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2311lectureoutline2.8 - • a sharp corner • a...

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MAC2311, Calculus I 2.8 The Derivative as a Function We will now look at finding the general derivative of a function. In other words, we will let the number a vary. Definition: The derivative of a function f(x) is defined as 0 ( ) ( ) ( ) lim h f x h f x f x h + - = , provided the limit exists. Alternative notations for the derivative include: ( ) ( ) ( ) ( ) x dy df d f x y f x Df x D f x dx dx dx = = = = = = Definition: A function f is differentiable at a if f’(a) exists. It is differentiable on an open interval (a, b) [or ( ) ( ) , , ( , ) a or a or -∞ -∞ ∞ ] if it is differentiable at every number in the interval. Theorem: If f is differentiable at a , then f is continuous at a . A function is NOT differentiable wherever its graph has:
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Unformatted text preview: • a sharp corner • a discontinuity • a vertical tangent line Suppose that f(x) is a continuous function, then… • when f(x) is increasing, f’(x) will be positive • when f(x) is decreasing, f’(x) will be negative • when f(x) is horizontal, f’(x) will be 0 The second derivative of f can be denoted by any of these… 2 2 2 2 ( ) ( ) d y d f x f x dx dx ′′ Similarly, with the subsequent derivatives: f ′′′ represents the third derivative of f y (4) represents the fourth derivative of f f (n)( x) represents the nth derivative of f(x) Try Exercises 2,6,8,22,28,34,36,42,52 on pp. 162-165 in the text....
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