differentiation & maxima & minima
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Differentiation is a method tocompute the rate at which a
dependent output y changes with
respect to the change in the
independent inputx.
This rate of change is called
the derivative ofywith respect
tox. In more precise language, the
dependence ofyuponxmeans
that yis a function ofx.
This functional relationship is
often denoted y=(x),
wheredenotes the function.
Ifxand yare real numbers, and ifthe graph ofyis plotted againstx,
the derivative measures
the slope of this graph at each
point.
http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Graph_of_a_functionhttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Graph_of_a_functionhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Function_(mathematics) -
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The simplest case is when yis a linear
function ofx, meaning that the graph of
yagainstxis a straight line. In this
case, y= (x) = mx+ b, for real numbers m
and b, and the slope m is given by
where the symbol (the uppercase form
of the Greek letter Delta) is an
abbreviation for "change in." This formulais true because
y y=(x x) = m (x x)
+ b = mx+ b + mx= y+ m x.
It follows that y= mx.
Tangent
m/Slope
This gives an exact value for the slope of astraight line. If the function is not linear (i.e.
its graph is not a straight line), however, then
the change in ydivided by the change
inxvaries: differentiation is a method to find
an exact value for this rate of change at any
given value of x.
http://en.wikipedia.org/wiki/Linear_functionhttp://en.wikipedia.org/wiki/Linear_functionhttp://en.wikipedia.org/wiki/File:Tangent_to_a_curve.svghttp://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/File:Tangent_to_a_curve.svghttp://en.wikipedia.org/wiki/File:Tangent_to_a_curve.svghttp://en.wikipedia.org/wiki/File:Tangent_to_a_curve.svghttp://en.wikipedia.org/wiki/File:Tangent_to_a_curve.svghttp://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Linear_functionhttp://en.wikipedia.org/wiki/Linear_function -
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The idea, illustrated by
Figures, is to compute the
rate of change as
the limiting value of
the ratio of thedifferencesy/ xas
xbecomes infinitely
small.
In Leibniz's notation, such an infinitesimal change
inxis denoted by dx, and the derivative ofywith
respect toxis written
suggesting the ratio of two infinitesimal quantities.
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Letbe a real valued function. In classical geometry, the tangent line to the graph of the
functionat a real number a was the unique line through the point (a,(a)) that
did notmeet the graph oftransversally, meaning that the line did not pass straightthrough the graph. The derivative ofywith respect toxat a is, geometrically, the slope of
the tangent line to the graph ofat a. The slope of the tangent line is very close to the
slope of the line through (a, (a)) and a nearby point on the graph, for
example (a + h,(a + h)). These lines are called secant lines. A value ofh close to zero gives
a good approximation to the slope of the tangent line, and smaller values (in absolute
value) ofh will, in general, give better approximations. The slope m of the secant line is thedifference between the yvalues of these points divided by the difference between
thexvalues, that is,
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This expression is Newton's difference quotient. The derivative is the value of the
difference quotient as the secant lines approach the tangent line. Formally,
the derivative of the function at a is the limit
of the difference quotient as h approaches zero, if this limit exists. If the limit exists,
thenis differentiable at a. Here (a) is one of several common notations for the
derivativeEquivalently, the derivative satisfies the property that
which has the intuitive interpretation that the tangent line to at a gives
the bestlinearapproximation
tonear a (i.e., for small h). This interpretation is the easiest to generalize to other
settings.
http://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Difference_quotienthttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Differentiable_functionhttp://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Differentiable_functionhttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Difference_quotienthttp://en.wikipedia.org/wiki/Isaac_Newton -
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Substituting 0 for h in the difference quotient causes division by zero, so the slope
of the tangent line cannot be found directly using this method. Instead, define Q(h)to be the difference quotient as a function ofh:
Q(h) is the slope of the secant line between (a, (a)) and (a + h,(a + h)). Ifis
a continuous function, meaning that its graph is an unbroken curve with no gaps,then Q is a continuous function away from h = 0. If the limit exists,
meaning that there is a way of choosing a value for Q(0) that makes the graph
ofQ a continuous function, then the functionis differentiable at a, and its
derivative at a equals Q(0).
In practice, the existence of a continuous extension of the difference quotient Q(h)to h = 0 is shown by modifying the numerator to cancel h in the denominator. This
process can be long and tedious for complicated functions, and many shortcuts are
commonly used to simplify the process.
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The squaring function (x) = x is differentiable at x= 3, and its derivative there is 6.
This result is established by calculating the limit as h approaches zero of the
difference quotient of(3):
The last expression shows that the difference quotient equals 6 + h when h 0 and is
undefined when h = 0, because of the definition of the difference quotient. However,
the definition of the limit says the difference quotient does not need to be defined
when h = 0. The limit is the result of letting h go to zero, meaning it is the value that 6
+ h tends to as h becomes very small:
Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so
its derivative atx= 3 is'(3) = 6.
More generally, a similar computation shows that the derivative of the squaring
function atx= a is'(a) = 2a.
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General Differentiation Formulas:
Where is any constant.1.
It is called Power Rule of Derivative.2.
3.
Power Rule for Function.4.
5.
6.
7.
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8.
It is called Product Rule.9.
It is called Quotient Rule.10.
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DerivativeofLogarithmFunctions:
1.
2.
3.
4.
Where log is denoted by ln.
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Derivative of Exponential Functions:
Where log is denoted by ln.
1.
2.
3.
4.
5.
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LOGARITHMIC DIFFERENTIATIONDifferentiate the following function with respect to x:
(log x)X
Solution:
Let y=(logx)X ,Then
Lets us take log on both sides,
log y=x.log(logx) [logeaX =xlogea]
Differentiating both the sides,1
/y .dy
/dx = x(1
/logx .1
/x) + log(log x).1 [Using Product Rule]1/y .dy/dx =
1/logx + log(log x)dy/dx = y{
1/logx + log(log x)}
Therefore,dy/dx = (log x)
X . {1/logx + log(log x)}
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The maximum and minimum of a function, known collectively
as extrema, are the largest and smallest value that the function takes at
a point either within a given neighborhood or on the function domain in
its entirety.
More generally, the maximum and minimum of a set (as defined in set
theory) are the greatest and least element in the set. Unbounded
infinite sets such as the set ofreal numbers have no minimum and
maximum.
A real-valued functionfdefined on a real line is said to have a maximum
point at the pointx, if there exists some > 0 such thatf(x) f(x) when
|xx| < . The value of the function at this point is calledmaximum of
the function. Similarly, a function has a minimum point atx, iff(x)
f(x) when |xx| < . The value of the function at this point is
called minimum of the function.
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Finding global maxima and minima is the goal ofmathematical optimization.
If a function is continuous on a closed interval, then by the extreme value
theorem global maxima and minima exist.
Furthermore, a global maximum (or minimum) either must be a local maximum(or minimum) in the interior of the domain, or must lie on the boundary of the
domain.
So a method of finding a global maximum (or minimum) is to look at all the local
maxima (or minima) in the interior, and also look at the maxima (or minima) of
the points on the boundary; and take the biggest (or smallest) one.
Local extrema can be found by Fermat's theorem, which states that they mustoccur at critical points.
One can distinguish whether a critical point is a local maximum or local minimum
by using the first derivative test or second derivative test.
For any function that is defined piecewise, one finds a maxima (or minima) by
finding the maximum (or minimum) of each piece separately; and then seeing
which one is biggest (or smallest).
http://en.wikipedia.org/wiki/Mathematical_optimizationhttp://en.wikipedia.org/wiki/Extreme_value_theoremhttp://en.wikipedia.org/wiki/Extreme_value_theoremhttp://en.wikipedia.org/wiki/Fermat's_theorem_(stationary_points)http://en.wikipedia.org/wiki/Critical_point_(mathematics)http://en.wikipedia.org/wiki/First_derivative_testhttp://en.wikipedia.org/wiki/Second_derivative_testhttp://en.wikipedia.org/wiki/Piecewisehttp://en.wikipedia.org/wiki/Piecewisehttp://en.wikipedia.org/wiki/Second_derivative_testhttp://en.wikipedia.org/wiki/First_derivative_testhttp://en.wikipedia.org/wiki/Critical_point_(mathematics)http://en.wikipedia.org/wiki/Fermat's_theorem_(stationary_points)http://en.wikipedia.org/wiki/Extreme_value_theoremhttp://en.wikipedia.org/wiki/Extreme_value_theoremhttp://en.wikipedia.org/wiki/Mathematical_optimization -
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The extreme value theorem states that if a real-valued functionfis continuous in
the closed and bounded interval [a,b], thenfmust attain its maximum and minimum
value, each at least once. That is, there exist numbers c and din [a,b] such that:
A related theorem is the boundedness theorem which states that a continuousfunctionfin the closed interval [a,b] is bounded on that interval. That is, there exist real
numbers m and M such that:
The extreme value theorem enriches the boundedness theorem by saying that not only is
the function bounded, but it also attains its least upper bound as its maximum and itsgreatest lower bound as its minimum.
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The global maximum of occurs atx= e.
The functionx2 has a unique global minimum atx= 0.
The functionx3 has no global minima or maxima. Although the first derivative (3x2) is
0 atx= 0, this is an inflection point.
The function has a unique global maximum atx= e.
The function x-x has a unique global maximum over the positive real numbers atx=
1/e.
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The functionx3/3 xhas first derivativex2 1 andsecond derivative 2x. Setting the
first derivative to 0 and solving forxgives stationary points at 1 and 1. From the
sign of the second derivative we can see that 1 is a local maximum and 1 is a local
minimum. Note that this function has no global maximum or minimum.
The function |x| has a global minimum atx= 0 that cannot be found by taking
derivatives, because the derivative does not exist atx= 0.
The function cos(x) has infinitely many global maxima at 0, 2, 4, , and infinitely
many global minima at , 3, . The function 2 cos(x) xhas infinitely many local maxima and minima, but no global
maximum or minimum.
The function cos(3x)/xwith 0.1 x 1.1 has a global maximum atx= 0.1 (a
boundary), a global minimum nearx= 0.3, a local maximum nearx= 0.6, and a local
minimum nearx= 1.0. (See figure at top of page.)
The functionx3
+ 3x2
2x 1 defined over the closed interval (segment) *4,2+ hastwo extrema: one local maximum atx= 1153, one local minimum atx= 1
153, a
global maximum atx= 2 and a global minimum atx= 4.
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For functions of more than one variable, similar conditions apply. For
example, in the (enlargeable) figure at the right, the necessary conditions
for a localmaximum are similar to those of a function with only one
variable. The first partial derivatives as to z (the variable to be maximized)
are zero at the maximum (the glowing dot on top in the figure). The
second partial derivatives are negative. These are only necessary, not
sufficient, conditions for a local maximum because of the possibility ofa saddle point. For use of these conditions to solve for a maximum, the
function z must also be differentiable throughout. The second partial
derivative test can help classify the point as a relative maximum or
relative minimum.
http://en.wikipedia.org/wiki/Partial_derivativeshttp://en.wikipedia.org/wiki/Saddle_pointhttp://en.wikipedia.org/wiki/Differentiablehttp://en.wikipedia.org/wiki/Second_partial_derivative_testhttp://en.wikipedia.org/wiki/Second_partial_derivative_testhttp://en.wikipedia.org/wiki/Second_partial_derivative_testhttp://en.wikipedia.org/wiki/Second_partial_derivative_testhttp://en.wikipedia.org/wiki/Differentiablehttp://en.wikipedia.org/wiki/Saddle_pointhttp://en.wikipedia.org/wiki/Partial_derivatives -
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In contrast, there are substantial differences between functions of one
variable and functions of more than one variable in the identificationof global extrema. For example, if a bounded differentiable
functionfdefined on a closed interval in the real line has a single
critical point, which is a local minimum, then it is also a global
minimum (use the intermediate value theorem and Rolle's theorem to
prove this by reductio ad absurdum). In two and more dimensions, thisargument fails, as the function
shows. Its only critical point is at (0,0), which is a local minimum with
(0,0) = 0. However, it cannot be a global one, because (4,1) = 11.
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Maxima and minima are more generally defined for sets. In general, if an ordered setS has
a greatest elementm, m is a maximal element. Furthermore, ifS is a subset of an ordered
setTand m is the greatest element ofS with respect to order induced by T, m is a least upper
bound ofS in T. The similar result holds for least element, minimal element and greatest lower
bound.
In the case of a general partial order, the least element (smaller than all other) should not be
confused with a minimal element (nothing is smaller). Likewise, a greatest element of
apartially ordered set (poset) is an upper bound of the set which is contained within the set,
whereas a maximal elementm of a posetA is an element ofA such that ifmb (for
any binA) then m = b. Any least element or greatest element of a poset is unique, but a posetcan have several minimal or maximal elements. If a poset has more than one maximal
element, then these elements will not be mutually comparable.
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