McMaster University Math 1A03. Fall 2010 Practice Midterm 2. Duration: 90
minutes. Instructors: L. Barto, D. Haskell, J. Marikova. 1) Find the general form of
the ...
McMaster University Math 1A03 Fall 2010 Practice Midterm 2 Duration: 90 minutes Instructors: L. Barto, D. Haskell, J. Marikova 1) Find the general form of the following indefinite integrals. Z √ a) x 1 − 9x2 dx Z √ b) 1 − 9x2 dx Z π/3 cos(2x) sin(3x) dx. 2) Find π/4 Z 2x4 + 7x2 − 4x + 3 dx. 3) Find the general form of the indefinite integral: 2x3 + 6x Z 2 x2 ex dx. 4) Consider 0
a) Use Simpson’s rule with n = 4 to approximate the value of the integral. b) Find an upper estimate for the error of the approximation given by the midpoint rule with n = 100. c) Find the precise value of the integral. 5) In the following cases decide whether there exists a continuous differentiable function f satisfying the assumptions. Do not explain your answer. (a) f has an absolute maximum at 1 and f 0 (1) = 1. (b) f 0 (1) = 0 and f has no local maximum or minimum at 1. (c) f 0 (x) > 2 for all x, f (1) = 2 and f (3) = 5. (d) f 00 (x) < 0 for all x, f (1) = 1, f (2) = 1 and f (3) = 3. (e) f 0 (x) < 0 for all x, f (1) = 1 and f (2) = 2. (f) f 0 (1) = 0, f 00 (1) = 3 and f has no local maximum or minimum at 1. (g) f 00 (x) < 0 for all x, f (1) = 1 and f (2) = 3. (h) f (1) = 0, f 0 (1) = 3 and f (3) = 1. 6) Find the local and global extremes of the function f (x) = |x|+2 sin x on the interval [−π, π]. Determine where the function is increasing/decreasing and concave upward/downward.
