# Lect16 - The Derivative as a Rate of Change Let y = f(x be...

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92.131 Lecture 16 1 of 8 Ronald Brent © 2009 All rights reserved. The Derivative as a Rate of Change Let ) ( x f y = be any function. Recall the difference quotient: h x f h x f ) ( ) ( + is the average rate of change of the function f over the interval ) , ( h x x + . Taking the limit at a point 0 x x = : h x f h x f x f h ) ( ) ( lim ) ( ' 0 0 0 0 + = is the instantaneous rate of change of the function f at the point 0 x x = .

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92.131 Lecture 16 2 of 8 Ronald Brent © 2009 All rights reserved. Motion along a line Suppose a particle moves along a straight line so that it’s position is ) ( t f s = . Then: D i s p l a c e m e n t : ) ( ) ( t f t t f s Δ + = Δ Average Velocity: t t f t t f t s v av Δ Δ + = Δ Δ = ) ( ) ( Instantaneous Velocity t t f t t f t s t d s d t v t t Δ Δ + = Δ Δ = = Δ Δ ) ( ) ( lim lim ) ( 0 0 S p e e d : t d s d t v = | ) ( | Acceleration: 2 2 ) ( t d s d t d v d t a = = J e r k : 3 3 2 2 ) ( t d s d t d v d t d a d t j = = =
92.131 Lecture 16 3 of 8 Ronald Brent © 2009 All rights reserved. Example: The height ) ( t s of a rock thrown vertically from the surface of the

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Lect16 - The Derivative as a Rate of Change Let y = f(x be...

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