Linear Approximation Calculator
Linear approximation is also known as a tangent line or tangent in geometry means a line or plane that intersects a curve or a curved surface at exactly one point.
What is the Linear Approximation Calculator?
'Linear Approximation Calculator' is an online tool that helps to calculate the value of linear approximation for a given function. Online Linear Approximation Calculator helps you to calculate the value of linear approximation for a given function in a few seconds.
Linear Approximation Calculator
How to Use Linear Approximation Calculator?
Please follow the steps below on how to use the calculator:
- Step1: Enter the function and point in the given input boxes.
- Step 2: Click on the "Calculate" button to find the value of linear approximation for a given function.
- Step 3: Click on the "Reset" button to clear the fields and enter a new function.
How to Find Linear Approximation?
Linear approximation is defined as the equation of a tangent line. We know that the slope of the tangent that is drawn to a curve y = f(x) at x = a is its derivative at that point. i.e., the slope of the tangent line is f'(a). Thus, the linear approximation formula is an application of derivatives.
The formula to calculate the linear approximation for a function y = f(x) is given by
L(x) = f(a) + f '(a) (x - a)
Where L(x) is the linear approximation of f(x) at x = a and f '(a) is the derivative of f(x) at x = a
Let us see an example to understand briefly.
Solved Examples on Linear Approximation Calculator
Example 1:
Find the linear approximation of f(x) = 2x2 at x = 3 and verify it using linear approximation calculator.
Solution:
Given: Function f(x) = 2x2
We have to find the linear approximation of f(x) at a = 3.
So f(a) = 2(3)2 = 18.
f '(x) = d/dx(2x2) = 4x
f '(a) = 4(3) = 12
Linear approximation L(x) = f(a) + f '(a) (x - a)
= 18 + (12)(x - 3)
= 18 + 12x - 36
L(x) = 12x - 18
Example 2:
Find the linear approximation of f(x) = 3x + 5 at x = 4 and verify it using linear approximation calculator.
Solution:
Given: Function f(x) = 3x + 5
We have to find the linear approximation of f(x) at a = 4.
So f(a) = 3(4) + 5 = 17.
f '(x) = d/dx(3x + 5) = 3
f '(a) = 3
Linear approximation L(x) = f(a) + f '(a) (x - a)
= 17 + (3)(x - 4)
= 17 + 3x - 12
L(x) = 3x + 5
Example 3:
Find the linear approximation of f(x) = 4x3 at x = 2 and verify it using linear approximation calculator.
Solution:
Given: Function f(x) = 4x3
We have to find the linear approximation of f(x) at a = 2.
So f(a) = 4(2)3 = 32.
f '(x) = d/dx(4x3) = 12x2
f '(a) = 12(2)2 = 48
Linear approximation L(x) = f(a) + f '(a) (x - a)
= 32 + (48)(x - 2)
= 32 + 48x - 96
L(x) = 48x - 64
Similarly, you can try the linear approximation calculator. to find the linear approximation for the given function:
- f(x) = x2 + 5x at x = -2
- f(x) = x3 + 2 at x = 5