Test Sample 1A, Math 1501F, Fall 2005
Problem 1. Let f (x) = 5x - 4 and g(x) = |x - 1| (a) Compute (f g)(2); (b) Find the largest possible domain for f g. Problem 2. Answer the following questions on limits. (a) Give a precise definition of the meaning of the statement limxc f (x) = L; (b) Give an - proof that limx-2 (x2 - 1) = 3. Problem 3. Compute the following limits, if they exist. Show your work. x2 - x + 2 ; x-2 x-1 x2 + x - 12 (b) lim ; x3- x-3 x2 + x - 2 (c) lim 2 ; x1 x - 2x + 1 tan(3x) (d) lim 2 . x0 2x + 5x (a) lim
Problem 4. Let
1, 2 x , f (x) = 1, x,
x0 0<x<1 1x<2 2 x.
Sketch the graph of f (x) and the classify discontinuities (if any) as being a removable discontinuity, a jump discontinuity, or an infinite discontinuity. Problem 5. Show that the equation x3 - 4x + 2 = 0 has three distinct roots in [-3, 3] and locate the roots between consecutive integers. Problem 6. Find the points where the tangent to the graph of f (x) = -x2 - 6 is parallel to the line y = 4x - 1. Problem 7. A stone is thrown upward from ground level. The initial speed is 32 feet per second. (a) In how many seconds will the stone hit the ground? (b) How high will it go? (c) With what minimum speed should the stone be thrown so as to reach a height of 36 feet?