Linear approximation estimates the value of a function near a given point by using the equation of its tangent line at that point, making it essentially a tangent line approximation and a common tool in methods like finite differences for constructing first-order solutions.
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.
Solved Examples
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
- Find the linear approximation of the function f(x) = cos x if the value of x is approaching
\pi/4 . - Find the linear approximation of the function f(x) =
e^x if the value of x is approaching 0. - Find the linear approximation of the function f(x) =
x^3 if the value of x is approaching 2. - Find the linear approximation of the function f(x) = √x if the value of x is approaching 9.
- Find the linear approximation of the function f(x) =
ln(x) if the value of x is approaching 2.