FINAL REVIEW ANSWERS WITH BRIEF HINTS 1 polar Coordinates Example 1 1 graph r 4 3 cos 4 Should look like a 4 blade propeller with the blades aligned with the x and y axis Example 1 2 graph r 1 2 cos Should look like the a 1 2 graph from the last homework the trisectrix but rotated 2 clockwise and scaled up by a factor of 2 Example 1 3 graph r 2 sin 3 Sould look like a 3 blade propeller Example 1 4 Convert xy 4 to polar coordinates r2 sin cos 4 or r2 8 csc 2 Example 1 5 Find the equation of the circle centered at 0 5 with radius 5 r 10 sin Example 1 6 Find the equation of the circle centered at 3 4 with radius 5 r 8 sin 6 cos Example 1 7 Find the area enclosed by r 2 sin Hint r is positive over the interval 0 Z 1 2 sin 2 d 0 2 Example 1 8 Find the area enclosed by r 3 2 cos Z 2 1 3 2 cos 2 d 11 2 0 2 parametric equations Example 2 1 eliminate t from x 1 2 sin t y 3 cos t 2 x 1 y 3 2 1 2 Example 2 2 eliminate t from x 1 t2 y t3 x 1 y 2 3 or y x 1 3 2 Example 2 3 find the slope of the tangent to the curve x t cos t y t sin t at the point where t Example 2 4 find the slope of the tangent to the curve x 1 t 2 y et at the point where t 1 4e 3 power series P Example 3 1 determine interval of convergence of n 1 1 n nxn Example 3 2 determine interval of convergence of 1 1 P 1 n xn Example 3 3 determine interval of convergence of 1 1 P 1 n xn n 1 n 1 n n2 1 1 Example 3 4 Approximate x 2 by its second order Taylor polynomial at center 1 and estimate the accuracy over the interval 0 9 x 1 1 1 2 FINAL REVIEW ANSWERS WITH BRIEF HINTS T2 x 1 2 x 1 3 x 1 2 f x T2 x 4x3 4 0 1 3 0 00677404 5 0 9 5 Example 3 5 Approximate sin x by its third order Taylor polynomial at center the interval 0 x 3 6 and estimate the accuracy over x 6 2 1 1 1 x T3 x 3 x 2 2 6 4 6 4 3 1 2 4 sin 4 x f x T2 x 24 24 6 Example 3 6 Maclaurin series for x 2 x and x3 1 x3 3 Method f X x x 2 xn 1 1 n n 1 2 x 1 x 2 n 0 2 X x3 x3n 3 1 x3 n 0 Example 3 7 Maclaurin series for R Z Z Z Z 2 ex dx R sin x2 dx 2 ex dx C sin x2 dx C cos x2 dx C R cos x2 dx and R sin x x dx Method b c d e X x2n 1 2n 1 n n 0 X 1 n n 0 X 1 n n 0 X x4n 3 4n 3 2n 1 x4n 1 4n 1 2n x2n 1 sin x 1 n dx C x 2n 1 2n 1 n 0 Example 3 8 If a power series diverges at 0 and 2 and converges at 1 then at 3 it converges diverges don t know It must diverge at 3 because the hypothesis implies that the interval of convergence is contained between 0 and 2 Example 3 9 If a power series centered at 2 converges at 4 then at 1 it converges diverges don t know and at 0 it converges diverges don t know Converges at 1 because it is inside the radius of convergence which is 2 Don t know at 0 because it could be on the radius of convergence Example 3 10 A power series may have 0 as its integerval of convergence True False False If radius of convergence is infinite then the interval of convergence must be 4 Sequences and Series P 1 Example 4 1 Classify n 1 n 3 n LCT P 1 Converges absolutely by limit comparison with the convergent geometric series 3n P n Example 4 2 Classify n 1 1 n n 1 DST Diverges by divergent series test P Example 4 3 Classify n 1 1 n n2n 1 AST LCT P Converges by Alternating Series Test but only conditionally since n 1 n2n 1 diverges by a limit comparison with P1 n n P Example 4 4 Classify n 2 1 n ln n AST IT P Converges by Alternating Series Test but only conditionally since n 1 ln ln x 1 n ln n R diverges by Integral test dx x ln x FINAL REVIEW ANSWERS WITH BRIEF HINTS Example 4 5 Classify 3 P n ln n n 2 1 n AST DCT AST IT P diverges by a direct comparison with Converges by Alternating Series Test but only conditionally since n 1 ln n n P 1 the p series n P 1 Example 4 6 Classify n 2 2 n 1 n RT Converges Absolutely by Root Test 0 1 P Example 4 7 Classify n 1 sin en n2 DCT P 1 sin en Converges Absolutely since n12 and n2 n2 is a convergent p series P n n Example 4 8 Classify n 1 1 n ln n e LCT RT P P n 1 Converges Absolutely since n 1 n ln n may be limit compared with en en and this converges by Root Test e 1 P Example 4 9 Evaluate n 1 n21 n telescoping X X 1 1 1 1 2 n n n n 1 n 1 n 1 Example 4 10 Evaluate P n 1 1 n 1 22n 5n 1 geometric 20 9 P Example 4 11 If n 1 an is C then what is limn cos an cos 0 1 P is AC CC D don t know Example 4 12 With the tests you know n 1 sin en n None of the tests deal with this sum so it is a don t know P P Example 4 13 If n 1 an is AC then n 1 a1n is AC CC D don t know P Diverges since limn a1n is by the hypothesis that n 1 an is AC P P 2 Example 4 14 If know P n 12an is CC then n 1 n an is AC CC D don t Diverges since if n 1 n an were to converge then limn n2 an 0 which means that an P P 1 equation were true then n 1 an would be AC by direct comparison with the p series n2 1 n2 If this last There were also questions regarding power series for …