# Math 116 Final Exam December 19, 2016

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3 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 3 3. [8 poits] For =,, 3,... cosider the sequece a give by a = (+)/ if is odd, a = if is eve. / 3 a. [ poits] Write out the first 5 terms of the sequece a. The first five terms are, 3, 4, 9, 8. b. [ poits] The series a is alteratig. I a setece or two, explai why the Alteratig Series Test caot be used to determie whether a coverges or diverges. The coditio a + < a does ot hold for all. (It does ot eve hold evetually.) c. [4 poits] The series a coverges. Show that it coverges, either by usig theorems about series, or by computig its exact value. Oe possible aswer is that the series is equal to the differece of two coverget geometric series: k= 3 k k= k = 3 3 =. Aother aswer uses the Compariso Test; for =,,..., let b =, ad otice that a b evetually. Sice b coverges by the p-test (p = ), a coverges by compariso. Hece the origial series coverges.

4 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 4 4. [5 poits] The followig series diverges: = + l(). Use theorems about ifiite series to show that the series diverges. Give full justificatio, showig all your work ad idicatig ay theorems or tests that you use. for all. Sice compariso. Oe solutio uses the Compariso Test. Notice that Alteratively, let a = Sice + l() + = diverges by the p-test (p = ), the origial series diverges by + l() ad b = a lim =. b for all, ad otice that = b coverges by the p-test (p = ), the origial series diverges by the Limit Compariso Test. 5. [5 poits] Let α > 0 be a costat. Compute the first 3 terms of the Taylor series of x f(x) = about x = 0. Write the appropriate coefficiets i the spaces provided. + αx 0 + x + α x +

5 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 6. [0 poits] After receivig a termiatio otice, The Iter has begu to read up o the global job market. A dubious pop-ecoomics book he is readig claims that the rate at which iters are hired or termiated i a large compay is purely a fuctio of the umber of iters at the compay. Specifically, it states that dh dt = g (H), where H(t) gives the umber of iters at a compay, i thousads, after t days, ad g(h) is a differetiable fuctio. A graph of g(h) (ot g (H)) is give i the book: g(h) H - a. [ poits] What are the uits of g (H)? The uits are thousads of iters per day. b. [3 poits] Are there ay stable equilibrium solutios of the differetial equatio? If so, what are they? Yes; the stable equilibrium solutios are H = ad H = 4.

6 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 6 6. (cotiued). Recall that the umber of iters i thousads H(t) satisfies dh dt = g (H), where a graph of g(h) (ot g (H)) is give below: g(h) H - c. [ poits] If a compay starts with 3,500 iters, what will happe to the umber of iters i the log ru? The umber of iters will approach 4,000 asymptotically from below. d. [ poit] Estimate the umber of iters at which the umber of iters is decreasig the fastest. The umber of iters is decreasig the fastest whe there are 4, 500 iters. e. [ poits] Suppose that a compay begis with 5,500 iters. If you used Euler s method to estimate how may iters there will be 5 days from ow, would you expect a uderestimate or a overestimate? Justify your aswer briefly. The correspodig solutio of the differetial equatio is cocave up, so we expect Euler s method to yield a uderestimate.

7 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 7 7. [7 poits] The Legedre equatio is a differetial equatio that arises i the quatum mechaical study of the hydroge atom. I oe of its forms, the Legedre equatio is ( x )y xy + y = 0. For this problem, let y be a solutio to the Legedre equatio satisfyig y( ) = ad y ( ) = 3. Assume that the Taylor series for y(x) about x = coverges to y(x) for all < x < 3. a. [4 poits] I the blak below, write dow P (x), the degree Taylor polyomial of y(x) ear x =. Your aswer should ot cotai the fuctio y(x) or ay of its derivatives. P (x) = ( + 3 x ) ( 4 x ) b. [3 poits] Compute the limit y(x) lim 3x x / (x. ) The limit is 4.