advertisement

Math 152 “FREE” Final Exam Review Compiled by Ben Aurispa and Amy Austin 1. Integrate the following. Z x √ (a) dx x−9 Z 8x − 7 (b) dx (x − 1)(x2 + 4) Z 1 √ dx (c) 2 x x2 − 4 Z (d) sec7 θ tan3 θ dθ (e) Z sin2 (ln x) dx x Z x7 cos(x4 ) dx Z ∞ 1 3 dx − (g) 3x + 4 x + 7 0 Z 1 1 (h) dx 4x − 1 0 Z ∞ 5 + e−2x dx diverges. 2. Show x 1 (f) 3. Set up integrals to find the following. (a) The area bounded by the curves y = √ x, y = x2 , x = 0, and x = 4. (b) The area bounded by the curves x = 2y 2 , x + y = 1. π (c) The volume of the solid found by rotating the region bounded by y = tan x, y = 0, x = 0 and x = about 4 x-axis. √ (d) The volume of the solid found by rotating the region bounded by y = x, x = 0 and y = 4 about the y-axis. (e) The volume of the solid found by rotating the region bounded by y = 4 − x2 and y = 3 about the line x = 4. (f) The volume of the solid found by rotating the region bounded by y = x2 and y = 2x + 3 about the line y = −2. (g) The volume of the solid whose base is the triangular region with vertices (0, 0), (3, 0), and (0, 4) and whose cross sections perpendicular to the y-axis are semicircles. (h) The length of the curve x = y 3 − y from y = 2 to y = 3. (i) The surface area of the solid found by rotating the curve y = √ x, 0 ≤ x ≤ 9 about the x-axis. (j) The surface area of the solid found by rotating the curve y = t3 , x = t2 , 1 ≤ t ≤ 3 about the y-axis. 4. Find the average value of the function f (x) = cos2 x on the interval [0, π4 ]. 5. A spring has a natural length of 3 ft. A force of 30 lbs is required to hold the spring stretched to a length of 5 ft. How much work is required to stretch the spring from 4 ft to 8 ft? 6. A trough is 20 m long and has semicicular ends with height 9 m. The trough has a 2 m spout at the top and is full of water. Set up an integral to find the work necessary to pump all the water out of the spout. 1 7. Determine whether the following series converge or diverge. Justify your answer. If the series converges, does it also converge absolutely? ∞ X (a) n2 √ n7 + 12 n=1 (b) ∞ X (2n)! (n!)2 n=0 (c) ∞ X (−1)n √ 3 n2 + 5 n=0 (d) ∞ X (−1)n+1 n ln n n=2 (e) ∞ X ln n n2 n=3 8. Find the sums of the following series, if possible. ∞ X 2 2 − (a) n+1 n n=3 (b) ∞ X 3(−2)n+1 5n n=2 (c) ∞ X (−1)n+1 42n+1 32n (2n)! n=0 9. Find the radius and interval of convergence for the series ∞ X 32n (3x − 1)n √ n n=3 10. Express the following functions as Maclaurin series. 7x 5 − x3 (b) x sin(3x5 ) (a) (c) x4 ln(1 + 4x2 ) 1 (d) (1 − 3x)2 11. Integrate the following as a power series. Z 6 x4 e−x dx 12. Find the Taylor series for the function f (x) = 1 centered at x = 1. (7x + 1)2 13. Consider the triangle formed by the vertices P (−1, −5, 2), Q(−4, −2, 1), and R(−1, 0, 2). (a) Find the angle at vertex Q. (b) Find the area of this triangle. (c) Find a unit vector perpendicular to this triangle. −−→ −→ (d) Find the vector and scalar projection of P Q onto P R 14. Find the volume of the parallelipiped determined by the vectors h1, 0, 2i, h0, 2, 3i, and h8, −5, 6i. 2