Basics of Calculus

Basics of Calculus

Calculus is a branch of mathematics that studies how things change. It helps us understand motion, growth, and areas under curves. Calculus is divided mainly into two parts: Differential Calculus and Integral Calculus.

Differential Calculus

Differential Calculus focuses on the concept of the derivative, which represents the rate at which a quantity changes. For example, it tells us how fast a car is moving at any instant.

The derivative of a function $f(x)$ with respect to $x$ is denoted as:

$\frac{df}{dx}\text{ or }{f}^{\prime }(x)$

The derivative is defined as the limit:

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$

Example

If $f(x)=x^2$, then the derivative is:

$f^{\prime }(x)=\frac{d}{dx}(x^2)=2x$

This means at any point $x$, the slope of the curve $y=x^2$ is $2x$.

Integral Calculus

Integral Calculus deals with the integral, which represents the accumulation of quantities, such as area under a curve.

The definite integral of a function $f(x)$ from $a$ to $b$ is written as:

$\underset{a}{\overset{b}{\int }}f(x)dx$

It calculates the total area between the graph of $f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$.

Example

Find the area under the curve $f(x)=x$ from $x=0$ to $x=3$:

$\underset{0}{\overset{3}{\int }}xdx={\left[\frac{x^2}{2}\right]}_0^3=\frac{3^2}{2}-\frac{0^2}{2}=\frac{9}{2}=4.5$

Summary

• Derivatives tell us how a function changes at any point (rate of change).
• Integrals tell us the total accumulation, like area under curves.
• Calculus is fundamental for physics, engineering, economics, and many sciences.

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Calculus opens the door to understanding the world through mathematics!