Chapter7 Calculus)

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Differential Calculus

Derivatives

It is possi ble to transform a continuous function of one or more independent varia bles into a derivative function.

In si mple cases the derivative can be interpreted as the slope. Of the curve descri bed by the origina l function.

First Derivative

The derivative of a function y=f(x) is written i n various ways.

Critical Points

Derivatives are used to locate critical points, or, extreme points (also known as minimum and maximum points)

as well as points ofcontraflexure of functions of one variabl e.

Critical Points

The first derivative f'(x) is cal culated to determine where the critical points are . The second derivative, f''(x) is

cal culated to determine whether a critical point is a local maximum, minimum, or inflection point.

Critical points are located where the first derivative is zero.

Test for a maximum: For a function (x) with an extreme point at x=, if the point is a maximum then:

Test for a minimum: for a function (x) with a critical point at x=, i f the point is a minimum then:

Test for a point of inflection: for a function(x) with a critical point at x=, i f the point is apoint of inflection then:

Partial Derivatives

If a function has more than one independent vari abl e(x1, x2, x3), a partia l derivative can be found, but only with

respect to one of the var iabl es. All other varia bles are treated as constants.

Curvature

The sharpness of a curve between 2 points on the curve can be defined a s the rate of change of the incl ina tion of

the curve with respect to the dis tance traveled al ong the curve.

The sharpness of the curve at one point is known as the curvature, K:

If the equation of th3e curve is given in rectangular coordinates, the curvature is defined as:

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Determining radius of curvature

The derivative or slope of an equation at any point is a measure of how much an equation is changing . The

relative sharpness of this curve is known as the curvature. The radius of curvature is the reciprocal of the absolute

value of the curvature. The circle represented by the radius of curvature is tangent to the original curve and the

center of the circle is one the concave side of the curve. The equation for the radius of curvature is:

L'Hopital's Rule

A limit is the value a function approaches when an independent variable approaches a target value . Suppose

the value ofy=x2 is desired asxapproaches 5.

The object of limit t heory is to determine the limit without having to evaluate the function at t he target. Asx

approaches the target value :

If(x) is undefined on one side, or if(x) is dis continuous at (x=) the limit does not exist at (x=).

L'Hopital's rule may be used only when the numerator and denominator of the expressi on both approach zero or

both approach i nfinity at the li mit point.

Integral Calculus

Fundamental Theorem of CalculusIntegration is the inverse operation of differentiation . There are 2 types of integral s: definite integrals, which are

restricted to a specific range of the independent variable, and indefinite integrals (antiderivatives), which a re

unrestricted. See http://www.karlscalculus.org/calc10_1.html .

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Methods of Integration

Integration by parts: If(x) and g(x) are functions , then the integral of(x) with respect to g(x) is:

(x)dg(x)= (x)g(x) - g(x)df(x)+C

Integration by substitution: an integrand (that i s diffi cult to integrate) a nd the corresponding differential are

replaced by equiva lent express ions with known solutions. Substitutions that may be used include trigonometric

substitutions:

Separation of rational fractions into partial fractions: The method of partial fra ctions is used to transform a

proper polynomial fraction of 2 polynomials into a sum of simpler expressions , a procedure known as resolution.

Once the general for ms of the partial fractions have been determined from i nspection, the method of

undetermined coefficients is us ed. The partial fractions are all c ross-multiplied to obtain Q(x) as the denominator,

and the coefficients are found by equating P(x) and the cross -multipl ied numerator.

Centroids and Moments of Inertia

Applications of integration include the determination of the centroid of an area and the area moment of inertia.

The centroid of an area is analogous to the center of gravity of a homogenous body.

The locati on of the centroid of the a rea bounded by the x-and y-axes and the math function y=(x) can be found

from:

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The quantity ofxdA is known as the first moment of the area orfirst moment with respect to the y-axis.

Similarly,ydA is known as the first moment of the area of the x-axis.

The moment of inertia (I) of an area is needed i n mechanics of materials questions.

(Ix)-used for moment of inertia x-axis

(Iy)-used for moment of inertia y-axis

The moment of inertia taken with respect to an axi s pass ing through the area's centroid is the centroidal moment

of inertia, (Ic). It is the smallest possible moment of inertia for the shape.

Transfer axis theorem-If moment of inertia is known for one axis, the moment of inertia can be calculated for a

parallel line. Used to evaluate moment of inertia of areas composed of 2 or more basic shapes.

Iparallel axis=Ic+Ad2

(d) is distance between centroidal axis and the second parallel axis

The integration method for bas ic s hapes is not need a s formulas for basic shapes can be obtained from tables.

Gradient Divergence and Curl

Gradient of a Scalar Function

The slope of a function is the change in one variable with respect t o a distance in a chosen direction. The

gradient vector function ("del f") gives the max rate of change of the function f(x, y, z.)

Divergence of a Vector Field

The divergence, div F, of a vector field F(x, y, z) is a scalar function. The divergence ofF can be interpreted as the

accumulation of flux. One of the uses of divergence is to determine whether flow (represented in direction and

magnitude by F) is compressible. Flow is incompressible if div F=0, since the substance is not accumulating.

The divergence of a vector fi eld is ascalar field. If (del F)=0, then the field is sa id to be a divergenceless field.

The physica l si gnificance of the divergence of avector field is the ra te at which "densi ty" exits a given region of

spac e. The defini tion of the divergence therefore foll ows naturall y by noting that, in the absence of the creation

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or destructi on of matter, the density within a region of space can change only by havi ng it flow i nto or out of the

region.

By measuri ng the net fl ux of content pas si ng through a surface surroundi ng the region of space, it is therefore

immediately possi ble to say how the density of the i nterior has changed. This property i s fundamental in physi cs,

where it goes by the name "principle of continui ty." When stated as a formal theorem, i t is cal led the divergence

theorem, al so known as Gauss's theorem. In fact, the defini tion in equation ( 1) is i n effect a statement of the

divergence theorem.

Divergence -- from Wolfram MathWorld

http://mathworld.wolfram.com/Divergence.html

Screen cli pping taken: 5/27/2011, 10:29 PM

Pasted from

Curl of a Vector Field

The curl, curl F, of a vector field F(x, y, z) is a vector field. The curl F can be interpreted as the vorticityper unit

area of flux. One of the uses of the curl is to determine whether flow (represented in direction and magnitudeby F) is rotational. Flow is irrotat ional if curl F=0.

The curl of a vector fi eld, denoted (cur l F) or (del F), i s defined as the vector fi eld havi ng magnitude equal to the

maximum "circ ulation" at each point and to be oriented perpendicula rly to this pl ane of circula tion for each

point. More precisely, the magnitude of (del F) is the li miting value of circulation per unit area. Written explici tly,

where the r ight side is a li ne integral around an infi nitesimal region of areaA that is al lowed to shrink to zero vi a a

li miting process and n is the unit normal vector to this region. If (del F)=0, then the field is sai d to be an

irrotational field.

The physic al significance of the curl of a vector field is the amount of "rotation" or angul ar momentum of the

contents of given region of space. It a ris es in flui d mechanics and elasticity theory. It i s al so fundamental in the

theory of electromagnetism, where it arises in two of the four Maxwell equations,

Pasted from

Laplacian of a Scalar Function

The Lapla cia n of a scalar function,=f(x, y, z) is the divergence of the gradient function. Thi s is essential ly thesecond derivative of a sca lar function. A function that s atisfi es Laplace's equation is known as a potential

function.

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Summary Review

Friday, May 27, 2011

11:23 PM

Manipulating vectors

A vector is a quantity that has magnitude and direction. The manipulation of scalar

values does not require any knowledge of the direction, only magnitude. The

manipulation of vectors requires special treatment of direction. A vector can

contain known or unknown, variable, values. Three common vectors are called the

unit vectors and are one unit of length long in each of the primary axis directions,

noted as i, j, and k. Any vector can then be written as a sum of each unit vector

multiplied by their component value. Once a vector is written in terms of unit

vectors, the individual components can be added to see how two vectors add or

subtract together. Another common vector manipulation is the calculation of the

proportion of one vector projected onto another vector and is called the dot

product. The dot product is a scalar value and is the sum of the componentsmultiplied together for each axis. The final vector manipulation is the cross product

and represents a vector that is perpendicular to the two vectors using the right hand

rule. The cross product is the determinant of a matrix using the units vectors as the

first row and the two vector components on the second and third rows.

Sequence progressions

A sequence is a set of numbers where the value of each element, noted as l, is

defined by an equation that uses the place in the sequence, noted as n, in the

calculation. The sequence diverges if the elements approach positive or negative

infinity. The sequence is converges if the elements approach a single value. A series

is the sum of terms of a sequence. There are four common series Arithmetic,

Geometric, Power Series, and Taylors Series. The Arithmetic series is where the

terms are defined as l = a + (n 1)d where a is the first term and d is the common

difference. The Geometric series is where the terms are defined as l = ar n 1where a

is the first term and r is the common ratio. The Power Series is the sum of all terms

of a polynomial of order n with coefficients on each term. The Taylor Series is the

sum of derivatives.

Derivatives of equations

The slope of most equations can be defined by taking the derivative of the originalequation. The derivative is a function that is derived from the original function,

f(x). There are many nomenclatures for derivatives including f(x), Df(x), or dy/dx.

The actual definition of a derivative uses the limit theory where you evaluate the

equation at some point and at some close other point divided by the distance

between the points along the x axis where the distance approaches zero. The rule of

polynomial derivatives takes the exponent and multiplies the coefficient on the

variable then reducing the exponent by one. There are many other derivative rules

but the most basic rule is the derivative of a constant is zero.


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Analyzing equations with derivatives

The first derivative of an equation is used to determine the slope of the original

equation and to determine where the critical points are located. The critical points

of an equation might be an extreme points or inflection points. The extreme points

may be high points otherwise known as maxima or low points otherwise known asminima. If the extreme points are extreme for the entire function, then they are

called global extrema otherwise they are just known as local extrema. The inflection

point is where the curve changes from increasing slope to decreasing slope or vice

versa. There is always an inflection point between successive extrema. The second

derivative of an equation is used to determine the exact nature of the extreme point

or inflection point.


The derivative or slope of an equation at any point is a measure of how much an

equation is changing. The relative sharpness of this curve is known as the

curvature. The radius of curvature is the reciprocal of the absolute value of thecurvature. The circle represented by the radius of curvature is tangent to the

original curve and the center of the circle is one the concave side of the curve. The

equation for the radius of curvature is:

LHpitals rule

Equations that have a single y value for any x value are special and are called

functions. The limit is the value of the function evaluated at a particular position. If

the functions are continuous at the point of interest then the limit exists and can

easily be evaluated. If the functions are not continuous at the point of interest then

the limit can not be readily obtained by evaluating the function. LHpitals Rule is

used for these functions that can not be evaluated and is stated as the limit of the

ratio of functions evaluated at some position is equal to the ratio of the derivatives

of the functions evaluated at that same position.

Methods of integration

There are four methods of integrating equations; observation, parts, substitution,

and partial integration. Integration by observation recognizes that many integral

rules are memorized or are easily observed by looking up the problem in integration

tables. Integration by parts recognizes that the original integral may be a broken

down into a product of functions and a simpler integral. Integration by substitution

recognizes that a portion of the function may be hard to integrate can be substituted

with a more easily integrated equivalent. Integration by partial integration

recognizes that a complicated equation may be split into a sum of more easily

integrated expressions. Integration of an equation may use a single technique or all


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of these techniques together.

Theorem of Integral Calculus

For every mathematical function there is an inverse operation. If derivatives find

the slope of a function then integrals find the function given the slope. Integrals can

be generic functions that are valid for any position along the function and are knownas indefinite integrals. Indefinite integrals do not have limits of integration and

always have a unknown constant signifying the fact that the derivative of a constant

is zero so you must leave open the possibility that the function can have a constant

added to it and would have the same derivative. Definite integrals have limits of

integration and do not have an unknown constant added to the answer. An

interesting geometric interpretation of a definite integral is that it is the area

between the curve and the axis. The fundamental theorem of integral calculus is

given for definite integrals and uses Riemann sums with widths that are

infinitesimally small:

Gradient, divergence, curl, and laplacian

For three dimensional functions, there are four interesting operations that help

analyze the behavior of a function using the del,, operator. The del operator is theaddition of partial derivatives in the three principal directions and is helpful in

many fluid applications. The first operation is the gradient of a scalar function using

the del operator on the function and finds the maximum rise of a function regardless

of the direction. The second operation is the divergence of a vector function and isthe dot product of the del operator and the function and can be used to check for

compressible fluids. The third operation is the curl of a vector function and the

cross product of the del operator and the function and can be used to determine the

presence and magnitude of rotational flow. The fourth operation is the Laplacian of

the scalar function and it is del squared of the function.

Centroids and moments of inertia

Since the integration of functions provides the area under the function then there

are manipulations to these integrals that allow the calculation of centroids and

moments. A centroid is the center of an area, that is to say that there is exactly as

much area to the left as there is to the right of the centroid along the x axis.

Likewise for the y axis centroid, the areas are the same size. A moment, in

mechanical systems, is torque or force at a distance. A moment in calculus is an area

at a distance. The order of the moment indicates the exponent on the distance.

Hence the first moment of area uses only x, whereas the second moment of area

uses x2. The second moment of area is sometimes referred to as the moment of

inertia. If the moment of inertia is calculated at the centroid, it is the minimum

value and called the centroidal moment of inertia. A shortcut calculation of moment

of inertia off the centroid is to add the area times the distance squared to the


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centroidal moment of inertia.

Homogeneous and nonhomogeneous differential equations

Homogeneous differential equations are differential equations that are only a sum of

a function and its derivatives that equal zero. The solution of homogeneous

equations is based on recognition of forms involving the value, or order, of thehighest directive. First order solutions are simple exponential form where second

order solutions are the sum of exponents which may oscillate or asymptotically

approach zero, known as damping. Nonhomogeneous differential equations are

similar to homogeneous differential equations except the sum does not equal zero.

The solution of nonhomogeneous differential equation require you solve the

homogeneous portion first, called the complimentary solution, and then the

nonhomogeneous portion, called the particular solution. The particular solution

may come from the method of undetermined coefficients and involves solving for

differential equation using some known initial value.


10/12

Manipulating vectors

A vector is a quantity that has magnitude and direction. The manipulation of scalar

values does not require any knowledge of the direction, only magnitude. Themanipulation of vectors requires special treatment of direction. A vector can

contain known or unknown, variable, values. Three common vectors are called the

unit vectors and are one unit of length long in each of the primary axis directions,

noted as i, j, and k. Any vector can then be written as a sum of each unit vector

multiplied by their component value. Once a vector is written in terms of unit

vectors, the individual components can be added to see how two vectors add or

subtract together. Another common vector manipulation is the calculation of the

proportion of one vector projected onto another vector and is called the dot

product. The dot product is a scalar value and is the sum of the components

multiplied together for each axis. The final vector manipulation is the cross product

and represents a vector that is perpendicular to the two vectors using the right hand

rule. The cross product is the determinant of a matrix using the units vectors as the

first row and the two vector components on the second and third rows.

Sequence progressions

A sequence is a set of numbers where the value of each element, noted as l, is

defined by an equation that uses the place in the sequence, noted as n, in the

calculation. The sequence diverges if the elements approach positive or negative

infinity. The sequence is converges if the elements approach a single value. A series

is the sum of terms of a sequence. There are four common series Arithmetic,

Geometric, Power Series, and Taylors Series. The Arithmetic series is where the

terms are defined as l = a + (n 1)d where a is the first term and d is the common

difference. The Geometric series is where the terms are defined as l = ar n1where a

is the first term and r is the common ratio. The Power Series is the sum of all termsof a polynomial of order n with coefficients on each term. The Taylor Series is the

sum of derivatives.

Derivatives of equations

The slope of most equations can be defined by taking the derivative of the original

equation. The derivative is a function that is derived from the original function,

f(x). There are many nomenclatures for derivatives including f(x), Df(x), or dy/dx.

The actual definition of a derivative uses the limit theory where you evaluate the

equation at some point and at some close other point divided by the distance

between the points along the x axis where the distance approaches zero. The rule of

polynomial derivatives takes the exponent and multiplies the coefficient on thevariable then reducing the exponent by one. There are many other derivative rules

but the most basic rule is the derivative of a constant is zero.

Analyzing equations with derivatives

The first derivative of an equation is used to determine the slope of the original

equation and to determine where the critical points are located. The critical points

of an equation might be an extreme points or inflection points. The extreme points

may be high points otherwise known as maxima or low points otherwise known as

minima. If the extreme points are extreme for the entire function, then they are

called global extrema otherwise they are just known as local extrema. The inflection

Summary ReviewFriday, May 27, 2011

11:23 PM

Calculus Page 1


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point is where the curve changes from increasing slope to decreasing slope or vice

versa. There is always an inflection point between successive extrema. The second

derivative of an equation is used to determine the exact nature of the extreme point

or inflection point.


The derivative or slope of an equation at any point is a measure of how much an

equation is changing. The relative sharpness of this curve is known as thecurvature. The radius of curvature is the reciprocal of the absolute value of the

curvature. The circle represented by the radius of curvature is tangent to the

original curve and the center of the circle is one the concave side of the curve. The

equation for the radius of curvature is:

LHpitals rule

Equations that have a single y value for any x value are special and are called

functions. The limit is the value of the function evaluated at a particular position. If

the functions are continuous at the point of interest then the limit exists and can

easily be evaluated. If the functions are not continuous at the point of interest then

the limit can not be readily obtained by evaluating the function. LHpitals Rule is

used for these functions that can not be evaluated and is stated as the limit of the

ratio of functions evaluated at some position is equal to the ratio of the derivatives

of the functions evaluated at that same position.

Theorem of Integral Calculus

For every mathematical function there is an inverse operation. If derivatives find

the slope of a function then integrals find the function given the slope. Integrals can

be generic functions that are valid for any position along the function and are known

as indefinite integrals. Indefinite integrals do not have limits of integration and

always have a unknown constant signifying the fact that the derivative of a constant

is zero so you must leave open the possibility that the function can have a constant

added to it and would have the same derivative. Definite integrals have limits of

integration and do not have an unknown constant added to the answer. An

interesting geometric interpretation of a definite integral is that it is the area

between the curve and the axis. The fundamental theorem of integral calculus is

given for definite integrals and uses Riemann sums with widths that are

infinitesimally small:

Methods of integration

Calculus Page 2


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There are four methods of integrating equations; observation, parts, substitution,

and partial integration. Integration by observation recognizes that many integral

rules are memorized or are easily observed by looking up the problem in integration

tables. Integration by parts recognizes that the original integral may be a broken

down into a product of functions and a simpler integral. Integration by substitution

recognizes that a portion of the function may be hard to integrate can be substituted

with a more easily integrated equivalent. Integration by partial integration

recognizes that a complicated equation may be split into a sum of more easily

integrated expressions. Integration of an equation may use a single technique or all

of these techniques together.

Centroids and moments of inertia

Since the integration of functions provides the area under the function then there

are manipulations to these integrals that allow the calculation of centroids and

moments. A centroid is the center of an area, that is to say that there is exactly as

much area to the left as there is to the right of the centroid along the x axis.

Likewise for the y axis centroid, the areas are the same size. A moment, in

mechanical systems, is torque or force at a distance. A moment in calculus is an area

at a distance. The order of the moment indicates the exponent on the distance.

Hence the first moment of area uses only x, whereas the second moment of area

uses x2. The second moment of area is sometimes referred to as the moment ofinertia. If the moment of inertia is calculated at the centroid, it is the minimum

value and called the centroidal moment of inertia. A shortcut calculation of moment

of inertia off the centroid is to add the area times the distance squared to the

centroidal moment of inertia.

Gradient, divergence, curl, and laplacian

For three dimensional functions, there are four interesting operations that help

analyze the behavior of a function using the del, , operator. The del operator is the

addition of partial derivatives in the three principal directions and is helpful in

many fluid applications. The first operation is the gradient of a scalar function using

the del operator on the function and finds the maximum rise of a function regardless

of the direction. The second operation is the divergence of a vector function and is

the dot product of the del operator and the function and can be used to check for

compressible fluids. The third operation is the curl of a vector function and the

cross product of the del operator and the function and can be used to determine the

presence and magnitude of rotational flow. The fourth operation is the Laplacian of

the scalar function and it is del squared of the function.

Homogeneous and nonhomogeneous differential equations

Homogeneous differential equations are differential equations that are only a sum of

a function and its derivatives that equal zero. The solution of homogeneous

equations is based on recognition of forms involving the value, or order, of thehighest directive. First order solutions are simple exponential form where second

order solutions are the sum of exponents which may oscillate or asymptotically

approach zero, known as damping. Nonhomogeneous differential equations are

similar to homogeneous differential equations except the sum does not equal zero.

The solution of nonhomogeneous differential equation require you solve the

homogeneous portion first, called the complimentary solution, and then the

nonhomogeneous portion, called the particular solution. The particular solution

may come from the method of undetermined coefficients and involves solving for

differential equation using some known initial value.

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