Calculus is an important section of Mathematics which involves a detailed study about different functions and behavior of those functions. Calculus is sub-divided into two important branches which are finding derivatives of functions and finding integrals of functions. Derivatives of a function gives the rate of change of one variable with respect to the other and integrals of functions analyses the area covered under the graph of a given function. In order to analyze math in advanced level, it is very important to understand calculus as it helps us understand functions in a much detailed manner.
Example 1: Use the derivative method to find the slope of the given straight line, f(x) = 5x + 2.
In order to find the derivative, we can use the Power rule of the Derivatives:
èd(xn)/dx = n * xn-1
We can distribute the derivative to both the terms:
èd(5x + 2)/dx è[ d(5x)/ dx ] + [ d(2)/ dx ]
Using the above formula, we get
è5(x1-1) + 0 = 5
Hence the slope or the derivative of the given straight line is 5.
Example 2: Calculate the integral of the function, f(x) = 3x2 – 8x.
Given function: f(x) = 3x2 – 8x.
In order to find the integral of the given function, we can use the formula:
è ∫xn dx= x(n+1)/ (n+1)
Applying the above formula for the given function, we get
∫(fx) dx = 3 * x2+1/(2 + 1) – 8 * x1+1/(1 + 1) + c
∫f(x)dx = 3x3/3 – 8x2/2 + c
We can simplify it as: ∫f(x)dx = x3 – 4x2 + c