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Calculus Problems

Calculus is a vast branch of mathematics, that is used for calculating the integrals and derivatives of functions. Calculus is generally taught at a higher level, only after one has full knowledge of the general algebra.

Calculus is basically divided into two broad categories: Differential Calculus and Integral Calculus. Differentiation refers to differentiating or separating things, whereas integration refers to putting the things all together. 
Thus, we can say integration is just the reverse process of derivative. Hence, calculus problems are those problems, which are concerned with finding derivatives, integration, and all other related problems, that are interrelated to derivatives and integration as their base.

The calculus problems include derivatives, mean value theorem, higher order derivatives, the riemann integral, differentiability and integrability including Fundamental Theorem of Calculus, First Mean Value Theorem, Second Mean Value Theorem, sequences and series of functions, difference equation, linear difference equation with constant coefficients and its solution as to find out the particular integral, etc. Calculus of finite differences includes the difference operator as del, the condition for the minimum (or maximum) of a function f (x), which is very useful for researchers. 

Given below are some solved examples in calculus.

Example 1:

Find the Derivative of $x^{5}$.

Solution:

We know, the derivative of $x^{n}$ is $n x^{n-1}$.
We can write the derivative of $x^{5}$ as $5x^{4}$.

Example 2:

Find the differentiation of $\frac{1}{x}$.

Solution:

Differentiation of $\frac{1}{x}$ equals to differentiation of $x^{-1}$.

This equals to $-1(x^{-1-1})$ which becomes $-x^{-2}$.

Example 3:

Find the integration of $3x^{2}$.

Solution:

We can write, integration of $3x^{2}$ equals to 3 times integration of $x^{2}$.
And, this becomes 3 times $\frac{x^{3}}{3}$ + c.
We can simplify this to get $x^{3}$ + c.

Example 4:

Find the integration of $e^{2x}$.

Solution:

Integration of $e^{2x}$ equals to integration of $\frac{e^{2x}}{\text{Derivative of 2x}}$ + c.
We can write this as $\frac{e^{2x}}{2}$ + c. 

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