Can you explain following concepts?

Question

The concept of limit in calculus.
The idea of differentiation...first,second and third order differentiation and partial differentiation.
The idea of Integration.
all of the above should be simply explained with their relation to two or more functions.Please be specific and do not give roundabout answers.I know the concepts but wish them expressed simplistically.You can include any website address as well but you must write your answer.

Answer 1

A function f(z) is said to have a limit lim_(z->a)f(z)==c if, for all epsilon>0, there exists a delta>0 such that |f(z)-c| lim_(z->a^-)==lim_(z^a)
(1)

or from above
lim_(z->a^+)==lim_(zva).
(2)

if the two are equal, then "the" limit is said to exist
lim_(z->a)==lim_(z->a^-)==lim_(z->a^+).
(3)

Limits are implemented in Mathematica as Limit[f, x->x0]. This command also takes options Direction (which can be set to +1 or -1), and Analytic, which computes symbolic limits for functions.

A lower limit h
lowerlim_(n->infty)S_n==lim_(n->infty)__S_n==h
(4)

is said to exist if, for every epsilon>0, |S_n-h|
An upper limit k
upperlim_(n->infty)S_n==lim_(n->infty)^_S_n==k
(5)

is said to exist if, for every epsilon>0, |S_n-k|
Indeterminate limit forms of types infty/infty and 0/0 can often be computed with L'Hospital's rule. Types 0.infty can be converted to the form 0/0 by writing
f(x)g(x)==(f(x))/(1/g(x)).
(6)

Types 0^0, infty^0, and 1^infty are treated by introducing a dependent variable
y==f(x)^(g(x))
(7)

so that
lny==g(x)ln[f(x)],
(8)

then calculating lim lny. The original limit then equals e^(limlny),
L==limf(x)^(g(x))==e^(limlny).
(9)

The indeterminate form infty-infty is also frequently encountered.


The derivative of a function represents an infinitesimal change in the function with respect to one of its variables.

The "simple" derivative of a function f with respect to a variable x is denoted either f^'(x) or
(df)/(dx),
(1)

often written in-line as df/dx. When derivatives are taken with respect to time, they are often denoted using Newton's Eric Weisstein's World of Biography overdot notation for fluxions,
(dx)/(dt)==x^..
(2)

The "d-ism" of Leibnitz's df/dt eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).

When a derivative is taken n times, the notation f^((n))(x) or
(d^nf)/(dx^n)
(3)

is used, with
x^.,x^..,x^...,
(4)

etc., the corresponding fluxion notation.

When a function f(x,y,...) depends on more than one variable, a partial derivative
(partialf)/(partialx),(partial^2f)/(partialxpartialy), etc.
(5)

can be used to specify the derivative with respect to one or more variables.

The derivative of a function f(x) with respect to the variable x is defined as
f^'(x)=lim_(h->0)(f(x+h)-f(x))/h,
(6)

but may also be calculated more symmetrically as
f^'(x)==lim_(h->0)(f(x+h)-f(x-h))/(2h),
(7)

provided the derivative is known to exist.

It should be noted that the above definitions refer to "real" derivatives, i.e., derivatives which are restricted to directions along the real axis. However, this restriction is artificial, and derivatives are most naturally defined in the complex plane, where they are sometimes explicitly referred to as complex derivatives. In order for complex derivatives to exist, the same result must be obtained for derivatives taken in any direction in the complex plane. Somewhat surprisingly, almost all of the important functions in mathematics satisfy this property, which is equivalent to saying that they satisfy the Cauchy-Riemann equations.

These considerations can lead to confusion for students because elementary calculus texts commonly consider only "real" derivatives, never alluding the the existence of complex derivatives, variables, or functions. For example, textbook examples to the contrary, the "derivative" (read: complex derivative) d|z|/dz of the absolute value function |z| does not exist because at every point in the complex plane, the value of the derivative depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than x==0 as
(d|x|)/(dx)=={-1 for x<0; undefined for x==0; 1 for x>0.
(8)

As a result of the fact that computer algebra programs such as Mathematica generically deal with complex variables (i.e., the definition of derivative always means complex derivative), d|x|/dx correctly returns unevaluated by such software.

If the first derivative exists, the second derivative may be defined as
f^('')(x)=lim_(h->0)(f^'(x+h)-f^'(x))/h
(9)

and calculated more symmetrically as
f^('')(x)==lim_(h->0)(f(x+2h)-2f(x+h)+f(x))/(h^2),
(10)

again provided the second derivative is known to exist.

Note that in order for the limit to exist, both lim_(h->0^+) and lim_(h->0^-) must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite Eric Weisstein's World of Biography wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

A three-dimensional generalization of the derivative to an arbitrary direction is known as the directional derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into "tangent maps."

Performing numerical differentiation is in many ways more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes.

Simple derivatives of some simple functions follow.
d/(dx)x^n=nx^(n-1)
(11)
d/(dx)lnx=1/x
(12)
d/(dx)sinx=cosx
(13)
d/(dx)cosx=-sinx
(14)
d/(dx)tanx=sec^2x
(15)
d/(dx)cscx=-cscxcotx
(16)
d/(dx)secx=secxtanx
(17)
d/(dx)cotx=-csc^2x
(18)
d/(dx)e^x=e^x
(19)
d/(dx)a^x=(lna)a^x
(20)
d/(dx)sin^(-1)x=1/(sqrt(1-x^2))
(21)
d/(dx)cos^(-1)x=-1/(sqrt(1-x^2))
(22)
d/(dx)tan^(-1)x=1/(1+x^2)
(23)
d/(dx)cot^(-1)x=-1/(1+x^2)
(24)
d/(dx)sec^(-1)x=1/(xsqrt(x^2-1))
(25)
d/(dx)csc^(-1)x=-1/(xsqrt(x^2-1))
(26)
d/(dx)sinhx=coshx
(27)
d/(dx)coshx=sinhx
(28)
d/(dx)tanhx=sech^2x
(29)
d/(dx)cothx=-csch^2x
(30)
d/(dx)sechx=-sechxtanhx
(31)
d/(dx)cschx=-cschxcothx
(32)
d/(dx)snx=cnxdnx
(33)
d/(dx)cnx=-snxdnx
(34)
d/(dx)dnx=-k^2snxcnx.
(35)

where sn(x)=sn(x,k), cn(x)=cn(x,k), etc. are Jacobi elliptic functions, and the product rule and quotient rule have been used extensively to expand the derivatives.

There are a number of important rules for computing derivatives of certain combinations of functions. Derivatives of sums are equal to the sum of derivatives so that
[f(x)+...+h(x)]^'==f^'(x)+...+h^'(x).
(36)

In addition, if c is a constant,
d/(dx)[cf(x)]==cf^'(x).
(37)

The product rule for differentiation states
d/(dx)[f(x)g(x)]==f(x)g^'(x)+f^'(x)g(x),
(38)

where f^' denotes the derivative of f with respect to x. This derivative rule can be applied iteratively to yield derivative rules for products of three or more functions, for example,
[fgh]^'=(fg)h^'+(fg)^'h==fgh^'+(fg^'+f^'g)h
(39)
=f^'gh+fg^'h+fgh^'.
(40)

The quotient rule for derivatives states that
d/(dx)[(f(x))/(g(x))]==(g(x)f^'(x)-f(x)g^'(x))/([g(x)]^2)
(41)

while the power rule gives
d/(dx)(x^n)==nx^(n-1).
(42)

Other very important rule for computing derivatives is the chain rule, which states that for y==y(u),
(dy)/(dx)==(dy)/(du).(du)/(dx),
(43)

or more generally, for z==z(x(t),y(t))
(dz)/(dt)==(partialz)/(partialx)(dx)/(dt)+(partialz)/(partialy)(dy)/(dt),
(44)

where partialz/partialx denotes a partial derivative.

Miscellaneous other derivative identities include
(dy)/(dx)==((dy)/(dt))/((dx)/(dt))
(45)
(dy)/(dx)==1/((dx)/(dy)).
(46)

If F(x,y)==C, where C is a constant, then
dF==(partialF)/(partialy)dy+(partialF)/(partialx)dx==0,
(47)

so
(dy)/(dx)==-((partialF)/(partialx))/((partialF)/(partialy)).
(48)

Derivative identities of inverse functions include
(dx)/(dy)=1/((dy)/(dx))
(49)
(d^2x)/(dy^2)=-(d^2y)/(dx^2)((dy)/(dx))^(-3)
(50)
(d^3x)/(dy^3)=[3((d^2y)/(dx^2))^2-(d^3y)/(dx^3)(dy)/(dx)]((dy)/(dx))^(-5).
(51)

A vector derivative of a vector function
X(t)=[x_1(t); x_2(t); |; x_k(t)]
(52)

can be defined by
(dX)/(dt)==[(dx_1)/(dt); (dx_2)/(dt); |; (dx_k)/(dt)].
(53)

The nth derivatives of x^nf(x) for n==1, 2, ... are
d/(dx)[xf(x)]=f(x)+xf^'(x)
(54)
(d^2)/(dx^2)[x^2f(x)]=2f(x)+4xf^'(x)+x^2f^('')(x)
(55)
(d^3)/(dx^3)[x^3f(x)]=6f(x)+18xf^'(x)+9x^2f^('')(x)+x^3f^(''')(x).
(56)

The nth row of the triangle of coefficients 1; 1, 1; 2, 4, 1; 6, 18, 9, 1; ... (Sloane's A021009) is given by the absolute values of the coefficients of the Laguerre polynomial L_n(x)

An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."

The Riemann integral of the function f(x) over x from a to b is written
int_a^bf(x)dx.
(1)

Note that if f(x)==1, the integral is written simply
int_a^bdx
(2)

as opposed to int_a^b1dx.

Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The process of computing an integral is called integration (a more archaic term for integration is quadrature), and the approximate computation of an integral is termed numerical integration.

There are two classes of (Riemann) integrals: definite integrals such as (1), which have upper and lower limits, and indefinite integrals, such as
intf(x)dx,
(3)

which are written without limits. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if F(x) is the indefinite integral for f(x), then
int_a^bf(x)dx==F(b)-F(a).
(4)

Since the derivative of a constant is zero, indefinite integrals are defined only up to an arbitrary constant of integration C, i.e.,
intf(x)dx==F(x)+C.
(5)

Wolfram Research maintains a web site http://integrals.wolfram.com/ that can find the indefinite integral of many common (and not so common) functions.

Differentiating integrals leads to some useful and powerful identities. For instance, if f(x) is continuous, then
d/(dx)int_a^xf(x^')dx^'==f(x),
(6)

which is the first fundamental theorem of calculus. Other derivative-integral identities include
d/(dx)int_x^bf(x^')dx^'==-f(x),
(7)

the Leibniz integral rule
d/(dx)int_a^bf(x,t)dt==int_a^bpartial/(partialx)f(x,t)dt
(8)

(Kaplan 1992, p. 275), its generalization
d/(dx)int_(u(x))^(v(x))f(x,t)dt==v^'(x)f(x,v(x))-u^'(x)f(x,u(x))+int_(u(x))^(v(x))partial/(partialx)f(x,t)dt
(9)

(Kaplan 1992, p. 258), and
d/(dx)int_a^xf(x,t)dt==1/(x-a)int_a^x[(x-a)partial/(partialx)f(x,t)+(t-a)partial/(partialt)f(x,t)+f(x,t)]dt,
(10)

as can be seen by applying (9) on the left side of (10) and using partial integration.

Other integral identities include
int_0^xdt_nint_0^(t_n)dt_(n-1)...int_0^(t_3)dt_2int_0^(t_2)f(t_1)dt_1==1/((n-1)!)int_0^x(x-t)^(n-1)f(t)dt
(11)
partial/(partialx_k)(x_jJ_k)==delta_(jk)J_k+x_jpartial/(partialx_k)J_k==J+rdel .J
(12)
int_VJd^3r=int_Vpartial/(partialx_k)(x_iJ_k)-int_Vrdel .Jd^3r
(13)
=-int_Vrdel .Jd^3r
(14)

and the amusing integral identity
int_(-infty)^inftyF(f(x))dx==int_(-infty)^inftyF(x)dx,
(15)

where F is any function and
f(x)==x-sum_(n==0)^infty(a_n)/(x+b_n)
(16)

as long as a_n>=0 and b_n is real (Glasser 1983).

Integrals with rational exponents can often be solved by making the substitution u==x^(1/n), where n is the least common multiple of the denominator of the exponents.

Answer 2

Concept of limit. Draw any two points on a curved line and connect the points with a straight line. The slope of the straight line serves as a rough proxy for the slope of the curve between the points. As the points are moved closer together they will become a better proxy and as the distance between the points aproaches zero the "limit" is the actual slope (tangent) of the curve at the point of interest.

The concept of differentiation (dy/dx?). Differentiation (using differences!) is used to find the slope of a curve (on a graph?) at a given point. If you draw a straight line through two points on a curve it is a rough proxy for the slope of the curve between the points. If you measure (using a ruler?) the vertical and horizontal differences between the points and divide them (dy/dx) they approximate in value the slope (tangent) of the curve. As the points are drawn closer and closer to the point on the curve of interest they become a better proxy (and both dy and dx get smaller). As dy and dx approach zero, the limit is the slope (tangent) to the curve.

There are often families of equations (of successively higher order) that can each be represented by curves on a graph and therefore differentiated to obtain the slope of the curve when that has useful meaning (or for study?). For example, you can measture a distance s with a tape or odometer. Traveling the distance in a given time s/t gives speed (or velocity if a vector with both magnitude and direction). Changing the velocity with respect to time (v/t) gives acceleration a. In free fall, acceleration is represented by acceleration constant g. Finding the slope at a point on the curve (sometimes the "curve" is a straight line) tells you the instantaneous rate of change at that point.

Partial differentiation can be demonstrated using a watermellon, a rather complex collection of curves on its 3-D surface. Cut the watermellon any way you wish into two pieces. Pick one of the pieces and note the cut (round or oval?) and you have a single curve that can be differentiate to find the slope of the curve at any point (if that is useful or for study). Need more information about the watermellon (or some other phenomenon?) make another cut and differentiate that.

Integration is the process of finding the complete (exact) area under a curve (or a portion of a curve between vertical limits rising from the x-axis). You can get the approximate answer graphically by dividing the area into vertical boxes using approximate heights of the boxes based on inspection of the curve then sum the areas of the boxes. The more boxes the closer will be your approximation. Calculus does the same thing mathematically using an infinite number of boxes. Sometimes the area gives useful information (or study practice). For example, if a moving point on the x-axis represents the passage of time and the y-axis represent the speed of a car at any moment (along x), the area under the curve (intergral) will represent the distance traveled. Any square within the area is speed x time (miles per hour x hours?) which is distance. The area under the curve from two hours to seven hours on the x-axis will represent the exact distance traveled in that interval.

Hope that helps a little.

Answer 3

1) The method name and the argument types 2) return ... it returns a variable to the calling method 3) you just call the method name (using its signature) 4) Not sure what you mean by this 5) A way of passing arguments into a method by just sending the values to the method directly 6) A method with the same name as another method, but a different signature (meaning it takes different arguments) 7) block is an area within brackets { }. Variables have a "scope" where they are available and accessable. Depending on where you declare the variable, the variable's scope changes. 8) Not sure what you meanby this.

Answer 4

the answers to your questions may be found in Courant and John's text on calculus.

simple explanations will come when YOU understand the material

Answer 5

Hey.. its way too early in the morning for that... !!!