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Lesson 2 6: Diﬀerentiability: Afunctionisdiﬀerentiable at a point if it has a derivative there In other words: The function f is diﬀerentiable at x if

AP Calculus Limits, Continuity, and Differentiability

Limits, Continuity, and Differentiability Reference Page With Associated Question Numbers Existence of a Limit at a Point (#5, 9, 13, 14, 17) A function f ()x has a limit Las xapproaches cif and only if the left-hand and right-hand limits at cexist and are equal 1 lim ( ) xc f x exists 2 lim ( ) xc f x exists 3 lim ( ) xc f x = lim ( ) xc f x

Complex differentiability - mathdrexeledu

Complex differentiability Let f = u+iv be a complex-valued function de ned in an open subset G of the complex plane, and let z0 = x0 +iy0 be a point of G: Complex ﬀtiability We say that f(z) is ﬀtiable at z0 if there exists f′(z 0) = lim z→z0 f(z)−f(z0) z −z0: Thus f is ﬀtiable at z0 if and only if there is a complex number c

DIFFERENTIABILITY IN SEVERAL VARIABLES: SUMMARY OF BASIC CONCEPTS

2 DIFFERENTIABILITY IN SEVERAL VARIABLES: SUMMARY OF BASIC CONCEPTS then f is diﬀerentiable In other words : (4) C1) Diﬀerentiable yet the converse is not true Example 6 The function f from Example 2 satis

01 Limits and Continuity and Differentiability

Discuss differentiability algebraically and graphically and know its relation to limits and continuity Recognize the limit definition of derivative and be able to identify the function involved and the point at which the derivative is evaluated For example, since , recognize that is simply the derivative of cos(x) at

NOTES ON QUANTITATIVE RECTIFIABILITY AND DIFFERENTIABILITY

NOTES ON QUANTITATIVE RECTIFIABILITY AND DIFFERENTIABILITY 3 It follows directly that the same is true for maps to Rk In fact, intervals like this are abundant: the notions of †–efﬁcient, †–coarse differentiable are scale-invariant, so both of these results apply to a Lipschitz function on any interval

The Derivative and the Tangent Line Problem

differentiability and continuity The derivative of f at c is provided this limit exists (see Figure 3 10) Differentiability and Continuity Alternative form of derivative As x approaches c, the secant line approaches the tangent line Figure 3 10 Note that the existence of the limit in this alternative form requires that the one-sided limits

Continuity and Differentiability - Classwork

Continuity and Differentiability - Classwork Back in our precalculus days, we dabbled in the concept of continuity We reached a very informal definition of continuity: a curve is continuous if you can draw it without taking your pencil from the paper This is a good "loose" definition but when one examines it closely, it is filled with holes