AP_Questions_-_Applications_of_Derivative (2)

AP_Questions_-_Applications_of_Derivative (2) -...

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Applications of the Derivative 1. Let d cx bx ax x ) x ( P 2 3 4 . The graph of ) x ( P y is symmetric with respect to the y-axis, has a relative maximum at (0, 1), and has an absolute minimum at . 3 , q [1974 #2] (a) Determine the values of a, b, c, and d, and using these values write an expression for ) x ( P . (b) Find all possible values of q. 2. A ball is thrown from the origin of a coordinate system. The equation of its path is 2 m 2 x 000 , 1 e mx y . Where m is positive and represents the slope of the path at the origin. [1974 #5] (a) For what value of m will the ball strike the horizontal axis at the greatest distance from the origin? (b) For what value of m will the ball strike at the greatest height on a vertical wall located 100 feet from the origin? 3. Given the function defined by x sin x y for all x such that 2 3 x 2 . [non-calculator] [1975 #4] (a) Find the coordinates of all maximum and minimum points on the given interval. Justify your answer. (b) Find the coordinates of all points of inflection on the given interval. Justify your answer. (c) Sketch the graph of the function. 4. Given the function defined by x sin e y for all x such that 2 x . [non-calculator] [1976 #5] (a) Find the coordinates of all maximum and minimum points on the given interval. Justify your answer. (b) Sketch the graph of the function. (c) Write an equation for the axis of symmetry for this graph. 5. Consider the function f defined by 3 2 1 x ) x ( f for all real numbers x. [1977 #2] (a) For what values of x is the function increasing? (b) Find the x and y-coordinates of the relative maximum and minimum points. Justify your answer. (c) For what values of x is the graph concave upward? (d) Using the information found in parts a – c above, sketch the graph of f. 6. Let g and h by any two twice-differentiable functions that are defined for all real numbers and that satisfy the following properties for all x: (i) 1 x h x g 2 2
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(ii) 2 x h x g (iii) 0 ) x ( h (iv) 0 ) 0 ( g [1978 #7] (a) Justify that ) x ( h ) x ( g ) x ( h for all x.
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