Question

Find the linear approximation to f(x) at x = x o. Graph the function and its linear approximation. f(x) = sin 3x, x 0, = 0

Answer 1

For values of x closer to x = a, we expect f(x) and fl(x) to have close values. Since fl(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 fl(x) is given by

fl(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 fl(x) is given by

fl(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) = ex, for x close to 0.

Solution to Example 3:

f '(0) = 1

Hence the linear approximation fl(x) is given by

fl(x) = e0 + f '(0) (x - 0) = 1 + x

Use your calculator to calculate ex and 1 + x for

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

and compare.

Answer 2

So what will this be? for this question