Difference between revisions of "Differentiable function"
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(New page: A function f(x) is '''differentiable''' at the point ''a'' if and only if, as ''x'' approaches ''a'' (which it is never allowed to reach), the value of the quotient: :<math>\frac{f(x) - f...) |
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| − | A function f(x) is '''differentiable''' at the point ''a'' if and only if, as ''x'' approaches ''a'' (which it is never allowed to reach), the value of the quotient: | + | A [[function]] f(x) is '''differentiable''' at the point ''a'' if and only if, as ''x'' approaches ''a'' (which it is never allowed to reach), the value of the quotient: |
:<math>\frac{f(x) - f(a)}{(x - a)}</math> | :<math>\frac{f(x) - f(a)}{(x - a)}</math> | ||
| − | approaches a | + | approaches a [[limit]]ing value that we call the [[Derivative (calculus)|derivative]] of the function f(x) at ''x=a''. |
| − | + | ||
| − | [[ | + | There is also the more rigorous <math>\epsilon-\delta</math> definition: a function f is said to be differntiable at point ''a'' if ∀<math>\epsilon>0</math> ∃<math>\delta>0</math> such that if |
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| + | ::<math> |x - a| < \delta\,</math> | ||
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| + | then | ||
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| + | ::<math>|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,</math>. | ||
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| + | [[Category:Calculus]] | ||
Latest revision as of 21:18, September 8, 2020
A function f(x) is differentiable at the point a if and only if, as x approaches a (which it is never allowed to reach), the value of the quotient:
- <math>\frac{f(x) - f(a)}{(x - a)}</math>
approaches a limiting value that we call the derivative of the function f(x) at x=a.
There is also the more rigorous <math>\epsilon-\delta</math> definition: a function f is said to be differntiable at point a if ∀<math>\epsilon>0</math> ∃<math>\delta>0</math> such that if
- <math> |x - a| < \delta\,</math>
then
- <math>|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,</math>.