 + y?. Use this to show that
lim (9” + W = 0.
(am—«0.0) r2 + y2
(5 pts) (3) Use the above results to ﬁnd the limit mm (1 +m+y)1/3 — 1 — 111/3 — y/3 +2my/9.
(x,y)~(o.o> $2 + y2 ii 1. Let f(;z;,y,z) = ezy — :rz.
(2 pts) 8.) Find the gradient V f .
(5 pts) b) Let P(t) be a moving point with position P(t) and velocity 17(t). Assume we know that at time t = 0, we have 77(0) = (1,1,2) and «3(0) = (—1,3, 1). Find gtma» H. (5 pts) 0) Let S = {(r,y,z) E R3 I f(:z:,y, z) = —1} be the level set of f passing through the
point Po(2,0,1). Find the equation of the tangent lane to S at P0. 09 $9 = (ﬁance): is) ﬁtczmﬂ: ﬁlm) — (9.4;,43 (1.3,0 2. (5 pts) a) Sketch the curve r = sin2 6 in polar coordinates.
(2 pts) b) Write a parametrization of your curve in the form 77(0) = (33(6), y(0)), for 0 S 6 S 27r.
(Yes, I want you to use 0 instead of t as the time variable.) (5 pts) c) When 0 = 7r/4, ﬁnd the position vector Fle=ﬂ4 and the velocity vector 17' [emf/4. Use 17 todeduce the speed of the moving point at 6 = 7r/4.
a) 5 O 77/4 1'71. 3&4 w 5 5‘4 3 ‘71... 9 r4? 74 l
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am} “An. y..aw'.s) 3. Let f(:z:,y) = ln(a:2 + (4 pts) a) Sketch the domain of f and brieﬂy indicate why the domain is not closed. (4 pts) b) Find the directional derivative of f at the point Po(3, 1) in the direction of the vector
A = (3,4). (Suggestion: do not simplify fractions; for example, keep 40/70 the way it is, and do
not reduce it to 4/7.) (4 pts) c) Approximately how much is f(3.03, 1.04)? This can be done either using (b) or by
a direct calculation. You may choose whichever method you prefer. f9 Dev/Min :9 1:; Rim)“ X1+17 ‘7 'chr‘ M \LEN'thM'ﬁ mam.
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154m : mo +0? "‘ 4. (3 pts) a) Use integration by parts twice to show that if n is a constant, then 2 , —x2 cos m: 2m sin m: Zoos n11:
msmnmda:=———————+——2—+ 3 +0
n n n (7 pts) b) Let f(:r) be given by f(z) {25, 0 period 27r. Your job is to sketch the grapm’of f and to compute ONLY the coefﬁcients bn
in the Fourier series f(x) = (10/2 + 2:11 0.72 cos m: + 22:1 bn sin ms. (2 pts) 0) At what points a: does the Fourier series of f not converge to f What is the value of the Fourier series at those points? ' {:M"
(*Lr XEE3N,3I] i m x) 05 '1. _ a. —casm< — L
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— —>9_{.’11‘ + 3 cha thX = ’1932‘ +— 311 SXJ(SM’\X)
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Xm, xm, ““Sm air. git$in» (this M‘Fww sum canwa ~1~ 90+) 443:) 0+ 1" _ 7" ice? x 5. We Wish to ﬁnd the maximum and minimum of f (x, y) = 22(312 — 1) in the half disk D deﬁned
' ‘by D = {(as,y)E R2 l 2:2 {y2 3 10,3; 2 0}. (Note that D is closed and bounded, so f attains 3.
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these points belongs to D. (3 pts) b) Test the top part of the boundary of D (this means the semicircle m2 + y: = 10
where y 2 0) for possible maxima and minima of f. Parametrize the boundary in the following
way: P(t) = (t, v10 — t2). Remember to specify the range of t for your parametrization. (3 pts) c) Test the bottom part of the boundary (this means the line segment) for possible
maxima and minima. (3 pts) d) Make a table of values that includes the points you found above, as well as the corners, and deduce the maximum and minimum values of f, as well as the points where they are
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"Am(L nu MWWM “(K4 (1 a“ D is (301)2— (‘5’ «Halal al~ Msgﬁ). 5 6. (3 pts) a) State the ﬁrst three terms in the Maclaurin series of (1+ 101/3. Express your answer
in the form (1 + 101/3 = a. + bu + cu2 + 0(u3). (In other words, ﬁnd speciﬁc enumbers a, b, and c.) (4 pts) b) Show that :3 + y = 0(A3), where As = \/$2 + 3/2. Use this to show that 3
lim (2: + y) = (mm40,0) 2:2 + y2 0’ (5 pts) 0) Use the above results to ﬁnd the limit hm (1+m+y)1/3—1—m/3—y/3+21y/9
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This note was uploaded on 03/01/2010 for the course MATH 201 taught by Professor Variousteachers during the Spring '10 term at American University of Beirut.
 Spring '10
 VariousTeachers
 Calculus

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