x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c

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1 . Evaluate the given integral (a) 3xe x2 dx 3 2 e x2 + c (b) 3 x ln xdx 2x 3/2 ln x 4 3 x3/2 + c (c) x + 5 x 2 + x 2 dx ln x ln x + c

2 (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( + x 2 ) 2 dx 3 2( + x 2 ) + c (f) 2x arctanxdx x 2 arctan x x + arctanx = π 2

3 2. Determine the area bounded by the curves f(x) = 4 x and g(x) = 3x +. (3x + 4 x ) dx = 3 2 x2 + x ln 4 4x = ln 4 3. Determine the area bounded by the x-axis, the y-axis, the line y = 3 and the curve y = x 2. 3 (y 2 + 2) dx = 3 3 y3 + 2y = 5

4 4. Calculate the volume obtained by rotating the region in the first quadrant bounded by f(x) = x 3 and g(x) = 2x x 2 about the x-axis. π(2x x 2 x 6 ) dx = ( 4 π(4x 2 4x 3 +x 4 x 6 ) dx = π = 7) 4π 5 5. Calculate the volume obtained by rotating the region in the first quadrant bounded by f(x) = x 3 and g(x) = 2x x 2 about the y-axis. 2π(2x 2 x 3 x 4 ) dx = 2π ( ) = 3π 3

5 6. A 6-lb boulder is suspended over a roof by a 4-ft cable that weighs lb/ft. How much work is required to raise the boulder with the cable over the roof, the distance of 4 ft? 6(4) + 4 xdx = 24 + ( 5x 2) 4 = = 4 ft-lb 7. A trough is filled with water and its vertical ends have the shape of a parabola with top length 8 ft and height 4 ft. Find the hydrostatic force on one end of the trough (4 y)(4 4 y) dy = 62.5 (6 y 4y 3/2 ) dy = 62.5 ( ) ( ) 2 = 62.5(256) lbs 5 5

6 8. Determine the arclength of the curve y = 2 x2 ln x on 2 x 4 4 ( ) 2 dy + = x 2 + dx 2 + ( 6x = x + ) 2 2 4x 4 2 ( x + ) dx = 4x 2 x ln x = ln 2 9. Evaluate the integral if it converges. Show divergence otherwise. (a) 3 2 (x ) 2 dx ( ) 3 2 = diverges x x (b) 2 + x 2 dx 2 arctanx x = π converges

7 . Solve the initial value differential equation explicitly for y(t): dy dt = 2(y )2 y() = 2. dy = 2 dt (y ) 2 y = 2t + c y = 2t + 2. Solve the initial value differential equation explicitly for y(x): dy dx = xy x y() =. dy = xdx y ln y = 2 x2 + c y = + 9e x2 /2

8 2. Solve the initial value first order linear differential equation: y = y + x y() = 2. I.Factor = e dx = e x e x y e x y = xe x (e x y) = xe x e x y = xe x e x + c y = x + 3e x 3. Solve the initial value first order linear differential equation: xy + y = 3x 2 y() = 2. y + x y = 3x I.Factor = e x dx = e ln x = x xy + y = 3x 2 (xy) = 3x 2 xy = x 3 + c y = x 2 + x

9 4. Use Euler s Method to approximate y() if dy dx = y + x with y() = and x = 2. x n y n dy/dx y n + (dy/dx)(δx) + (/2) /2 3/2 2 3/2 + 2(/2) 5/2 5. Use Simpsons Rule to approximate 3 dx with n = 4. x ( ( ( ( (2) 3) 2) 5) ) =

10 6. Solve the initial value second order homogeneous differential equation: y + 2y + y = y() = 2 y () = 4. r 2 + 2r + = r = r 2 =. y = Me x + Nxe x M = 2 y = 2e x + Ne x Nxe x N + 6 y = 2e x + 6xe x 7. Solve the initial value second order homogeneous differential equation: y + 4y = y() = y () = 3. r = r = ±2i y = M cos (2t) + N sin (2t) M = y = 2 sin (2t) + 2N cos (2t) N = 3 2 y = cos (2t) + 3 sin (2t) 2

11 8. Solve the initial value second order nonhomogeneous differential equation using the method of undetermined coefficients. y + 5y + 4y = sin x y() = y () =. y p y p y p = A sin x + B cosx = A cosx B sin x = A sin x B cos x 4A sin x + 4B cosx + 5A cosx 5B sin x A sin x B cosx = sin x 3A 5B = 5A + 3B = r 2 + 5r + 4 = A = 3 6 B = 5 6 y p = 3 6 sin x cosx r = 4, r 2 = therefore y c = Me 4x + Ne x y = Me 4x + Ne x 3 6 sin x cosx M + N = 5 6 4M N = 3 6 M = 2 48 N = 7 48 y = 24 e 4x 7 48 e x 3 6 sin x cos x

12 9. Tell whether the series converges or diverges and justify your answer by showing reason by a valid test. (a) n= ( ) n 3n + Converges; Alternating with b n = 3n+ decreasing and lim n b n =. (b) n= 2 3n 5 n Diverges; Geometric with r = 8 5 > (c) n= 4 n + Diverges; Comparison to 4 dx = lim 4 ln (x + ) = x + x (d) n= 3 n Converges; Comparison to dx = lim x arctanx = 3π x 2

13 2. Determine the given sum: (a) 5 2 n n= 3 n /3 2 3 = (b) ( ) n n= n! e = e (c) n= 2 n 2 + 4n + 3 n= ( n + ) = 3 n (d) n= 4 n! 4e

14 2. Determine the Taylor Series about x = for: (a) f(x) = + x ( ) n x n n= (b) g(x) = ( + x) 2 n= n( ) n x n (c) h(x) = + x 2 ( ) n x 2n n= (d) k(x) = arctan x n= ( ) n 2n + x2n+

15 22. Determine the fourth degree Taylor Polynomial about x = for the function f(x) = + x. + x = c + c x + c 2 x 2 + c 3 x 3 + c 4 x 4 2 ( + x) /2 = c + 2c 2 x + 3c 3 x 2 + 4c 4 x 3 4 ( + x) 3/2 = 2c c 3 x + 4 3x ( + x) 5/2 = 3 2c c 4 x 5 6 ( + x) 7/2 = 4 3 2c 4 + x + 2 x 8 x2 + 6 x x4 23. Determine the fourth degree Taylor Polynomial about x = for the function f(x) = ln x. ln x = c + c (x ) + c 2 (x ) 2 + c 3 (x ) 3 + c 4 (x ) 4 x = c + 2c 2 (x ) + 3c 3 (x ) 2 + 4c 4 (x ) 3 x 2 = 2c c 3 (x ) + 4 3(x ) 2 2 x 3 = 3 2c c 4 (x ) 6 x 4 = 4 3 2c 4 ln x (x ) 2 (x )2 + 3 (x )3 (x )4 4

16 24. Determine the interval and radius of convergence of the given series: n= (x ) n 3 n. 2 < x < 4 R = Determine the interval and radius of convergence of the given series: n= (x ) n. n x < 2 R =

17 26. Given points P(, 4, 6) and Q( 3, 6, 7) and R( 4, 7, 6), (a) determine the angle θ between PQ and PR. PQ = 2, 2, and PR = 3, 3, 2 cosθ = ( 9)( 62) = θ = 9 (b) Determine the equation of the plane which contains the points P, Q, and R. i j k = 27 i 27 j + 2 k 9x + 9y 4z = 3 (c) Determine the volume of the parallelopiped formed by the vectors: and OR where O is the origin (,, ). OP, OQ = ( 36 49) 4(8 + 28) + 6( ) =

18 27. Change coordinates: (a) from rectangular coordinates to cylindrical coordinates. i. P( 3, 3, 6) (3 2, 3π4 ), 6 ii. z = 4x 2 + 4y 2 z = 2r (b) from spherical coordinates to rectangular coordinates. i. P(4, π/3, π/4) ( 2, 6, 2 2) (c) from rectangular to spherical coordinates. i. x 2 + y 2 + z 2 = 9 ρ = 3

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