2.2C Derivative as a Rate of Change

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Presentation transcript:

2.2C Derivative as a Rate of Change

Average Velocity: the slope of the secant line between [a, b] on a position curve s(t).

Instantaneous Velocity: the slope of the tangent line to a point (a, s(a)) on a position curve.

That is … f(x) = the slope of the tangent line at x s(a) = the velocity at t = a | s(a)| = the speed at t = a

Ex 1: t: min 36 38 40 42 44 s(t): Heartbeats 2530 2661 2806 2948 3080 Use the data to calculate the slope of the secant line between each of the given points and the point: (42, 2948).

Ex 1: Estimate the slope of the tangent line @ t = 42 min Find the equation of the tangent line @ t = 42 min

Ex 2: C(x): where C = cost in dollars & x = fabric in yards What are the units of C(x)? What does C(1000) = 9 mean?

Ex 3: The displacement (meters) of a particle moving in a straight line is given where t is seconds.

Ex 3: a) Find the average velocity. on the interval [4, 4 Ex 3: a) Find the average velocity on the interval [4, 4.5] b) Find the instantaneous velocity at x = 4

2.2C pg. 117 # 89 – 95 odds, # 97 – 100 all, 107, 113

Ex 4: a) Find the velocity & speed of the particle after 2 seconds. s: meters t: seconds

Ex 4: b) Will the particle ever stop? If so, when? s: meters t: seconds

Ex 4: c) When is the particle moving away from its starting point? s: meters t: seconds

The derivative of f at x is f’(x):