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math14 - 2 θ y = 13(sin θ-sin θ cos θ(Solution-39 4-39...

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Practice Exam for midterm III 1. Find the radius of convergence and the interval of convergence of n =1 ( - 1) n x n n +4 . (Solution: Radius of convergence is 1 and interval of convergence is ( - 1 , 1].) 2. Find a power series representation centered at 0 for f ( x ) = x 9+ x 2 . (Solution: f ( x ) = n =0 ( - 1) n x 2 n +1 9 n +1 3. Find the first 5 terms in the Taylor series representation centered at a = 1 for f ( x ) = x . (Solution: 1 + 1 2 ( x - 1) - 1 8 ( x - 1) 2 + 1 16 ( x - 1) 3 - 5 128 ( x - 1) 4 + · · · ) 4. Use Taylor series to evaluate the integral sin x x dx . (Solution: n =0 ( - 1) n x 2 n +1 (2 n +1)!(2 n +1) + C ) 5. Eliminate the parameter t to find a Cartesian equation of the curve x = 10 ln(9 t ) y = t (Solution: y = e x/ 10 9 or x = 10 ln(9 y 2 )) 6. Find an equation of the tangent line at the point corresponding to t = 1 for the curve x = e t y = t - ln( t 9 ) (Solution: ( y - 1) = - 16 e ( x - e )) 7. Find the points on the curve where the tangent is horizontal: x = 13(cos
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Unformatted text preview: 2 θ ) , y = 13(sin θ-sin θ cos θ ) (Solution: (-39 / 4 ,-39 √ 3 / 4), (-39 / 4 , 39 √ 3 / 4)) 8. Find the area of the surface obtained by rotating the curve about the x-axis x = a cos 3 θ, y = a sin 3 θ, ≤ θ ≤ π (Solution: 12 πa 2 / 5.) 9. Find the length of the curve x = t 1+ t , y = ln(1 + t ), 0 ≤ t ≤ 2. (Solution:-√ 10 / 3 + ln(3 + √ 10) + √ 2-ln(1 + √ 2).) 10. Find the slope of the tangent line to the polar curve r = 1 /θ at θ = π . (Solution:-π ) 11. Find the area bounded by the curve r = √ sin θ and lies in the sector 0 ≤ θ ≤ 2 π/ 3. (Solution: 3 / 4) 12. Find the length of the polar curve r = 7 cos θ for 0 ≤ θ ≤ 3 π/ 4. (Solution: 21 π/ 4) 1...
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