Calculus Cheat Sheet Part 2

Calculus Cheat Sheet Part 2 - 15 16a f(x exists on the...

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15. Suppose that ) ( x f exists on the interval ( 29 b a , 1. If 0 ) ( x f in ( 29 b a , , then f is concave upward in ( 29 b a , . 2. If 0 ) ( < x f in ( 29 b a , , then f is concave downward in ( 29 b a , . To locate the points of inflection of ) ( x f y = , find the points where 0 ) ( = x f or where ) ( x f fails to exist. These are the only candidates where ) ( x f may have a point of inflection. Then test these points to make sure that 0 ) ( < x f on one side and 0 ) ( x f on the other. 16a. If a function is differentiable at point a x = , it is continuous at that point. The converse is false, in other words, continuity does not imply differentiability. 16b. Linear Approximations The linear approximation to ) ( x f near 0 x x = is given by ) )( ( ) ( 0 0 0 x x x f x f y - + = for x sufficiently close to 0 x . 17. L’Hôpital’s Rule If ) ( ) ( lim x g x f a x is of the form or 0 0 , and if ) ( ) ( lim x g x f a x exists, then ) ( ) ( lim ) ( ) ( lim x g x f a x x g x f a x = . 18. Inverse function 1. If g f and are two functions such that x x g f = )) ( ( for every x in the domain of g and x x f g = )) ( ( for every x in the domain of f , then f and g are inverse functions of each other. 2. A function f has an inverse if and only if no horizontal line intersects its graph more than once. 3. If f is either increasing or decreasing in an interval, then f has an inverse. 4. If f is differentiable at every point on an interval I , and 0 ) ( x f on I , then ) ( 1 x f g - = is differentiable at every point of the interior of the interval ) ( I f and ) ( 1 )) ( ( x f x f g = .
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