For values of x closer to x = a, we expect f(x) and f_{l}(x) to have close values. Since f_{l}(x) is a linear function we have a linear approximation of function f.

This approximation may be used to linearize non algebraic functions such as sine, cosine, log, exponential and many other functions in order to make their computation easier. Examples are presented below.
**Example 1:** Find the linear approximation of f(x) = tan x, for x close to 0.

**Solution to Example 1:**

We first compute f '(0)

f '(x) = sec ^{2} x

f '(0) = sec ^{2} (0) = 1

Hence the linear approximation f_{l}(x) is given by

f_{l}(x) = f(0) + f '(0) (x - 0) = x

The above result means that tan x ≈ x for x close to 0 when x is in **RADIANS**.

Put your calculator to **RADIANS** and calculate tan x for the following values of x.

x = 0 , x = 0.001 , x = 0.01, x = 0.1, x = 0.2, x = 0.3 and x = 0.5

Note compare tan x and x. Conclusion.

**Example 2:** Find the linear approximation of f(x) = ln x, for x close to 1.

**Solution to Example 2:**

We first compute f '(1)

f '(x) = 1 / x

f '(1) = 1

Hence the linear approximation f_{l}(x) is given by

f_{l}(x) = ln 1 + f '(1) (x - 1) = x - 1

The above result means that

ln x ≈ x - 1 for x close to 1.

Use your calculator to calculate ln x and x - 1 for

x = 1 , x = 1.001 , x = 1.01, x = 1.1, x = 1.5

Note compare ln x and x - 1. Conclusion.

**Example 3:** Find the linear approximation of f(x) = e^{x}, for x close to 0.

**Solution to Example 3:**

f '(0) = 1

Hence the linear approximation f_{l}(x) is given by

f_{l}(x) = e^{0} + f '(0) (x - 0) = 1 + x

Use your calculator to calculate e^{x} and 1 + x for

x = 0 , x = 0.001 , x = 0.01, x = 0.1 and x = 0.5

and compare.

Linear approximation is one of the simplest approximations to transcendental functions that cannot be expressed algebraically. However there are other more powerful methods that give better algebraic approximations to these functions.

More on applications of differentiation

applications of differentiation