'01 Fall - Exam 1
(1) (15 pts) (a) The following is a table of values for the function f (x) = 2x/(x2 + 1). Compute the slopes of the secant lines through each of these points and the point (0, 0). Use your table to estimate the slope of the tangent line to the graph y = f (x) at (0, 0). x 0.1 0.01 0.001 0.0001 f (x) 0.198020 0.019998 0.002000 0.000200 (b) Using your slope estimate from problem (a) above, write down the equation of the tangent line to y = f (x) at the point (0, 0). (2) (15 pts) Given the function f (x) = x/(x2 - x - 2) (a) Find the equations of all horizontal and vertical asymptotes of the graph. (b) Draw the graph of y = f (x), showing all the asymptotes. (3) (16 pts) f Let (x) = x - 1. (a) Calculate the derivative f (x) directly from the definition. (b) Find the domain of f (x) and of f (x). (4) (24 pts) Consider the function 3x, -2 x < -1 g(x) = 2 + x2 , -1 x < 0 x e + 1, 0 x 1 (a) (b) (c) (d)
Find all points x at which g(x) is discontinuous. Explain. Find all points x at which g(x) is not differentiable. Explain. Draw the graph, clearly labelling these points....