Higher Mathematics

Jai Ganesh · 2022-12-05 00:39:38

Second Derivative

Finding Maxima and Minima using Derivatives.

Jai Ganesh · 2022-12-06 00:36:39

Concave Upward and Downward

Inflection Points.

Jai Ganesh · 2022-12-09 00:45:33

Arc Length

Arc length is the distance between two points along a section of a curve.

Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. A rectifiable curve has a finite number of segments in its rectification (so the curve has a finite length).

If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function)

, then the curve is rectifiable (i.e., it has a finite length).

The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.

Arc Length

Jai Ganesh · 2022-12-12 02:14:44

Implicit Differentiation.

Implicit differentiation

In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.

To differentiate an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y and then differentiate. Instead, one can totally differentiate R(x, y) = 0 with respect to x and y and then solve the resulting linear equation for dy/dx to explicitly get the derivative in terms of x and y. Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.

Jai Ganesh · 2022-12-16 17:58:23

Partial Derivatives

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

Jai Ganesh · 2023-01-01 00:03:19

Proof of the Derivatives of sin, cos and tan

Trigonometric Identities.

Jai Ganesh · 2023-01-01 22:47:38

Taylor Series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

Definition

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number a is the power series

where n! denotes the factorial of n. In the more compact sigma notation, this can be written as

where

denotes the nth derivative of f evaluated at the point a. (The derivative of order zero of f is defined to be f itself and [math]{(x - a)}^0 and 0! are both defined to be 1.)

When a = 0, the series is also called a Maclaurin series.[

Jai Ganesh · 2023-01-06 00:06:25

Intermediate Value Theorem

Intermediate Value Theorem

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.

This has two important corollaries:

1. If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).
2. The image of a continuous function over an interval is itself an interval.