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2311lectureoutline2.7

# 2311lectureoutline2.7 - f’(a is lim h f a h f a f a h...

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MAC2311, Calculus I 2.7 Derivatives and Rates of Change We stated earlier that the slope of a secant line through f(x) at x and a can be used to estimate the slope of a tangent line using ( ) ( ) f x f a m x a - = - . We will now use what we have learned about limits to find the slope of a tangent line exactly. Definition: The tangent line to the curve y= f(x) at the point P(a,f(a)) is the line through P with slope ( ) ( ) lim x a f x f a m x a - = - provided that this limit exists. Another expression for the slope of a tangent line to f(x) at the point P(a,f(a)) is 0 ( ) ( ) lim h f a h f a m h + - = . This expression is sometimes easier to use. Definition: The derivative of a function f at a number a , denoted by
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Unformatted text preview: f’(a), is ( ) ( ) ( ) lim h f a h f a f a h → +-′ = provided this limit exists. An equivalent statement: ( ) ( ) ( ) lim x a f x f a f a x a →-′ =-. In General: • The derivative of a function at a, denoted , f’(a), gives us the slope of the tangent line at a . • The derivative of a function f(x) at a, denoted f’(a), also gives us an instantaneous rate of change. • If s(t) represents the position of an object at time t , then the s’(t) gives us the velocity of the object at time t –- the instantaneous rate of change in s(t) at time t . Try Exercises 4, 10ab, 16ab, 18, 30, 32, 34, 38, 46a on pages 150-152 in the textbook....
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