Math 181 - Test #3 Info and Review Exercises Fall 2018, Prof. Beydler Test Info Date: Wednesday, November 28, 2018 Will cover sections 10.1-10.4, 11.1-11.7. You ll have the entire class to finish the test. This will be a 2-part test. Part 1 will be no calculator. Part 2 will be scientific calculator only. No notes, no books, no phones, no smart watches during the test. There will be a seating chart for the test. Where to get help as you re studying: o Office hours o TMARC, LAC, or other tutoring centers o E-mail me at dbeydler@mtsac.edu Here are some of the formulas/concepts that you ll need to know: Parametric Curves dy = dy/dt dx dx/dt d 2 y = dy /dt dx 2 dx/dt t=b Net area = y dx = g(t)f (t)dt t=a t=b Arc length = ( dx 2 dt ) + ( dy 2 dt ) dt t=a t=b Surface area = 2πy ( dx 2 dt ) + ( dy 2 dt ) dt t=a t=b or 2πx ( dx 2 dt ) t=a + ( dy dt ) 2 dt Polar Coordinates x = r cos θ, y = r sin θ r 2 = x 2 + y 2, tan θ = y x dy dx = dy/dθ dr dx/dθ = sin θ + r cos θ dθ dr cos θ r sin θ dθ Horizontal tangent lines will happen when dy dx = 0 and 0. dθ dθ Vertical tangent lines will happen when dx dy = 0 and 0. dθ dθ If both dx dy = 0 and = 0 at θ = θ dθ dθ 0, then you ll have to check lim and possibly use L Hospital. dx θ=β 1 Area = 2 r2 dθ θ=α θ=β Arc length = r 2 + ( dr 2 dθ ) dθ θ=α θ θ0 dy
Page 2 of 22 Know these polar graphs well!! Circle: r = a sin θ and r = a cos θ Rose: r = a sin nθ and r = a cos nθ Limacon: r = a ± b sin θ and r = a ± b cos θ If a > b, there is no inner loop. If a < b, there is an inner loop. If a = b, it s called a cardioid (heart-shaped).
Series Geometric series: 1 n p n=1 ar n 1 = a + ar + ar 2 + converges to p-series: n=1 converges if p > 1, and diverges if p 1 a 1 r if r < 1, diverges if r 1 Does a series converge or diverges? Here are the tests we ve learned so far Test for Divergence: If lim a n does not exist or if lim a n 0, then n=1 a n diverges. n n Page 3 of 22 The Integral Test: Suppose that a n = f(n), where f(x) is continuous, positive, and decreasing for all x N. N Then n=n a n and f(x) dx both converge or both diverge. The Comparison Test: Suppose a n and b n have nonnegative terms, and N is some integer. If a n b n for all n > N and if b n converges, then the smaller a n also converges. If b n a n for all n > N and if b n diverges, then the bigger a n also diverges. The Limit Comparison Test: Suppose a n and b n have positive terms for all n N (N is some integer). a 1. If lim n = c > 0, then a n b n and b n both converge or both diverge. n a 2. If lim n = 0 and b n b n converges, then a n converges. n a 3. If lim n = and b n b n diverges, then a n diverges. n Alternating Series Test (AST): 1. b n > 0 2. b n+1 b n for all n N 3. b n 0 n=1 The Absolute Convergence Test If a n converges, then a n converges. ( 1) n 1 b n = b 1 b 2 + b 3 b 4 + converges if: The Ratio Test Suppose that lim a n+1 n a n = L. If L < 1, then a n converges absolutely. If L > 1 (or L infinite), then a n diverges. If L = 1, then test inconclusive (try something else). The Root Test n Suppose that lim a n = L n If L < 1, then a n converges absolutely. If L > 1 (or L infinite), then a n diverges. If L = 1, then test inconclusive (try something else).
Page 4 of 22 How well does a partial sum approximate the infinite sum? Remainder Estimate for the Integral Test: Suppose that a n = f(n), where f(x) is continuous, positive, and decreasing for all x n. If a n converges, then n+1 f(x) dx R n n f(x) dx Alternating Series Estimation Theorem Suppose we have an alternating series s = n=1 ( 1) n 1 b n where b n > 0, b n+1 b n, and b n 0. Then, R n = s s n b n+1 I ll give you these formulas if you need them: sec x dx = ln sec x + tan x + C csc x dx = ln csc x + cot x + C 3 : s n + f(x) n+1 dx a n s n + n f(x) dx
Page 5 of 22 Review Exercises Note: If you write up solutions to all of the review exercises listed below, and hand them in at the test, you can earn up to 2% extra credit towards your test! It is important to understand that these review exercises are not guaranteed to cover all of the potential problems on the test. Please review the notes and homework problems to fully prepare for the test. Types of problems that will appear on Part 1 are labeled NC (for No Calculator). 1. Given the following parametric equations/intervals of a particle in the xy-plane, find the related Cartesian equation and graph it. Then, indicate the portion of the graph traced by the particle and the direction of motion. a. x = 2 + cos t, y = 3 + 2 sin t, 0 t 2π b. x = cos 2t, y = sin t, π 2 t π 2 2. Find an equation for the line tangent to the curve x = 1 2 t2 + 1, y = 1 3 t3 t at the point where t = 2.
Page 6 of 22 3. Find d2 y dx2 for x = t + cos t, y = 1 + sin t. (NC) 4. Find the length of the curve x = t 2, y = 2t 3/4, 1 t 16 5. Find the area of the surface generated by revolving x = e t cos t, y = e t sin t, 0 t π about the x-axis. 2
Page 7 of 22 6. Find the area of the region a. enclosed by r = 3 sin 2θ. b. inside r = 4 + 4 cos θ and outside r = 6.
Page 8 of 22 c. inside r = 1 + sin θ and outside r = sin θ. d. inside both r = 1 + cos θ and r = 3 cos θ.
Page 9 of 22 e. inside both r = 1 and r = 2 sin θ f. within the inner loop of r = 2 + 4 sin θ.
Page 10 of 22 7. Find the length of the curve r = e 2θ, 0 θ 2. 8. Find the length of the curve r = 1 + sin 2θ, 0 θ π 2.
Page 11 of 22 9. Find the slope of the curve r = cos θ at θ = π. 3 10. Find the values of θ in [0,2π) where the tangent line of r = 1 sin θ is horizontal or vertical.
Page 12 of 22 11. Determine whether each sequence converges or diverges. If it converges, find the limit. (NC) a. a n = tan 1 (ln 1 n ) b. a n = tan 1 n n c. a n = 2n3 n 3 +1 d. a n = 3n4 +2n 5n 7 n 2 +1 e. a n = n 1/n f. a n = ln n ln(n + 1)
Page 13 of 22 g. a n = ln(n+2) n h. a n = 5n n! i. a n = n sin 1 n j. a n = sin2 n+n n 2 k. a n = 2n 3 n+2
Page 14 of 22 12. Assume that the following sequence converges and find its limit. a 1 = 0, a n+1 = 5 + 4a n 13. Assume that the following sequence converges and find its limit. 2, 2 + 1, 2 + 1 2 2+ 1, 2 + 1 2+ 1, 2 2+ 1 2 14. Determine whether each series is convergent or divergent. Be sure to state any test that you use and show your reasoning. If you use the Integral Test or Alternating Series Test, be sure to state the conditions of the test and (if necessary) show why the conditions are met. (NC) a. 2n 2 3n+1 n=0 5n 6 +4n+2
Page 15 of 22 b. ln ( n n=1 2n+1 ) c. ( 1) n n=1 n+1 d. 1 n=2 n(ln n) 2
Page 16 of 22 4 n+1 e. n=2 (If it converges, what does it converge to?) 5 n+2 ( 2) n f. n=1 (If it converges, what does it converge to?) 3 2n+1 π n 1 g. n=1 (If it converges, what does it converge to?) 3 n+2
Page 17 of 22 n n=1 h. 3 i. 6 n 12 n=2 3n 2 +11 j. ( 1) n +4 n=1 n 2 +n+1 k. n=1 ( 1) n+1 ne n
Page 18 of 22 l. ( 1) n n n=1 n 3 +2 15. Find the sum of each series. a. (tan 1 n tan 1 (n + 1)) n=1 b. 2 n=1 n(n+2)
Page 19 of 22 16. Express 7.2345 as a ratio of integers. 1 17. Use n=1 to answer the parts below. (2n+1) 3 a. Estimate the error in using s 5 as an approximation to the series true sum. b. How many terms are needed to make sure that the sum is accurate to within 0.000005? c. Use 3 with n = 5 to give an improved estimate of the series sum (better than s 5 ).
Page 20 of 22 1 n(ln n) 2 18. Use n=2 to answer the parts below. a. Estimate the error in using s 6 as an approximation to the series true sum. b. How many terms are needed to make sure that the sum is accurate to within 0.05? c. Use 3 with n = 6 to give an improved estimate of the series sum (better than s 6 ). ( 1) n 19. Find the sum of the series n=1 correct to 4 decimal places. 3 n n!
Page 21 of 22 ( 1) n+1 20. Find the sum of the series n=1 correct to 4 decimal places. n 5 21. Determine whether each series converges absolutely, converges conditionally, or diverges. Be sure to show your reasoning and state any test(s) used. (NC) a. ( 1+2n n=1 3n+2 )2n b. n! n=1 n n
Page 22 of 22 c. n=2 (tan 1 n) n d. ( 1) n+1 n n=1 n 2 +2 e. 1 3 5 (2n 1) n=1 4 7 10 (3n+1)