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lecture12

# lecture12 - x 2 x> 1 6 ± What do you note about...

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Lecture 12: Derivatives and Rates of Change (Sections 2.7 and 2.8) Def. The derivative of a function f at x = a is de ned to be if the limit exists. Derivative and Slope of a Tangent line The equation of the tangent line to y = f ( x ) at x = a :

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ex. Find the equation of the tangent line to f ( x ) = x 2 2 x + 1 at (2 ; 1). 6 - ?
The derivative as a function Def. Given y = f ( x ), f 0 ( x ) = The derivative is itself a function of f . Its domain: Other notations for the derivative: Process of nding the derivative is called

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ex. Find the function f 0 ( x ) for f ( x ) = x 2 x . What is its domain?
NOTE: for a given x -value, f 0 ( x ) gives the slope of the tangent line to the graph of f at the point ( x; f ( x )). ex. Given the graph of f ( x ), sketch a possible graph of its derivative. 6 - ? y = f ( x ) 6 - ? y = f 0 ( x )

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Def. A function f is di erentiable at x = a if f 0 ( a ) exists. It is di erentiable on an open interval if it is di erentiable at each number in the interval. ex. Find each interval for which f ( x ) = x 2 x is di erentiable.
ex. Let f ( x ) = ( 2 x x 1 x 2 x > 1 . 1) Find f 0 (1) if possible. 2) Find a formula for f 0 ( x ).

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ex. Sketch the graph of f ( x ) =

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Unformatted text preview: x 2 x > 1 . 6-? ± What do you note about continuity and di±erentia-bility of f ( x ) at x = 1? Theorem: If f is di±erentiable at x = a , then f is continuous at x = a . When is a function not di±erentiable at a point? 6-? ± 6-? ± 6-? ± Def. The graph of a function f ( x ) has a vertical tangent line at x = a if ex. Find f ( ± 1) if f ( x ) = ( x + 1) 1 = 3 . ex. Let f ( x ) = j x j . Use the limit de±nition to show that f ( x ) is not di²erentiable at x = 0. ex. Let g ( x ) = x j x j . Show that g ( x ) is both con-tinuous and di±erentiable at x = 0. Higher Derivatives If y = f ( x ) is di±erentiable, we can ²nd its deriva-tive, a new function called f 00 ( x ). The limit de²nition: In the same way, the derivative of f 00 ( x ) is f 000 ( x ), and in general, we denote the n th derivative of f as f n ( x ). Other notation:...
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lecture12 - x 2 x> 1 6 ± What do you note about...

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