Have a Safe Winter Break

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1 SI: Math 122 Final December 8, 2015 EF: Name 1-2 / / / / / / / / / /20 Directions: Total / No books, notes or Keshara in any word problems. 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. 2. 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. You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. Have a Safe Winter Break

2 1. (10 points) (a) Compute arctan x dx (b) Compute sin 3 x cos 3 x dx 2. (10 points) (a) Compute dx x 2 9 x 2 (b) Compute (cosh 5 x)(sinh x) dx

3 3. (10 points) (a) Compute 5x 1 x 2 1 dx (b) Compute 0 1 dx x(x + 4) 4. (10 points) Find the arc length for the curve y = x x 2 on [1, 2]

4 5. (10 points) Find the center of mass of the region bounded by y = 9 x2 and y = 0 6. (10 points) Find the force on a vertical flat plate in the form of a trapezoid shown below in water. (ρ = 62.4 lb/ft 3 ) 2ft 9ft 3ft 12ft

5 7. (10 points) Find the solution of (a) e x2 y = xe y, y(0) = 0 (b) dy dx 3y = e3x sin x y(0) = 0 8. (10 points) On a deserted island there are duck bill platypuses. If after one week, there are 200, and after 2 weeks, there were 500. Assume the platypuses follows the logistic growth equation: dy dt = ky(1 y A ) with carrying capacity of How many were there originally?

6 9. (10 points) A tank contains 20 gallons of pure water. Water containing 2 pounds of dissolved yogurt per gallon enters the tank at 4 gallons per minute. The well stirred mixture drains out at 4 gallons per minute How much yogurt is dissolved in the tank after 10 minutes? 10. (10 points) (a) Compute the limit of the following sequence: { e n + ( 3) n } 5 n (b) Find the sum of the series: ( ) 1 1 n + 1 n + 3

7 11. (10 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) sin n cos n n (b) n 3 n (10 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) n (ln n)3 ( 1) n n=2 (b) ( 1) n (n!)2 e n (2n)!

8 13. (10 points) Find the radius and interval of convergence for 2 2n x n n 14. (10 points) Use the first three non-zero terms for the Maclaurin series for f(x) = cos(x 3 ) to estimate 1/2 0 cos(x 3 )dx

9 15. (10 points) For the curve x = 3 2 t2 y = t t3 4 (a) Find the equations of the tangent lines at the point ( ) 3 2, 3 4. (b) The arc length for 0 t (10 points) Find the area of the region outside r = 3 sin θ and intside r = 3. (Be sure to draw a picture.)

10 17. (10 points) For vectors a = 3, 3, 2 and b = 1, 0, 4, find (a) a b. (b) cos θ where θ is the angle between a and b. (c) a b. (d) Find the area of the parallelogram with vertices (0, 0, 0), (3, 3, 2), (1, 0 4) and (4, 3, 2). 18. (10 points) Is the line that passes through (2, 3, 3) and (3, 1, 8) parallel, intersect or skew to the line that passes through ( 3, 1, 0) and ( 1, 2, 1).

11 19. (10 points) Find the equation of the plane through P (2, 1, 3), Q(3, 3, 5), and R(1, 3, 6). 20. (10 points) True or False, indicate whether the following statements are true or false by circling the appropriate letter. A statement which is sometimes true and sometimes false should be marked false. a) x 2 dx = 4 3 T F b) = 0 T F c) If the power series a nx n converges at x = 2 then it converges at x = 1. T F d) The line x = 1 + t, y = 1 t, z = 3 + 2t and the plane 4x 2y + 6z = 3 are parallel. T F e) I enjoyed the final exam in Math 122. T F

12 FORMULA PAGE sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 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) = S = tan α + tan β 1 tan α tan β sin 2 1 cos 2x x = 2 cos cos 2x x = 2 sin 2x = 2 sin x cos x cos 2x = cos 2 x sin 2 x (sin x) = cos x (cos x) = sin x (tan x) = sec 2 x (sec x) = sec x tan x (csc x) = csc x cot x (cot x) = csc 2 x (e x ) = e x (ln x) = 1 x (arcsin x) 1 = 1 x 2 (arctan x) = x 2 (arcsec x) 1 = x x 2 1 S = b a b a 1 + (f (x)) 2 dx (dx ) 2 + dt ( ) dy 2 dt dt S = F = β α b a r 2 + (r ) 2 dθ ρ h(x) w(x) dx (sinh x) = cosh x (cosh x) = sinh x sinh x = ex e x 2 cosh x = ex + e x 2 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 2 csc x dx = ln csc x cot x + C e x = 1+x+ x2 2! +x3 3! + = n=0 n=0 x n, for all x n! sin x = x x3 3! +x5 5! + = ( 1) n x2n+1, for all x (2n + 1)! cos x = 1 x2 2! +x4 4! + = n=0 1 1 x = 1+x+x2 +x 3 +x 4 + = y = A 1 e kt C ( 1) n x2n, for all x (2n)! x n, 1 < x < 1 n=0 C = y 0 y 0 A D = ax 1 + by 1 + cz 1 d a 2 + b 2 + c = 2

13 NAME STATEMENT COMMENTS Geometric Series ar n n=0 Converges if r < 1 and has sum S = Diverges if r 1 a 1 r p-series 1 n p Converges if p > 1, diverges if p 1 n-th Term Test or Divergence Test If lim n a n 0, the a n diverges. If lim n a n = 0, a n may or may not converge. Integral Test Let a n be a series with positive terms, and let f(x) be the function that results when n is replaced by x in the formula for a n. If f is decreasing and continuous for x 1, then a n and f(x) dx both converge or both diverge. 1 Use this test when f(x) is easy to integrate. Comparison Test Let a n and b n be a series with positive terms such that if a n < b n and b n converges then or if b n diverges then a n di- b n < a n and verges a n converges

14 NAME STATEMENT COMMENTS Limit Comparison Test Let a n and terms such that b n be a series with positive a n lim = ρ n b n Ratio Test If 0 < ρ <, then both series converge or both diverge. Let a n be a series with positive terms and suppose a n+1 lim n a n = ρ a) Series converges if ρ < 1. b) Series diverges if ρ > 1. c) No conclusion if ρ = 1. Try this test when a n involves factorials or n- th powers. Root Test Let a n be a series with positive terms and suppose lim n n an = ρ a) Series converges if ρ < 1. b) Series diverges if ρ > 1. c) No conclusion if ρ = 1. Try this test when a n involves nth powers Let if a n be a series with alternating terms Alternating Series Test and lim a n = 0 n This test applies only to alternating series. a n > a n+1 the series converges conditionally