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[[Image:4 fonctions du second degré.svg .3. right .3. thumb .3. 200px .3. The second derivative of a QUADRATIC FUNCTION is CONSTANT.]] ..
{{Petició de traducció|en|Second derivative|Usuari:Amical-bot/Matemàtiques/en|--[[Usuari:Gomà|Gomà]] ([[Usuari Discussió:Gomà|disc.]]) 17:50, 10 març 2010 (CET)}}
In CALCULUS , the ((((( second derivative ))))) of a FUNCTION &fnof ; is the DERIVATIVE of the derivative of &fnof ;. Roughly speaking , the second derivative measures how the rate of change of a quantity is itself changing ; for example , the second derivative of the position of a vehicle with respect to time is the instantaneous ACCELERATION of the vehicle , or the rate at which the VELOCITY of the vehicle is changing. ..
..
On the GRAPH OF A FUNCTION , the second derivative corresponds to the CURVATURE or concavity of the graph. The graph of a function with positive second derivative curves upwards , while the graph of a function with negative second derivative curves downwards. ..
..
(==) Notation (==) ..
{{ Details .3. Notation for differentiation}} ..
The second derivative of a function <math>f(x)\!</math> is usually denoted <math>f''(x)\!</math>. That is: ..
: <math>f'' = (f')'\!</math> ..
When using LEIBNIZ'S NOTATION for derivatives , the second derivative of a dependent variable (( y )) with respect to an independent variable (( x )) is written ..
: <math>\frac{d^2y}{dx^2}.</math> ..
This notation is derived from the following formula: ..
: <math>\frac{d^2y}{dx^2} \ ,=\ , \frac{d}{dx}\left(\frac{dy}{dx}\right).</math> ..
..
(==) Example (==) ..
Given the function ..
: <math>f(x) = x^3 ,\!</math> ..
the derivative of &fnof ; is the function ..
: <math>f'(x) = 3x^2.\!</math> ..
The second derivative of &fnof ; is the derivative of &fnof ;&prime ; , namely ..
: <math>f''(x) = 6x.\!</math> ..
..
(==) Relation to the graph (==) ..
[[Image:Animated illustration of inflection point.gif .3. 400px .3. thumb .3. A plot of <math>f(x) = \sin(2x)</math> from <math>-\pi/4</math> to <math>5\pi/4</math>. The tangent line is blue where curve is concave up , green where the curve is concave down , and red at inflection points (<math>0</math> , <math>\pi/2</math> , and <math>\pi</math>).]] ..
..
(===) Concavity (===) ..
The second derivative of a function &fnof ; measures the ((((( concavity ))))) of the graph of &fnof ;. A function whose second derivative is positive will be CONCAVE UP (sometimes referred to as convex) , meaning that the TANGENT line will lie below the graph of the function. Similarly , a function whose second derivative is negative will be CONCAVE DOWN (sometimes called simply &ldquo ;concave&rdquo ;) , and its tangent lines will lie above the graph of the function. ..
..
(===) Inflection points (===) ..
{{ main .3. Inflection point}} ..
If the second derivative of a function changes sign , the graph of the function will switch from concave down to concave up , or vice versa. A point where this occurs is called an ((((( inflection point ))))). Assuming the second derivative is continuous , it must take a value of zero at any inflection point , although not every point where the second derivative is zero is necessarily a point of inflection. ..
..
(===) Second derivative test (===) ..
{{ main .3. Second derivative test}} ..
The relation between the second derivative and the graph can be used to test whether a STATIONARY POINT for a function (i.e. a point where <math>f'(x)=0\!</math>) is a LOCAL MAXIMUM or a LOCAL MINIMUM. Specifically , ..
* If <math>\ f^{\prime\prime}(x) < 0</math> then <math>\ f</math> has a local maximum at <math>\ x</math>. ..
* If <math>\ f^{\prime\prime}(x) > 0</math> then <math>\ f</math> has a local minimum at <math>\ x</math>. ..
* If <math>\ f^{\prime\prime}(x) = 0</math> , the second derivative test says nothing about the point <math>\ x</math> , a possible inflection point. ..
The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity , but with a negative acceleration. Clearly the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time , the velocity will become negative and the vehicle will reverse. The same is true for the minimum , with a vehicle that at first has a very negative velocity but positive acceleration. ..
..
(==) Limit (==) ..
It is possible to write a single LIMIT for the second derivative: ..
: <math>f''(x) = \lim_{h->0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.</math> ..
The expression on the right can be written as a DIFFERENCE QUOTIENT of difference quotients: ..
: <math>\frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = \frac{\frac{f(x+h) - f(x)}{h} - \frac{f(x) - f(x-h)}{h}}{h}.</math> ..
This limit can be viewed as a continuous version of the SECOND DIFFERENCE for SEQUENCES. ..
..
(==) Quadratic approximation (==) ..
Just as the first derivative is related to LINEAR APPROXIMATIONS , the second derivative is related to the best QUADRATIC APPROXIMATION for a function &fnof ;. This is the QUADRATIC FUNCTION whose first and second derivatives are the same as those of &fnof ; at a given point. The formula for the best quadratic approximation to a function &fnof ; around the point (( x )) &nbsp ;=&nbsp ;(( a )) is ..
: <math>f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2.</math> ..
This quadratic approximation is the second-order TAYLOR POLYNOMIAL for the function centered at (( x )) &nbsp ;=&nbsp ;(( a )) . ..
..
(==) Generalization to higher dimensions (==) ..
..
(===) The Hessian (===) ..
{{ main .3. Hessian matrix}} ..
The second derivative generalizes to higher dimensions through the notion of second PARTIAL DERIVATIVES. For a function &fnof ;:((((( R )))))<sup>3</sup>&nbsp ;&rarr ;&nbsp ;((((( R ))))) , these include the three second-order partials ..
..
: <math>\frac{\part^2 f}{\part x^2} , \ ; \frac{\part^2 f}{\part y^2} , \text{ and }\frac{\part^2 f}{\part z^2}</math> ..
..
and the mixed partials ..
..
: <math>\frac{\part^2 f}{\part x \ , \part y} , \ ; \frac{\part^2 f}{\part x \ , \part z} , \text{ and }\frac{\part^2 f}{\part y \ , \part z}.</math> ..
..
These fit together into a SYMMETRIC MATRIX known as the ((((( Hessian ))))). The EIGENVALUES of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the SECOND PARTIAL DERIVATIVE TEST.) ..
..
(===) The Laplacian (===) ..
{{ main .3. Laplace operator}} ..
Another common generalization of the second derivative is the ((((( Laplacian ))))). This is the differential operator <math>\nabla^2</math> defined by ..
: <math>\nabla^2 f = \frac{\part^2 f}{\part x^2}+\frac{\part^2 f}{\part y^2}+\frac{\part^2 f}{\part z^2}.</math> ..
The Laplacian of a function is equal to the DIVERGENCE of the GRADIENT. ..
..
(==) References (==) ..
(===) Print (===) ..
* {{ Citation ..
.3. last = Anton ..
.3. first = Howard ..
.3. last2 = Bivens ..
.3. first2 = Irl ..
.3. last3 = Davis ..
.3. first3 = Stephen ..
.3. date = February 2 , 2005 ..
.3. title = Calculus: Early Transcendentals Single and Multivariable ..
.3. place = New York ..
.3. publisher = Wiley ..
.3. edition = 8th ..
.3. isbn = 978-0471472445 ..
}} ..
* {{ Citation ..
.3. last = Apostol ..
.3. first = Tom M. ..
.3. date = June 1967 ..
.3. title = Calculus , Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra ..
.3. publisher = Wiley ..
.3. edition = 2nd ..
.3. volume = 1 ..
.3. isbn = 978-0471000051 ..
}} ..
* {{ Citation ..
.3. last = Apostol ..
.3. first = Tom M. ..
.3. date = June 1969 ..
.3. title = Calculus , Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications ..
.3. publisher = Wiley ..
.3. edition = 2nd ..
.3. volume = 1 ..
.3. isbn = 978-0471000075 ..
}} ..
* {{ Citation ..
.3. last = Eves ..
.3. first = Howard ..
.3. date = January 2 , 1990 ..
.3. title = An Introduction to the History of Mathematics ..
.3. edition = 6th ..
.3. publisher = Brooks Cole ..
.3. isbn = 978-0030295584 ..
}} ..
* {{ Citation ..
.3. last = Larson ..
.3. first = Ron ..
.3. last2 = Hostetler ..
.3. first2 = Robert P. ..
.3. last3 = Edwards ..
.3. first3 = Bruce H. ..
.3. date = February 28 , 2006 ..
.3. title = Calculus: Early Transcendental Functions ..
.3. edition = 4th ..
.3. publisher = Houghton Mifflin Company ..
.3. isbn = 978-0618606245 ..
}} ..
* {{ Citation ..
.3. last = Spivak ..
.3. first = Michael ..
.3. author-link = Michael Spivak ..
.3. date = September 1994 ..
.3. title = Calculus ..
.3. publisher = Publish or Perish ..
.3. edition = 3rd ..
.3. isbn = 978-0914098898 ..
}} ..
* {{ Citation ..
.3. last = Stewart ..
.3. first = James ..
.3. date = December 24 , 2002 ..
.3. title = Calculus ..
.3. publisher = Brooks Cole ..
.3. edition = 5th ..
.3. isbn = 978-0534393397 ..
}} ..
* {{ Citation ..
.3. last = Thompson ..
.3. first = Silvanus P. ..
.3. date = September 8 , 1998 ..
.3. title = Calculus Made Easy ..
.3. edition = Revised , Updated , Expanded ..
.3. place = New York ..
.3. publisher = St. Martin's Press ..
.3. isbn = 978-0312185480 ..
}} ..
..
(===) Online books (===) ..
..
* {{ Citation ..
.3. last = Crowell ..
.3. first = Benjamin ..
.3. title = Calculus ..
.3. year = 2003 ..
.3. url = http://www.lightandmatter.com/calc/ ..
}} ..
* {{ Citation ..
.3. last = Garrett ..
.3. first = Paul ..
.3. year = 2004 ..
.3. title = Notes on First-Year Calculus ..
.3. url = http://www.math.umn.edu/~garrett/calculus/ ..
}} ..
* {{ Citation ..
.3. last = Hussain ..
.3. first = Faraz ..
.3. year = 2006 ..
.3. title = Understanding Calculus ..
.3. url = http://www.understandingcalculus.com/ ..
}} ..
* {{ Citation ..
.3. last = Keisler ..
.3. first = H. Jerome ..
.3. year = 2000 ..
.3. title = Elementary Calculus: An Approach Using Infinitesimals ..
.3. url = http://www.math.wisc.edu/~keisler/calc.html ..
}} ..
* {{ Citation ..
.3. last = Mauch ..
.3. first = Sean ..
.3. year = 2004 ..
.3. title = Unabridged Version of Sean's Applied Math Book ..
.3. url = http://www.its.caltech.edu/~sean/book/unabridged.html ..
}} ..
* {{ Citation ..
.3. last = Sloughter ..
.3. first = Dan ..
.3. year = 2000 ..
.3. title = Difference Equations to Differential Equations ..
.3. url = http://synechism.org/drupal/de2de/ ..
}} ..
* {{ Citation ..
.3. last = Strang ..
.3. first = Gilbert ..
.3. year = 1991 ..
.3. title = Calculus ..
.3. url = http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm ..
}} ..
* {{ Citation ..
.3. last = Stroyan ..
.3. first = Keith D. ..
.3. year = 1997 ..
.3. title = A Brief Introduction to Infinitesimal Calculus ..
.3. url = http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm ..
}} ..
* {{ Citation ..
.3. last = Wikibooks ..
.3. title = Calculus ..
.3. url = http://en.wikibooks.org/wiki/Calculus ..
}} ..
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[[Category:Mathematical analysis]] ..
[[Category:Differential calculus]] ..
[[Category:Functions and mappings]] ..
[[Category:Linear operators in calculus]] ..
paraulesenllacos ..
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QUADRATIC FUNCTION ..
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CONSTANT ..
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CALCULUS ..
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FUNCTION ..
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DERIVATIVE ..
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ACCELERATION ..
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VELOCITY ..
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GRAPH OF A FUNCTION ..
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CURVATURE ..
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LEIBNIZ'S NOTATION ..
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CONCAVE UP ..
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TANGENT ..
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CONCAVE DOWN ..
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STATIONARY POINT ..
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LOCAL MAXIMUM ..
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LOCAL MINIMUM ..
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LIMIT ..
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DIFFERENCE QUOTIENT ..
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SECOND DIFFERENCE ..
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SEQUENCES ..
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LINEAR APPROXIMATIONS ..
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QUADRATIC APPROXIMATION ..
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QUADRATIC FUNCTION ..
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TAYLOR POLYNOMIAL ..
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PARTIAL DERIVATIVES ..
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SYMMETRIC MATRIX ..
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EIGENVALUES ..
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SECOND PARTIAL DERIVATIVE TEST ..
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DIVERGENCE ..
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GRADIENT ..
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