Math 113 (Calculus 2) Section 12 Exam 4

Transcription

1 Name: Row: Math Calculus ) Sectio Exam November 00 Istructios:. Work o scratch paper will ot be graded.. For questio ad questios 0 through 5, show all your work i the space provided. Full credit will be give oly if the ecessary work is show justifyig your aswer. Please write eatly.. Questios through 9 are short aswer. Fill i the blak with the appropriate aswer. You do ot eed to show your work. 4. Should you have eed for more space tha is allotted to aswer a questio, use the back of the page the problem is o ad idicate this fact. 5. Simplify your aswers. Expressios such as l), e 0, siπ/), etc. must be simplified for full credit. 6. Calculators are ot allowed. For Istructor use oly. # Possible Eared # Possible Eared a b c d e f Total 00

2 Uless idicated, each problem is worth 5%.. 0% Show your work.) Determie whether each series coverges absolutely, coverges coditioally, or fails to coverge. State ad justify your coclusio ext to the series. a) b) ) l Does ot coverge absolutely by Compariso Test with Coverges coditioally by the Alteratig Series Test. = ) si Coverges absolutely by the Limit Compariso Test with =. c) k= ) k k k + Does ot coverge absolutely by Limit Compariso Test with coditioally by the Alteratig Series Test. =. Coverges d) e) l = ) Coverges absolutely by the root test. The limit of the th root goes to zero. =! Coverges absolutely by the Ratio Test. The limit of the ratios goes to zero. f) k= )! Coverges absolutely by the Ratio Test. The limit ratio is e.

3 Questios through 9 are short aswer. Fill i the blak with the appropriate aswer. You do ot eed to show your work.. If fx) = x cos x, fid the 00th derivative evaluated at zero; i.e., fid f 00) 0). fx) = x cos x = x x5! + x7 4! + x99 96! x0 98! is zero, f 00) 0) = 0 +. Sice the coefficiet of x00. Evaluate the sum. π! + π4 4! π6 6! + π8 8! = k=0 ) k π k k)! cos π = 4. Fid the radius of covergece. = x 5 r = 5 5. Fid the coefficiet of x 4 i the Maclauri series for cos x + x Fid the the first four o-zero terms of Taylor series for si x about x = π/6. + x π ) x π ) x π ) 6 6! 6 7. Fid the Taylor series for px) = x + x + x + about x = ; i.e., write px) as a polyomial i x +. px) = x + ) x + ) + x + )

4 8. Fid the Maclauri series for + x ad give the iterval of covergece. x + x x + ) o, ) 9. The Taylor polyomial of degree 4 for fx) = e x expaded about a = 0 is T 4 x) = + x + x + 6 x + 4 x4. Use the Taylor Iequality to estimate the accuracy of the approximatio fx) T 4 x) whe x lies i the iterval [-0.5, 0.5]. error e 5 5! Problems 0-5. Show your work for full credit. 0. Fid the sum k + 4k + k= k + 4k + = k + k + S = S = ad lim S = + = 5 6. Evaluate the sum. x + x + 4 x x 5 + = d dx x = d dx x = + x + x + x + )x d dx x = x) = + x + x + 4x + x) = x) = + x + 4 x + 5 4x + x x) = x + x + 4 x x 5 +

5 . Evaluate the followig limit: lim e x x e x = + x + x! + x! lim lim e x x = lim + x + =!! + so ex = + x + x6! + x9! + + x + x6! + x9! + x = lim =! + x9! + =. Fid the the first four o-zero terms of a power series expasio for the fuctio si x expaded about x = 0. Let y = si x. The dy dx = = x x ) ) x ) = +! dy dx = + x + y = C + x + x x ) +! x4 + 5 ad! + x 5! 5 + 5! ) ) x )! By the Biomial Theorem )! ) ) 5 x ) + So x 7 7 But C = 0 because si 0 = Fid the iterval of covergece. x ) = x By the Ratio Test, the series coverges for coverges absolutely at the edpoits because [ = covergece is, ]. < or < x <. The series coverges. So the iterval of si x 5. Evaluate the idefiite itegral dx as a ifiite series. x ) si x x dx = x! + x4 5! x6 7! + dx = C + x x! + x5 5 5! x7 7 7! +