Review Problems Math 122 Midterm Exam Midterm covers App. G, B, H1, H2, Sec , 8.9,

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1 Review Problems Math Midterm Exam Midterm covers App. G, B, H, H, Sec , 8.9, Review the Cocept Check problems: Page 6/ -, Page 690/- 0 PART I: True-False Problems Ch. 8. Page 6 True-False Quiz Problems 8. Ch. 9. Page 69 True-False Quiz Problems 6. Additioal True-False Problems.. If the series a coverges, the the sequece {a } coverges.. If lim a = 0, the a coverges.. If a coverges, the a coverges. 4. If {a } coverges, the { a } coverges.. If { a } coverges, the {a } coverges. 6. If {a } coverges but {b } diverges, the {a + b } diverges. 7. If {a } coverges but {b } diverges, the {a b } diverges. 8. If 0 a b for ad {b } coverges, the {a } coverges. 9. If a b b for ad lim b =, the {a } coverges. 0. If {a } is bouded the {a } is coverget.. If 0 a b ad b coverges, the a coverges.. If a coverges, the lim e a =.. If a is absolutely coverget, the a coverges. 4. If f(x) = (x ) + (x ) + coverges for all x, the f () =.. For ay vectors u ad v i V, u v is a vector. 6. For ay vectors u ad v i V, u v is a vector. 7. For ay vectors u, v ad w i V, (u v) w is a vector. 8. If u + v = u + v, the u v = For ay u i V, u u = For ay u i V, u u = 0.. If two lies are perpedicular to a third lie, the they are parallel.. If two lies are parallel to a third lie, the they are parallel.

2 . If two plaes are perpedicular to a lie, the they are parallel. 4. If two plaes are parallel to a lie, the they are parallel.. The curve i polar coordiates give by r = si θ + cos θ is a circle. PART II: Multiple-Choide Problems. Exactly oe of the followig sequeces diverges, Which is it? { } { } 4 + { } { }! (l ) { cos }. Exactly oe of the followig sequeces diverges, Which is it? { si } { cos } { si } { e / } { } ( ) +. Exactly oe of the followig series diverges, Which is it? ( ) 4 = (l ) = ( ) l 4. Exactly oe of the followig series diverges, Which is it? si ( ) ( ) l si = l!. The series ( r) for 0 < r < coverges to r + r r r r + r 6. The sum of the geometric series ( π) is 4 π 4 + π π 4 + π + π Noe of the above. π 4 + π 7. The series ( + ) coverges to Noe of the above.

3 8. The sum of the series ( ) is The sum of the series! is π e l 0. The sum of the series ( ) π + is ( + )!6 +! 0 / π/6 /. The radius of covergece of the series / / 0. The iterval of covergece of the series l + x is (x ) is [/, 4/] (/, 4/) (/, 4/] [/, 4/) Noe of the above.. The Maclauri series for cos(x ) is ( ) x+ + ( ) x4 ()! ( ) x + ( + )! ( ) x 4+ ( + )! ( ) x ()! 4. The d degree Taylor polyomial of the fuctio f(x) = l x at a = is x (x ) x (x ) x x x x x. If a =, b =, ad the agle betwee a ad b is π/, the a b = 6. If a =, b =, ad a b =, the the agle betwee a ad b is either 0 or π either π/6 or π/6 either π/ or π/ either π/4 or π/4 π/

4 7. The scalar projectio comp v u of u =,, oto v = 0,, 4 is The vector projectio proj u v of v = 0,, 4 oto u =,, is ,, 4,, + 4 4,, + 4,, 9. The area of the triagle with vertices at the poits P (,, ), Q(, 0, ) ad R(,, 0) is The volume of the parallelepiped determied by the vectors a =,,, b =, 0, ad c =,, 0 is 0 4. The distace from the poit P (,, ) to the lie through the poits Q(, 0, ) ad R(,, 0) is The distace from the poit P (,, ) to the lie r(t) = 0,, + t,, is 0. The distace from the poit P (,, ) to the plae through the poits Q(,, ), R(4,, 0), ad S(,, ) is 4. The distace from the poit P (,, ) to the plae x y + z = is. If θ is the agle betwee the plaes x y + z = ad x y z =, the cos θ is 6 4

5 6. The distace betwee the plaes x y + z = ad x 6y + 4z = is Exactly oe of the followig vectors is parallel to the lie described by x = + t, y = t, z = t. Which is it?, 0,, 0, i j + 4k i + j + k, 0, 8. Exactly oe of the followig vectors is ormal to the plae described by 4x 6y = z. Which is it?,, 4i 6j k i j + k 4i + 6j + k 4, 6, 9. Exactly oe of the followig equatios i cylidrical coordiates completely describes the coe x + y z = 0. Which is it? r = z r = z r = z r = z r cos θ = z 0. Exactly oe of the followig equatios i spherical coordiates completely describes the ellipsoid x + y + z =. Which is it? ρ ( + si φ si θ) = ρ ( + cos φ si θ) = ρ ( + si φ cos θ) = ρ ( + cos φ cos θ) = ρ ( + cos φ) = PART III. Essay Problems. Evaluate the itegrals (x + ) (x ) dx x + x + (x ) (x + ) dx x + x (x ) (x + ) dx + 0x + 48 dx x + 6x. Fid the area of oe loop of the curve r = si θ.. Fid the area of the regio that lies iside the curve r = si θ ad outside the curve r =. 4. Fid the equatio of the taget lie to the curve r = + si θ at the poit θ = π 4.. Fid the poits o the curve r = cos θ where the taget lie is horizotal or vertical. 6. Show that the followig equatio represets a circle ad fid its ceter ad radius: x + y 4x + 0y + = Fid the foci of the ellipse x + 9y = 9.

6 8. Determie whether the sequece coverges. Fid the limit if it is coverget. a = a = l( + ) a = e + a = cos e a = l( + ) l( ) (F) a = ( + /) 9. Let the sequece {a } satisfy If {a } has a limit, fid the limit. a = 0, a + = 6 + a,. 0. Determie whether the series is coverget, or diverget, or absolutely coverget. si cos(π) e ( ) (l ) ( ) + e + ( )! ( + ) (F) ( + ) ( ). Use the partial sum s to estimate the sum of the series s as a approximatio of the sum of the series.. Use the partial sum s to esiitmate the sum of the series usig s as a approximatio of the sum of the series., ad estimate the error usig ( ), ad estimate the error. Fid the radius of covergece ad iterval of covergece of the power series. ( x) + ( + ) x + (x )! 4. Fid the Taylor series of the fuctio at a = 0. x ( x) l( + x) si(x ) x e x.. Fid the Maclauri series of e x ad approximate Fid the Taylor polyomial T (x) of the fuctio e si x at a = 0. 0 e x dx correct to withi a error of 6

7 7. Let T (x) be the degree Taylor polyomial of e x at a = 0. Use the Taylor iequality to fid a boud for R (x) = e x T (x) for x [0, 0.]. 8. Fid the sum of the series! ( ) π ( + )! 9. Let a ad b be vectors i V. Suppose a =, b =, ad the agle betwee a ad b is θ = π/4. Fid a b a + b a b (a) b comp a b (F) proj b a (G) The area of the parallelogram determied by a ad b. 0. Suppose that a =,,, b = i + k ad c =,,. Fid a (b c) (a c) b a (b c) b (c a) The agle betwee a ad b (F) The area of the parallelogram determied by a ad b (G) The volume of the parallelepiped determied by a, b ad c.. Give the poits P (,, ), Q(,, ), R(,, ), ad S(,, ), fid The area of the triagle with vertices P, Q, R The legth of the lie segmet P Q The agle betwee the vectors P Q ad P R. Parametric equatios for the lie that passes through the poits Q ad R The distace form the poit P to the lie that passes through Q ad R (F) A scalar equatio for the plae through the poits Q, R, ad S (G) The distace from the poit P to the plae through the poits Q, R, ad S (H) The volume of the parallelepiped with adjacet edges P Q, P R, ad P S (I) The distace betwee the lie through the poits P, Q ad the lie through the poits R ad S (J) Symmetric equatios for the lie through the poit P ad ormal to the plae through the poits R ad S (K) The agle betwee the plae through the poits P, Q, R ad the plae through the poits P, Q, S.. Fid the value(s) of t such that the vectors t, t, ad t +,, are orthogoal. 7

8 . Fid the work doe by a force F = i j+k that moves a particle from the poit P (,, ) to the poit Q(4,, ). 4. Fid the work doe by a force of 0N applied at a agle of π/6 to the movig directio i movig a object m.. Fid the lie (its vector equatio, parametric equatios, ad symmetric equatios) that passes through the poit P (,, ) ad is perpedicular to the plae x y + z = Fid the lie (its vector equatio, parametric equatios, ad symmetric equatios) that passes through the poits P (,, ) ad Q(,, ). 7. Fid the lie (its vector equatio, parametric equatios, ad symmetric equatios) that passes through the poit P (,, ) ad is perpedicular to the vectors,, 0 ad, 0,. 8. Fid the lie (its vector equatio, parametric equatios, ad symmetric equatios) of the itersectio of the plaes z = x + y ad y = x + z Fid a scalar equatio of the plae through the poit P (, 0, ) with ormal vector =,,. 0. Fid a scalar equatio of the plae that passes through the poit P (,, ) ad is perpedicular to the lie (x ) = y = z 4.. Fid a scalar equatio of the plae that cotais the lie (x ) = y = z 4 parallel to the vector,,. ad is. Fid a scalar equatio of the plae that passes through the poit P (,, ) ad is parallel to the vectors,, 0 ad, 0,.. Show that the lies L : {x = + t, y = t, z = t} ad L : {x = t, y = t, z = + t are skew. Fid the plae that cotais the lie L ad is parallel to the lie L. Determie the distace betwee L ad L. 4. Fid the parametric equatios of the lie of itersectio ad the the agle betwee the two plaes x + y = z + ad z = x + y.. The cylidrical coordiates of a poit are (, π/, ). Fid the rectagular ad spherical coordiates of the poit. 8

9 6. The rectagular coordiates of a poit are (,, ). Fid the cylidrical ad psherical coordiates of the poit. 7. The spherical coordiates of a poit are (8, π/4, π/6). Fid the rectagular ad cylidrical coordiates of the poit. 8. Write the equatio i cylidrical coordiates ad i spherical coordiates. x + y + z = 4 x + z = 4. 9

Math 21C Brian Osserman Practice Exam 2

Math 21C Brian Osserman Practice Exam 2 Math 1C Bria Osserma Practice Exam 1 (15 pts.) Determie the radius ad iterval of covergece of the power series (x ) +1. First we use the root test to determie for which values of x the series coverges