A linear approximation is a mathematical term that refers to the use of a linear function to approximate a generic function. It is commonly used in the finite difference method to create first-order methods for solving or approximating equations. The linear approximation formula is used to get the closest estimate of a function for any given value.

Linear Approximation Formula

The linear approximation of a function is defined as using a line to approximate the function’s value at a given position. The concept of the tangent line is linked with the curve of a function with a point on it. The value of the function at any point that is very close to the provided point can be approximated using the equation of the tangent line if the equation of the tangent line is found at a given point. This notion is known as linear approximation, and it is also known as tangent line approximation because it is done using the tangent line.

Formula

Suppose a tangent line is drawn to the curve y = f(x) at the point (a, f(a)).

The equation of tangent is the required linear approximation formula. It can be derived by using the point-slope form as its slope is the derivative of function f(x) at x = a, that is, f ‘(a).

L(x) = f(a) + f ‘(a) (x – a)

where,

L(x) is the linear approximation of the function f(x) for x = a,

f'(a) is the first derivative of f(x) for x = a.

Sample Problems

Problem 1. Find the linear approximation of the function f(x) = x³ if the value of x is approaching 2.

Solution:

We have, f(x) = x³.

Now, f'(x) = d(f(x))/dx

= 3x²

For a = 2,

f(a) = 2³ = 8

f'(a) = 3 (2)² = 3 (4) = 12

Using the formula we have,

L(x) = f(a) + f ‘(a) (x – a)

= 8 + 12 (x – 2)

= 8 + 12x – 24

= 12x – 16

Problem 2. Find the linear approximation of the function f(x) = √x if the value of x is approaching 4.

Solution:

We have, f(x) = √x.

Now, f'(x) = d(f(x))/dx

= 1/(2√x)

For a = 4,

f(a) = √4 = 2 

f'(a) = 1/(2√4) = 1/4  

Using the formula we have,

L(x) = f(a) + f ‘(a) (x – a)

= 2 + (1/4) (x – 4)

= 2 + (x – 4)/4

= (x + 4)/4

Problem 3. Find the linear approximation of the function f(x) = sin x if the value of x is approaching π/3.

Solution:

We have, f(x) = sin x.

Now, f'(x) = d(f(x))/dx

= cos x

For a = π/3,

f(a) = sin π/3 = √3/2 

f'(a) = cos π/3 = 1/2  

Using the formula we have,

L(x) = f(a) + f ‘(a) (x – a)

= √3/2 + (1/2) (x – π/3)

= (3 (√3 + x) – π)/6

Problem 4. Find the linear approximation of the function f(x) = log x if the value of x is approaching 1.

Solution:

We have, f(x) = log x.

Now, f'(x) = d(f(x))/dx

= 1/x

For a = 1,

f(a) = log 1 = 0

f'(a) = 1/1 = 1

Using the formula we have,

L(x) = f(a) + f ‘(a) (x – a)

= 0 + 1 (x – 1)

= x – 1

Problem 5. Find the linear approximation of the function f(x) = tan x if the value of x is approaching π/3.

Solution:

We have, f(x) = tan x.

Now, f'(x) = d(f(x))/dx

= sec² x

For a = π/3,

f(a) = tan π/3 = √3

f'(a) = sec² π/3 = 4  

Using the formula we have,

L(x) = f(a) + f ‘(a) (x – a)

= √3 + (4) (x – π/3)

=  (3 (√3 + 4x) – 4π)/3

Practice Problems

1. Find the linear approximation of the function f(x) = cos x if the value of x is approaching [Tex]\pi/4[/Tex].
2. Find the linear approximation of the function f(x) = [Tex]e^x[/Tex] if the value of x is approaching 0.
3. Find the linear approximation of the function f(x) = [Tex]x^3 [/Tex]if the value of x is approaching 2.
4. Find the linear approximation of the function f(x) = √x if the value of x is approaching 9.
5. Find the linear approximation of the function f(x) = [Tex]ln(x) [/Tex]if the value of x is approaching 2.
6. Find the linear approximation of the function f(x) = 1/x if the value of x is approaching 1.
7. Find the linear approximation of the function f(x) = [Tex]x^2 + 3x + 2 [/Tex]if the value of x is approaching -1.
8. Find the linear approximation of the function f(x) = [Tex]arctan(x) [/Tex]if the value of x is approaching 0.
9. Find the linear approximation of the function f(x) =[Tex] sin(x^2) [/Tex]if the value of x is approaching 1.
10. Find the linear approximation of the function f(x) = [Tex]cos(ln(x))[/Tex] if the value of x is approaching 1.

FAQs on Linear Approximation Formula

1.What is a linear approximation?

A linear approximation is a method of approximating a function near a given point using the tangent line at that point. It provides a way to estimate the value of the function based on its derivative.

2.When is linear approximation useful?

Linear approximation is useful when you need a quick estimate of a function’s value near a particular point. It is especially helpful when the function is complex and difficult to compute directly.

3.How do you find the linear approximation of a function?

To find the linear approximation, you need to determine the function’s value and its derivative at a specific point. Then use the formula,
L(x) = f(a) + f ‘(a) (x – a), where a is the point of approximation.

4.What is geometric interpretation of a linear approximation?

Geometrically, the linear approximation represents the equation of the tangent line to the function at the point of interest. It approximates the function’s behavior near that point.

5.Can linear approximation be used for all functions?

Linear approximation can be used for any differentiable function. However, the accuracy of the approximation depends on the function’s behavior and the distance from the point of approximation.