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Calculus

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Calculus! Many students get fear with this branch of mathematics, as it is considered as one of the difficult topics of math. The word Calculus is derived from Latin word, which means calculating something and used for counting anything. Calculus covers a wide area of modern mathematics. It mainly focuses on limits, functions, derivatives, integrals, and infinite series. Calculus plays a very important role in modern mathematics. It has two branches Differential Calculus and Integral Calculus.
Calculus have various applications as follows: Differential calculus persist velocity, acceleration optimization and slope of the curve, while integral calculus contains area, volume, center of mass, work, pressure and arc length.
Calculus provides tools to solve many problems. One of the problem is paradox, which comes in limit and infinite series. If we see the fundamental theorem of calculus, then we find that the differential and integration operation are opposite to each other or we can say that inverse to each other. Calculus have so many applications, and is widely used in different fields like computer science, statistics, economics business, physical science etc.

Calculus Topics

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Given below are the important topics commonly seen in calculus.

Chain rule: Chain rule is a formula for computing the derivative of the composition of two or more functions.
If l and m are two functions, then the chain rule is
$\frac{dl}{dy}$ = $\frac{dl}{dm}$ . $\frac{dm}{dy}$

where, 'l' differentiate with respect to m and m differentiate with respect to y.

Continuity: A function with no gaps, jumps or undefined points. A continuous function is a function in which small changes in the input results to small changes in the output.

Definite Integral: Let f be a function which is continuous on the closed interval [a, b]. The definite integral of f from a to b is defined to be the limit
$\int_{a}^{b}$ f(x) dx = $\lim_{n\rightarrow \infty}$ $\sum_{i=1}^{n}$f($x_{i})$ $\Delta x$

Derivative: Derivative is a measure of how a function changes as its input changes.

Discontinuity: If a function is not continuous at a point in its domain, then it is said to have a discontinuity.

Extreme Value Theorem: Helps in determining the possible maximum and minimum values of a function on certain intervals.
If a function f(x) is continuous on a closed interval [a, b], then f(x) has both a maximum and minimum values on [a, b].

Higher Order Derivatives: Any derivative beyond the first derivative is referred as higher order derivative.

Indefinite Integral: An integral without upper and lower limits.

Limit: The value of a function approaches as the variable approaches some point. If the function is not continuous, the limit could be different from the value of the function at that point.

L Hospital's Rule: Uses derivatives to evaluate limits involving indeterminate forms.

$\lim_{x\rightarrow c}$ $\frac{f(x)}{g(x)}$ = $\lim_{x\rightarrow c}$$\frac{f'(x)}{g'(x)}$

Maximum: A function f(x) has relative maximum value at x = a, if f(a) is greater than any value immediately preceding or following.

Mean Value Theorem: If f(x) is defined and continuous on the interval [a, b] and differentiable on (a, b), then there is atleast one number c in the interval (a, b), such that

f'(c) = $\frac{f(b) - f(a)}{b-a}$

Minimum: A function f(x) has relative minimum value at x = b, if f(b) is less than any value immediately preceding or following.

Newton's Method: Method for finding successively better approximations to the roots of a real-valued function.

Slope: Slope is the ratio of the rise divided by the run between two points on a line. It describes its steepness, incline or grade.

Power Rule: Power rule is an important differentiation rule. Polynomials are differentiated using the rule.
$\frac{d}{dx}$ x$^{n}$ = nx$^{n - 1}$ ; n$\neq$ 0

Quotient Rule: Method of finding the derivative of a function that is the quotient of two other functions for which derivative exist.
Quotient rule is given as
$\frac{d}{dx}$ ($\frac{g(x)}{h(x)})$ = $\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^{2}}$

Calculus Problems

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Given below are some of the problems in calculus.

Solved Examples

Question 1: Is the following function continuous?
  f(x) = $\begin{Bmatrix}
x^{2}-1&for\ x<2 \\
x^{3}+5 & for\ x\geq2
\end{Bmatrix}$
Solution:
The given function is a junction of two continuous functions. So, we worry about discontinuity, where the functions meet i.e, at x = 2

$\lim_{x\rightarrow 2^{-}}$f(x)= $\lim_{x\rightarrow 2^{-}}(x^{2}-1)$ = 4 - 1 = 3
$\lim_{x\rightarrow 2^{+}}$f(x)= $\lim_{x\rightarrow 2^{+}}(x^{3}+5)$ = 8 + 5 = 13

Since $\lim_{x\rightarrow 2^{-}}$ f(x) $\neq$  $\lim_{x\rightarrow 2^{+}}$
f(x) does not exist and f is not continuous at x = 2.

Question 2: Solve $\lim_{x\rightarrow 3}$ $\frac{3x^{2}+3x -5}{3-2x}$
Solution:
$\lim_{x\rightarrow 3}$ $\frac{3x^{2}+3x -5}{3-2x}$ = $\frac{\lim_{x\rightarrow 3}(3x^{2}+3x-5)}{\lim_{x\rightarrow 3}(3-2x)}$

= $\frac{\lim_{x\rightarrow 3}3x^{2}+\lim_{x\rightarrow 3}3x - \lim_{x\rightarrow 3} 5}{\lim_{x\rightarrow 3}3 -\lim_{x\rightarrow 3}2x }$

= $\frac{3(3)^{2}+ 3(3)-5}{3 - 2(3)}$

= $\frac{27+9-5}{3-6}$

=  - $\frac{31}{3}$