Differentiation & Maxima & Minima

8/1/2019 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.
http://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Difference_quotienthttp://en.wikipedia.org/wiki/Difference_quotienthttp://en.wikipedia.org/wiki/Leibniz's_notationhttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Leibniz's_notationhttp://en.wikipedia.org/wiki/Difference_quotienthttp://en.wikipedia.org/wiki/Difference_quotienthttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/File:Lim-secant.svghttp://en.wikipedia.org/wiki/File:Secant-calculus.svghttp://en.wikipedia.org/wiki/File:Tangent-calculus.svg


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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,
http://en.wikipedia.org/wiki/Transversality_(mathematics)http://en.wikipedia.org/wiki/Secant_linehttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Approximationhttp://en.wikipedia.org/wiki/Approximationhttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Secant_linehttp://en.wikipedia.org/wiki/Secant_linehttp://en.wikipedia.org/wiki/Transversality_(mathematics)


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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.
http://en.wikipedia.org/wiki/Substitution_property_of_equalityhttp://en.wikipedia.org/wiki/Division_by_zerohttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Division_by_zerohttp://en.wikipedia.org/wiki/Substitution_property_of_equality


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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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http://en.wikipedia.org/wiki/File:Extrema_example_original.svg


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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.
http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Domain_(mathematics)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Domain_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)


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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.
http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Closed_sethttp://en.wikipedia.org/wiki/Bounded_functionhttp://en.wikipedia.org/wiki/Bounded_functionhttp://en.wikipedia.org/wiki/Closed_sethttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)


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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.
http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Inflection_pointhttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Inflection_pointhttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/File:Xth_root_of_x.svg


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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.
http://en.wikipedia.org/wiki/Second_derivativehttp://en.wikipedia.org/wiki/Second_derivative


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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.
http://en.wikipedia.org/wiki/Intermediate_value_theoremhttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Reductio_ad_absurdumhttp://en.wikipedia.org/wiki/Reductio_ad_absurdumhttp://en.wikipedia.org/wiki/Reductio_ad_absurdumhttp://en.wikipedia.org/wiki/Reductio_ad_absurdumhttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Intermediate_value_theorem


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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.
http://en.wikipedia.org/wiki/Ordered_sethttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Maximal_elementhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Minimal_elementhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Partial_orderhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Upper_boundhttp://en.wikipedia.org/wiki/Upper_boundhttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Partial_orderhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Minimal_elementhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Maximal_elementhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Ordered_set


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Differentiation & Maxima & Minima

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