SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone.. You should show your work, and explain how you arrived at your answers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. 3. You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. 4. Numerical experiments do not count as justification. For example, computing the first few terms of a series is not enough to show the terms decrease. Computing the first few partial sums of a series is not enough to show the series converges or diverges. Plugging in numbers is not enough to justify the computation of a limit. You may, of course use numerical experiments to help guide your work, but they do not count as justification for answers. 5. On this test, explanations count. If I can t follow what you are doing, you will not get much credit. 6. You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam.
1. (10 points) (a) Compute the limit of the following sequences: { n( } n + 1 n) (b) Determine if the following series converges or diverges, and if it converges, find the sum: n=0 ( 5 ) n 3
. (15 points) For each of the following series, determine if it converges or diverges. For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. (a) n 3 n 5 + 10n 4 + 3n 3 + 7 (b) (n!) n (n)! (c) n n (3n + 5) n
3. (15 points) Determine if the following series converge absolutely, converge conditionally, or diverge. For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. (a) ( 1) n (n + 1) n + n 1 (b) n= ( 1) n n(ln n) (c) n= ( 1) n n + 1
4. (10 points) Consider the power series: n= (a) Find the radius of convergence. (x 3) n n ln n (b) Find the interval of convergence. [Hint: Check the endpoints.] 5. (10 points) (a) Find the Maclaurin Series for f(x) = Hint: ( ) d 1 dx 1 x = 1 (1 x) x (1 x) (b) Find n n Hint: Let x = 1
6. (15 points) For the curve given parametrically by x = cos 3 t y = sin 3 t (a) Find dy dx at t = π 4. (b) Find the equation of the tangent line at t = π 4. (c) Find the length of the curve for 0 t π.
7. (15 points) (a) Sketch a graph of the area of the region inside both r = 3 cos θ and r = sin θ. (b) For the the area of region inside both r = 3 cos θ and r = sin θ. Fill in the boxes. A = ( ) dθ + ( ) dθ
8. (10 points) (a) Find the equation of the ellipse obtain by translating the ellipse: ( ) x 8 ( ) y + 4 + = 1 so the center is at the origin. 6 3 (b) Find the equation of the parabola that has vertex at (3, 1) and focus at (3, 4).
FORMULA PAGE sin θ + cos θ = 1 tan θ + 1 = sec θ 1 + cot θ = csc θ sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β cos(α + β) = cos α cos β sin α sin β cos(α β) = cos α cos β + sin α sin β tan(α + β) = tan α + tan β 1 tan α tan β sin 1 cos x x = cos 1 + cos x x = sin x = sin x cos x (sin x) = cos x (cos x) = sin x (tan x) = sec x (sec x) = sec x tan x (csc x) = csc x cot x (cot x) = csc x (e x ) = e x cosh x = ex + e x x n+1 = x n f(x n) f (x n ) f (c) = f(b) f(a) b a f(c) = 1 b f(x)dx b a a sec x dx = ln sec x + tan x + C sec 3 x dx = 1 [sec x tan x + ln sec x + tan x ]+C csc x dx = ln csc x cot x + C e x = 1+x+ x! +x3 3! + = k=0 k=0 x k, for all x k! sin x = x x3 3! +x5 5! + = ( 1) k x k+1, for all x (k + 1)! cos x = 1 x! +x4 4! + = ( 1) k xk, for all x (k)! k=0 1 1 x = 1+x+x +x 3 +x 4 + = 1 + 1 = x k, 1 < x < 1 k=0 (ln x) = 1 x (arcsin x) 1 = 1 x (arctan x) = 1 1 + x (arcsec x) = 1 x x 1 (sinh x) = cosh x (cosh x) = sinh x sinh x = ex e x
NAME STATEMENT COMMENTS Geometric Series ar k k=0 Converges if r < 1 and has sum S = Diverges if r 1 a 1 r p-series 1 k p Converges if p > 1, diverges if p 1 Divergence Test If lim k a k 0, the a k diverges. If lim a k = 0, a k n may or may not converge. Integral Test Let a k be a series with positive terms, and let f(x) be the function that results when k is replaced by x in the formula for a k. If f is decreasing and continuous for x 1, then a k and f(x) dx both converge or both diverge. 1 Use this test when f(x) is easy to integrate. Comparison Test Let a k and b k be a series with positive terms such that if a k < b k and b k converges then a k converges or if b k a k di- verges < a k and b k diverges then
NAME STATEMENT COMMENTS Limit Comparison Test Let a k and terms such that b k be a series with positive a k lim = ρ k b k Ratio Test If 0 < ρ <, then both series converge or both diverge. Let a k be a series with positive terms and suppose a k+1 lim k a k = ρ a) Series converges if ρ < 1. b) Series diverges if ρ > 1. c) No conclusion if ρ = 1. Try this test when a k involves factorials or k- th powers. Root Test Let a k be a series with positive terms and suppose lim k k ak = ρ a) Series converges if ρ < 1. b) Series diverges if ρ > 1. c) No conclusion if ρ = 1. Try this test when a k involves kth powers Let if a k be a series with alternating terms Alternating Series Test and lim a k = 0 k This test applies only to alternating series. a k > a k+1 the series converges conditionally