ࡱ>  "y bjbj 0,{{2  D N^qssssss$d-www:qwqww1!D&.]0 ,11,ww  : AP Calculus AB 3.4 Velocity and Other Rates of Change Objectives: Instantaneous Rates of Change; Motion along a line; Sensitivity to Change; Derivatives in Economics Procedure: Instantaneous Rates of Change: Definition 1: Instantaneous Rate of Change: The (instantaneous) rate of change of f with respect to x at a is the derivative Example 1: Enlarging circles: Find the rate of change of the area A of a circle with respect to its radius r. Evaluate the rate of change of A at r = 5 and at r = 10. If r is measured in inches and A is measured in square inches, what units would be appropriate for dA/dr? Motion along a Line: Suppose that an object is moving along a coordinate line (say an s-axis) so that we know its position s on that line as a function of time t: The __________________ of the object over the time interval from  EMBED Equation.3  and the ____________________________ of the object over that time interval is Definition 2: Instantaneous Velocity: The instantaneous velocity is the derivative of the position function  EMBED Equation.3  with respect to time. At time t the velocity is Definition 3: Speed: Speed is the absolute value of velocity. Example 2: Reading a velocity graph: A student walks around in front of a motion detector that records her velocity at 1-second intervals for 36 seconds. She stores the data in her graphing calculator and uses it to generate the time-velocity graph shown below. Describe her motion as a function of time by reading the velocity graph. When is her speed a maximum? Definition 4: Acceleration: Acceleration is the derivative of velocity with respect to time. If a bodys velocity at time t is  EMBED Equation.3 , then the bodys acceleration at time t is Free-fall Constants: English units: Metric units: Example 3: Modeling vertical motion: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of  EMBED Equation.3  ft after t seconds. How high does the rock go? What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? What is the acceleration of the rock at any time t during its flight (after the blast)? When does the rock hit the ground? Example 4: Studying particle motion: A particle moves along a line so that its position at any time t e" 0 is given by the function  EMBED Equation.3  where s is measured in meters and t is measured in seconds. Find the displacement of the particle during the first 2 seconds. Find the average velocity of the particle during the first 4 seconds. Find the instantaneous velocity of the particle when t = 4. Find the acceleration of the particle at t = 4. Describe the motion of the particle. At what values of t does the particle change directions? 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