Difference between revisions of "User:Temperal/The Problem Solver's Resource9"
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| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 9}} | | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 9}} | ||
==<span style="font-size:20px; color: blue;">Derivatives</span>== | ==<span style="font-size:20px; color: blue;">Derivatives</span>== | ||
− | + | This page will cover derivatives and their applications, as well as some advanced limits. The Fundamental Theorem of Calculus is covered on the [[User:Temperal/The Problem Solver's Resource10|integral page]]. | |
+ | ===Definition=== | ||
+ | |||
+ | *<math>\displaystyle\frac{df(x)}{dx}=\lim_{x->x_0}\frac{f(x_0)-f(x)}{x_0-x}</math>, where <math>f(x)</math> is a function continuous in <math>L</math>, and <math>x_0</math> is an arbitrary constant such that <math>x_0\subset L</math>. | ||
+ | |||
+ | *Multiple derivatives are taken by evaluating the innermost first, and can be notated as follows: <math>\frac{d^2f(x)}{dx^2}</math>. | ||
+ | |||
+ | *The derivative of <math>f(x)</math> can also be expressed as <math>f'(x)</math>, or the <math>n</math>th derivative of <math>f(x)</math> can be expressed as <math>f^{(n)}(x)</math>. | ||
+ | ===Rolle's Theorem=== | ||
+ | If <math>f(x)</math> is differentiable in the open interval <math>(a,b)</math>, continuous in the closed interval <math>[a,b]</math>, and if <math>f(a)=f(b)</math>, then there is a point <math>c</math> between <math>a</math> and <math>b</math> such that <math>f'(c)=0</math> | ||
+ | ====Extension: Mean Value Theorem==== | ||
+ | If <math>f(x)</math> is differentiable in the open interval <math>(a,b)</math> and continuous in the closed interval <math>[a,b]</math>, then there is a point <math>c</math> between <math>a</math> and <math>b</math> such that <math>f(b)-f(a)=f'(c)\cdot(b-a)</math>. | ||
+ | ===L'Hopital's Rule=== | ||
+ | <math>\lim \frac{f(x)}{g(x)}=\lim \frac{f'(x)}{g'(x)}</math> | ||
+ | |||
+ | Note that this inplies that <math>\lim \frac{f(x)}{g(x)}=\lim \frac{f^{(n)}(x)}{g^{(n)}(x)}</math> for any <math>n</math>. | ||
+ | ===Taylor's Formula=== | ||
+ | Let <math>a</math> be a point in the domain of the function <math>f(x)</math>, and suppose that <math>f^{(n+1)}(x)</math> (that is, the <math>n+1</math>th derivative of <math>f(x)</math>) exists in the neighborhood of <math>a</math> (where <math>n</math> is a nonnegative integer). For each <math>x</math> in the neighborhood, | ||
+ | |||
+ | <cmath>f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+...+\frac{f^{(n)}(a)}{n!}(x-a)^n+\frac{f^{(n+1)}(a)}{n!}(x-a)^{n+1}</cmath> | ||
+ | |||
+ | where <math>c</math> is in between <math>x</math> and <math>a</math>. | ||
[[User:Temperal/The Problem Solver's Resource8|Back to page 8]] | [[User:Temperal/The Problem Solver's Resource10|Continue to page 10]] | [[User:Temperal/The Problem Solver's Resource8|Back to page 8]] | [[User:Temperal/The Problem Solver's Resource10|Continue to page 10]] | ||
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Revision as of 10:30, 6 October 2007
DerivativesThis page will cover derivatives and their applications, as well as some advanced limits. The Fundamental Theorem of Calculus is covered on the integral page. Definition
Rolle's TheoremIf Extension: Mean Value TheoremIf L'Hopital's Rule
Note that this inplies that Taylor's FormulaLet
where |