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c Math 151 WIR, Spring 2013, Benjamin Aurispa Math 151 Week in Review 5 Sections 3.2 & 3.3 1. Differentiate the following functions. √ 1 + π2 (a) f (x) = 9x + 3 x + √ 53x (b) f (x) = (5x5 − 7x3 + 9x + (c) h(x) = 5x2 + 7x x3 − x (d) g(t) = 4t4 + 3t − 2 √ t √ 5)(3x6 − 10x2 + e5 + cos 3) 1 c Math 151 WIR, Spring 2013, Benjamin Aurispa 2. Find the equation of the tangent line to the graph of f (x) = x2 x at x = 1. +5 3. Given the graphs of f (x) and g(x) below calculate the following. (a) h′ (1) where h(x) = g(x)[f (x) + x2 ] (b) u′ (3) where u(x) = f (x) xg(x) 2 c Math 151 WIR, Spring 2013, Benjamin Aurispa 4. Consider the function f (x) = 2x(x2 − 1). (a) Find the values of x for which the tangent line to the graph of f is horizontal. (b) Find the values of x for which the tangent line to the graph of f is parallel to the line 8x − 2y = 9. 5. For what values of a and b is the line y = 3x + b tangent to the graph of f (x) = ax2 when x = −2. 6. At what points on the graph of f (x) = −x2 + 4 does the tangent line also pass through the point (1, 7)? 3 c Math 151 WIR, Spring 2013, Benjamin Aurispa 7. Find f ′ (x) for the function below. Where is f not differentiable? 4x + 11 6 − x2 f (x) = −2x + 7 2 x −8 if if if if x ≤ −2 −2<x<1 1≤x≤3 x>3 8. Given f (x) below, find the values of a and b that make f differentiable everywhere. f (x) = ( ax + b x2 − x if x ≤ 3 if x > 3 9. From the edge of a 80 ft building, a ball is thrown straight up into the air with a speed of 64 ft/s. After t seconds, its height from the ground is given by the function s(t) = −16t2 + 64t + 80. (a) What is the maximum height the ball reaches? (b) What is the ball’s velocity when it hits the ground? 4 c Math 151 WIR, Spring 2013, Benjamin Aurispa 10. An object moves in a straight line with position after t seconds given by the function s(t) = t3 −9t2 +24t where distance is measured in ft. Determine the total distance traveled by the object in the first 6 seconds. 11. (a) Determine the rate at which the area of a circle is changing with respect to the radius when r = 2 inches. (b) If the radius of the circle is changing at a rate of 3 inches/min, determine the rate at which the area is increasing after 2 minutes. 5