1. Compute the following derivatives. (Simplify your answers when possible.) x (a) f� (x) where f(x) = 1 − x2 (b) f� (x) where f(x) = ln(cos x) − 1 sin2 (x) 2 x (c) f(5)(x), the fifth derivative of f, where f(x) = xe 2. Find the equation of the tangent line to the “astroid” curve defined implicitly by the equation x2/3 + y2/3 = 4 at the point (− √27, 1). 3. A particle is moving along a vertical axis so that its position y (in meters) at time t (in seconds) is given by the equation y(t) = t 3 − 3t + 3, t ≥ 0. Determine the total distance traveled by the particle in the first three seconds. 4. State the product rule for the derivative of a pair of differentiable functions f and g using your favorite notation. Then use the DEFINITION of the derivative to prove the product rule. Briefly justify your reasoning at each step. 5. ⎧ ⎪⎨ ⎪⎩ Does there exist a set of real numbers a, b and c for which the function f(x) = tan−1(x) x ≤ 0 ax2 + bx + c, 0 < x < 2 x3 − 1×2 + 5, x ≥ 2 4 is differentiable (i.e. everywhere differentiable)? Explain why or why not. (Here tan−1(x) denotes the inverse of the tangent function.) 6. Suppose that f satisfies the equation f(x + y) = f(x) + f(y) + x2y + xy2 for all real numbers x and y. Suppose further that f(x) lim = 1. x→0 x (a) Find f(0). (b) Find f� (0). (c) Find f� (x). MIT OpenCourseWare http://ocw.mit.edu 18.01SC Single Variable Calculus�� Fall 2010 �� For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.