Math 2B Practice Final Exam Z 1. Suppose that Z 1 (a) 6f (x)dx 1 Z −1 1 4 Z 1 [2f (x) − 3x2 ]dx (b) −1 Z 4 f (x)dx (c) −1 d 2. Evaluate dx Z x2 sin(x) t3 tan(t)dt 4 f (x)dx = −2, evaluate the following integrals: f (x)dx = 6 and 3. Evaluate the following integrals: Z (a) 6x2 tan−1 (x)dx Z (b) 1 dx x ln(3x) Z (c) sin5 (θ) cos2 (θ)dθ Z √ (d) x2 − 25 dx x 2 Z (e) 0 Z (f) 0 1 dx (x − 2)2 ∞ z2 1 dz + 3z + 2 h πi 4. Find the average value of f (x) = sec2 (x) on 0, 4 5. Find the Maclaurin series for f (x) = (1 − x)−2 and find its radius of convergence. 11. Find thelimit of each sequence below if it exists. 2 n (a) an = +3 3 (b) bn = n3 e−n (c) cn = tan−1 (ln(n)) 12.Let R be the region bounded by the curves y = x2 + 1 and y = 3 − x (a) Find the area of R (b) Find the volume when R is rotated around y = 5 (c) Find the volume of the solid with base R and square cross-sections perpendicular to the x-axis. 13. Determine whether the following series are convergent or divergent and state any test used. ∞ X n (a) n3 + 1 n=1 ∞ X (−1)n √ (b) n+1 n=1 (c) ∞ X 2n n=1 n2 (d) ∞ X n2 e−n n=1 (e) ∞ X n=1 (f) 3n 1 −1 ∞ X cos(πn) n=1 (2n)!