MATH 19 MIDTERM II FALL 2016 BROWN UNIVERSITY SAMUEL S. WATSON

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1 zyllt ZCZCEH) Name: Key MATH 19 MIDTERM II FALL 2016 BROWN UNIVERSITY SAMUEL S WATSON 1 (a) (5 points) What expression would you substitute for y when using the method of undetermined coefficients to find a particular solution of the differential equation y 0 (t)y(t) t 3 t? You do not actually have to solve for the coefficients In other words, do the first step of solving this DE and stop before you substitute into the differential equation to find the coefficients At3tbt2cted (b) (10 points) Find the general solution of the differential equation y 00 (t) 2y 0 (t) 3y(t) 6e 2t homogeneous solution : y "le ) ) 3yH)O ytt ) Ae3tBEt since I 2 3 Xe {31} particular solution : yh Cet y ' 'tt ) 2g 'H ) 462 3yH 3ce2t this equals yltl 364 Get if C 2, so we get Ae3tBet 2ft as our general solution

2 so ' 2 Determine whether the following series converge absolutely, converge conditionally, or diverge For each example, clearly state which convergence test or tests you are using Be sure to address the hypotheses of that test, if the test has any hypotheses (a) (8 points) 1 Â arccos n! n1 As utan arcos Is continuous areas 0 And ( 0) Hz and arccos ( at) 0 the series fails the nth term test and therefore, so diverges (b) (8 points) 7(n!) Â 3 (2n 1)! (4n)! n1 All terms are positive, e a ten so we "* ae 5Y" may apply the ratio test IE#mlniggCnttzu@ut4)(4ut3)l4ue2)l4uH ) x, so the series diverges

3 Is continuous and sin 6) O Also (c) (8 points)  n2 3 n(ln(n)) 4 fext Eu Is positive & decreasing, so we integral test : UluX o sie#xese as < a x o may apply the 3 thus 2 need ' converges (d) (8 points) i  n1 heavies 1 ( 1) n1 sin pn and sin is an Increasing, converges by the alternating series test, In positive function on [ 0, 'Tz] is decreasing So sink ) is positive & decreasing silka ) 0 since sure

4 blk zra 3 (15 points) Determine whether the following improper integral converges, and if so determine its value: Z px 1 dx 1 x Eat a table dx fake fief be#eddx also r bra 'T bukxsta fee ( Ali) za ab the Al )] lucia bbjzfzrb b) Yrz " # ) Feelin ( 42 ) a he the improper integral diverges

5 ( n( i UZ 4 Suppose that (a n ) n1 is a sequence of real numbers with the property that for all n 1, 2, 3, a 1 a 2 a n n 1 n (a) (8 points) Determine whether the infinite series N,auµbi g,an  a n converges, and if so determine its value n1 IF L (b) (4 points) Find a n Express your answer as a single, simplified fraction in terms of n An Autau, an, t 1 1) u # u I C D 2 # n 2 ) UZ Zn 1 2h u(u ) )

6 ZX 5 (12 points) By calculating derivatives of f and using the formula for a Taylor polynomial, calculate the secondorder Taylor polynomial of the function f (x) x 2 centered at c 1 fat 2 f (e) #III the " 1 the Taylor series Is R(x) ftp#xu1ecxdelxt51zcxt)tcxi5 (b) (2 points) How well does this polynomial approximate the original function? (This seems like a vague question, but it does have a clear answer) exactly : )H XZ I

7 6 The Koch Snowflake is a fractal curve, meaning that it continues to look wiggly no matter how far you zoom in on it: It is created by starting with an equilateral triangle with sidelength 1 and by repeating the following process: Step 1: Divide each line segment in the figure into three equallength pieces Step 2: For each line segment from Step 1, draw an equilateral triangle whose base is the middle segment from Step 1 Step 3: For each triangle you draw in Step 2, erase its base The result of the nth iteration of this procedure is shown below for n 0, 1, 2, 3 The Koch Snowflake is defined to be the set of all points which are eventually drawn and then never erased as this procedure is repeated for all n 0, 1, 2, n : N Ettgwub " (a) (3 points) What is the length of the curve (that is, the sum of the lengths of all the line segments) after the nth iteration? Express your answer in terms of n # of segments is 34 " as shown above length of teetotal is 34 each " 5 segment " " 3(4 Is 5h So

8 (b) (3 points) What is the length of the snowflake itself? In other words, what is the limit as n! of the length of the curve after n iterations? Express your answer as an element of the interval (0, ] leg 3t Tx (c) (3 points) What is the area inside the curve after the nth iteration? You may express your answer as an unsimplified sum of n 1 terms, using either summation notation or ellipsis notation Hint: You may want to count the number of new triangles added at the nth iteration and figure out the area of each of these triangles to calculate the area added at the nth step no not hz n3 istoward ( If original triangle ) at gt 3 3 ( sgueu triangles) ( 12 pink triangles ) gtn 34 " " ', # 344

9 (d) (3 points) What is the area inside the snowflake? (same idea as for (b), except with area instead of length) the limit as n x of the area of the ath iterate is if # In FEE p r Hath, E It 4g of 34k erected ft

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