AP Calculus Slides October 17, 2007
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The document provides information about the height of a ball thrown upwards over time. It can be summarized as: 1) The height of a ball t seconds after being thrown is given by an equation involving t and the ball's initial upward velocity. 2) The document asks questions about determining the ball's starting height, initial velocity, whether its height was increasing or decreasing at a given time, and the time and height at which it reached its maximum. 3) It also provides information about the derivative of a function and asks questions relating the derivatives of related functions.
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AP Calculus Slides October 17, 2007
- 1. The height of a ball (in feet) t seconds after it is thrown is given by: it's (upward) velocity at time t is: (a) From what height was the ball thrown? (b) What is the ball's initial velocity? Was it thrown up or down? How can you tell? (c) Was the ball's height increasing or decreasing at time t = 2? (d) At what time did the ball reach it's maximum height? How high was it then? (e) How long was the ball in the air?
- 2. The graph of a function ƒ is shown below. Rank the values of ƒ'(-3), ƒ'(-2), ƒ'(0), ƒ'(1) and ƒ'(4) in increasing order.
- 3. Suppose that: and that ƒ'(x) exists for all x. (a) Explain how the graphs of ƒ and g are related. (b) How is the graph of g' related to the graph of ƒ'? Explain. (c) If ƒ'(1) = 5, what is g'(1)? (d) Let h(x) = ƒ(x) + k where k is constant. How is the graph of h' related to the graph of ƒ'?
- 4. Suppose that ƒ'(3) = -2. (a) Assume that ƒ is an even function. Explain why ƒ'(-3) = 2. (b) Now assume that ƒ is an odd function. What is ƒ'(-3)?
- 5. Suppose that ƒ'(x) = 1 for all x. Find an algebraic expression for ƒ(x) assuming that: (a) ƒ(0) = 2 (c) ƒ(0) = -2 (b) ƒ(1) = 2 (d) ƒ(-1) = -2
- 6. Suppose that ƒ(a) = b and ƒ'(x) = c for all x. Find an algebraic expression for ƒ(x) HOMEWORK