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Math 1210-007 Fall 2013 Practice Midterm 1 Name : UID : Question 1 2 3 4 5 6 Total Points Out of 10 10 10 10 10 10 60 1. Compute the following limits (a) x−4 lim √ x→4 x−2 (b) lim x→5 x2 sin x − 7x + 10 (c) lim x→0 tan 4x x 2. (a) Give definition for continuity of a function f (x) at a point a. (b) Give definition for differentiability of a function f (x) at a point a. (c) Show that the function f (x) = 1 if x = 0 x sin x1 if x 6= 0 is continuous but not differentiable at 0. 3. (a) Show that x5 + x3 − 1 = 0 has a real solution between 0 and 1. (b) Find the vertical and horizontal asymptotes for the function √ . x x2 − 4 4. Find the derivatives for the following functions (no need to simplify your answer): (a) f (x) = sin(t2 + 1) (b) g(x) = 5x2 + x − 1 x−3 5. Take first, second and third derivative of the following function (a) h(x) = 2x3 − 5x + 10 (b) ξ(x) = cos 5x 6. Find the equation of the tangent line to the graph y 2 = x3 − x √ at (2, 6).