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Differentiability and continuity

Differentiability and continuity

We explore the notion of differentiability at a point and its relationship with continuity, using the definition of a derivative. If a function is differentiable at x = c, then it is continuous at x = c. However, a function being continuous does not guarantee that it is differentiable, as demonstrated with an absolute value function.

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Definition of the Derivative

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Differentiability at a point: graphical

Differentiability at a point: graphical

2016年8月18日

Differentiability at a point: algebraic (function isn't differentiable)

Differentiability at a point: algebraic (function isn't differentiable)

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Derivatives of inverse functions: from equation

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Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

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SOLVED: For Problems 7-21, verify that the given function is a solu- tion to the given differential equation (C1 and c2 are arbitrary constants) , and state the maximum interval over which the solution is valid7. Y6) = C1e-Sr czesr, Y"' - 25y =0. 8. y(r) = C1cos 2x €2 sin 2x, Y" 4y=0.9. Y(x) #cet c2e-2r Y Y -2 =010. y(r) = Y=- X 4- 11. ylr) = c1xl/ Y-r 2 12. y(r) = e-* sin 2x , Y" 2y Sy=0. 13. y(x) = €1 cosh 3x c2 sinh 3x , y" -9y = 0. 14. y(r) =c1X 3 cr-! , xy" Sxy' 3y = 0. 15. y

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