Math 113 Fall 2005 key Departmental Final Exam
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1 Math 3 Fall 5 key Departmental Final Exam Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.. Fill in the blanks with the correct answer. (a) The integral cos(x + ) equals sin(x + ) + C (b) The integral sec x tan x equals sec(x) + C (c) The integral (d) The integral (e) The integral (f) The integral (g) The integral (h) The integral + x equals tan x π 4 equals x sin x π tan x equals x equals x x 3 (i) Give the limit of the sequence write DIVERGENT. e equals divergent integral x + x equals + x + C sec (x) tan(x) x + C {( ) n } n (j) State the integration by parts formula: u(x)v (x) u(x)v(x) u (x)v(x) (k) Give a limit definition of the improper integral sin x lim ɛ ɛ x as n if it is convergent, otherwise sin x x (l) State the (m)-th term of the MacLaurin series for sin x x ( ) m (m + )! xm (m) The integral cot x equals ln(sin(x)) + C
2 . True/False: Write T if statement always holds, F otherwise. Let a n n a n be an arbitrary series. (a) F : need a n If {a n } is a positive decreasing sequence then ( ) n a n converges. (b) T: Divergence test If a n converges then a n. (c) F : +... If the partial sums of a n are bounded, then a n converges. Problems 3 through 9 are multiple choice. Each multiple choice problem is worth 3 points. In the grid below fill in the square corresponding to each correct answer. 3. The most appropriate first step to integrate x would be 3x 3 x (a) Integration by parts (d) Other (non trigonometric) substitution (b) Partial fractions (e) Differentiate the integrand (c) Trigonometric Substitution (f) None of these 4. The series x + x 4 + x6 + x8 6 + n x (n+) (a) x +x (e) x (sin x + cos x ) (b) x tan x (f) sin x + cos x (c) e x + (g) None of these n! converges to the function (d) x e x 5. The improper integral xe x converges to (a) (e) (b) /e (f) e (c) / (g) None of these (d) (h) It doesn t converge
3 6. The length of the curve y cosh x from x to x is (a) sinh (e) (b) cosh (f) a real number in (,) (c) cosh cosh (g) Imaginary (d) (h) None of these 7. The area enclosed by the polar curve r 3 + sin θ is (a) 5π (e) 4.5π (b) 4π (f) 9π (c) 9π (g) 9π (d) π/4 (h) None of these: 9π/ 8. The interval of convergence of the power series n (5x 3) n is (a) ( 3/5, 3/5) (e) (/5, 4/5) (i) None of the above (b) ( 5/3, 5/3) (f) (/5, ) (c) (, ) (g) (, ) (d) (, ) (h) (, ) n 9. The coefficient of x 3 in the series expansion of ( + x) /4 is (a) (e) 4 3 3! 5 96 (i) None of the above (b) 4 3 3! 384 (f) 4 3 3! 7 8 (c) ! 64 (g) ! (d) ! 5 8 (h) ! The answers to the multiple choice MUST be entered on the grid on the previous page. Otherwise, you will not receive credit. 3
4 Part II: Written Solutions For problems 8, write your answers in the space provided. Neatly show your work for full credit.. (a) Evaluate the integral t e t dt. Let u t, dv e t dt, then du t dt, v e t, t e t dt t e t te t dt Let u t, dv e t dt, then du dt, v e t, t e t dt e (te t e (e (e )) e ) e t dt (b) Expand in partial fraction form x + 3 x. x + 3 x x + 4 x + 4 (x )(x + ) + x + x (c) Evaluate the integral x + 3 x. x + ln ( ) x x + 4
5 . Evaluate the integral. Let z tan(x/), then 4 3 sin x dz z, sin x + z + z 4 3 sin x dz 4 3 z + z +z 4z 6z + 4 dz ( z 3 4 ) ( 7 ) dz + 4 z 3z + dz Let z tan t, then dz 4 4 sec t dt and 4 3 sin x dt 7 7 t + C ( ( 4 tan 7 z ( 4 ( x ) tan 7 (tan )) + C )) + C ( ( x ) tan 7 (4 tan 3) ) + C 7 5
6 . The region bounded by y x and y x is revolved about the y-axis ; find the volume of the solid generated. Intersection of curves: ) x x x, points of intersection: (, ), (, Disc method: / π ( ) y / [ πy y dy π y dy π 4 y ] / 3 y3 π 48 Shell method: / πx(x x ) π [ x ] / x3 π Find the area of the surface of revolution generated by revolving the curve y x, x 4, about the x-axis. Area π x π x + ( ) d x + 4x π + 4x 4 ( + 4x)3/ 3 π ( 7 3/ ) 6 4 6
7 4. Find the centroid of the region bounded by the curves y + x, x and y + x. Express you answer in terms of unevaluated integrals. (Note: You should simplify the integrands as much as possible.) The curves y + x and y + x intersects at x. Area of region Coordinates of centroid ( x, ȳ) : A + x + x x x( + x + x ) A ȳ A ( ) ( + x + + x + x + x ) A (( + x) ( + x )) A x ( A) 5. If a region in the first quadrant, with area π and centroid at the point (, ), is revolved around the line x 5, find the resulting volume of revolution. By the first theorem of Pappus, V π ra. Now A π, r ( 5) 6, so V π 7
8 6. Determine whether each infinite series is absolutely convergent, conditionally convergent, or divergent. Give reasons for your conclusion. (a) n ln n 3n + 7 Divergent: comparison with harmonic series n3 ln n 3n + 7 n3 3n + 7 n3 6n 6 n3 n (b) (3 n 5 n ) n Absolutely convergent: geometric series (3 n 5 n ) n 3 n n n 5 n ) /3 /3 /5 /5 4 4 (c) n ( ) n n ln n Conditionally convergent: alternating series related to decreasing sequence of positive terms and comparison test { } Series is convergent since the sequence is a decreasing sequence of positive n ln n ( ) n terms, hence the alternating series is convergent. n ln n n Series is not absolutely convergent since the integral is divergent. x ln x (d) n ( ) n n ln(n) Divergent: divergence test ( ) n n lim n ln(n) lim ( ) n n /n ± 8
9 7. (a) Determine the power series expansion of tan x. so tan x x 3 x3 + 5 x5 7 x7 + 9 x9 + + ( )n n + xn+ + tan x C + x x4 + ( ) n 3 x6 + (n + )(n + ) xn+ + (b) Find first two nonzero terms of the Taylor series of ln( + sin x) at x π. What is the remainder after these terms? From Maclaurin expansion, As sin π, so ln( + z) z z + 3 z3 4 z4 + + ( )n+ z n + n ln( + sin x) sin x sin4 x + 3 sin6 x + Also the Taylor series of sin x about x π is Thus sin x (x π) + 6 (x π)3 (x π)5 + sin x (x π) 3 (x π)4 + Hence ln( + sin x) (x π) 3 (x π)4 (x π) 5 6 (x π)4 + ( (x π) 3 (x π)4 + ) + The remainder is given by x π d 6 5! ln( + 6 sin t) (x t) 5 dt where ξ is a number between x and π. or 6! d 6 ln( + 6 sin x) (x π) 6 xξ 9
10 8. Given the polar curve r θ, θ 3/, (a) sketch the curve; 3 radian is slightly less than π radian or 9. ( note: 3 radian is about 86 ) (b) find the area swept out by the curve; (c) find the arc length. Area 3/ 3/ r dθ θ4 dθ ( ) 5 3 Arc length 3/ 3/ 3/ r + ( ) dr dθ dθ θ4 + 4θ dθ θ θ + 4 dθ ( t + 4 ) 3/ 3/ End
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