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Basic Differentiation Rules and Rates of Change Part 3 (Section 2-2)

Basic Differentiation Rules and Rates of Change Part 3 (Section 2-2). Derivative can be used to determine rate of change of one variable with respect to another. population growth rate production rates water flow rates velocity acceleration object moving in a straight line

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Basic Differentiation Rules and Rates of Change Part 3 (Section 2-2)

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  1. Basic Differentiation Rules and Rates of Change Part 3 (Section 2-2)

  2. Derivative can be used to determine rate of change of one variable with respect to another. • population growth rate • production rates • water flow rates • velocity • acceleration • object moving in a straight line • up/right = positive direction • down/left = negative direction • Position function – function, s, gives the position (relative to the origin) of an object as a function of time, t

  3. Displacement: of the object over the interval from t to t + ∆t is Average Velocity: of the object over the time interval is

  4. Example 1 Use the position function s(t) = -16t2 + v0t + s0 for free falling objects. If a billiard ball is dropped from a height of 100 feet, its height s at time t where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals. a) [1, 2] b) [1, 1.5]

  5. Instantaneous Rates of Change: of f with respect to x at a is the derivative provided the limit exists.

  6. Example 2 • Find the rate of change of the area A of a circle with respect to the radius r. • b) Evaluate the rate of change of A at r=5 and at r=10. • c) If r is measured in inches and A is measured in square inches, what units would be appropriate for ?

  7. Instantaneous Velocity: is the derivative of the position function s=f(t) with respect to time. At time t is velocity is • Velocity shows direction of motion… • Positive is moving forward/increasing • Negative is moving backward/decreasing Speed: the absolute value of velocity

  8. Example 3 At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by • s(t) = -16t2 + 16t + 32 • where s is measured in feet and t is measured in seconds. • When does the diver hit the water? • What is the diver’s velocity at impact?

  9. Example 4 A particle moves along a line so that its position at any time t≥0 is given by the function s(t)=t2+1, where s is measured in meters and t is measured in seconds. • Find the displacement during the first 10 seconds. • Find the average velocity during the first 10 seconds. • Find the instantaneous velocity when t=5.

  10. Example 5 A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s=160t-16t2 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?

  11. Example 5 A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s=160t-16t2 after t seconds. c) When does the rock hit the ground?

  12. HW # 28 pgs 116-118 (73, 77, 79, 89, 91, 93-97 all, 99, 103, 104, 109, 113, 114)

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