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2.2 The Derivative as a Function

# 2.2 The Derivative as a Function - = 1 1 U x x x =< ≥...

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Derivative as a Function Section 2.2

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Definition of the Derivative of a Function The derivative of f at x is given by provided the limit exists. 0 ( ) ( ) '( ) lim h f x h f x f x h + - =
Notation for Derivatives ) ( ' x f dx dy ' y [ ] ) ( x f dx d [ ] y D x df dx

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(Alternate)Derivative at a Point The derivative of the function f at the point x = a is the limit provided the limit exists. ( ) ( ) '( ) lim x a f x f a f a x a - = -
Higher Derivatives

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Jerk l Jerk is the derivative of acceleration. l It is the 3rd derivative of the position function. l If a body’s position at time t is s(t), the body’s jerk at time t is 3 ( ) s da d s j t dt dt = =
Cases Where a Function is not Differentiable l A corner, where the one-sided derivatives differ. l A cusp, where the slopes of the secant lines approach from one side and from the other side. ( ) f x x = -∞ 2 3 ( ) f x x =

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l A vertical tangent, where the slopes of the secant lines approach either from both sides. l A discontinuity (which will cause one or both of the one-sided derivatives to be nonexistent. or -∞ 3 ( ) f x x = ( )

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Unformatted text preview: = ( ) 1, 1, U x x x = -< ≥ Thm. 2.1 Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c. It is possible for a function to be continuous at x = c and not be differentiable at x = c. Thus, continuity does not imply differentiability. One-Sided Derivatives l A function y = f(x) is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits exist at the endpoints. l The usual relationship between one-sided and two-sided limits holds for derivatives. A function has a (two-sided) derivative at a point iff the function’s right-hand and left-hand derivatives are defined and equal at that point. Joke Time What would America be called if we all drove pink cars? A pink carnation! Why should you never play cards in the jungle? It’s full of cheetahs!...
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2.2 The Derivative as a Function - = 1 1 U x x x =< ≥...

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