help? linear approximations?

Question

Answer 1

4)

Step 1:

The function is  and .

Linear approximation of the function is .

Consider .

Differentiate on each side with respect to .

Substitute in above function.

.

.

Substitute in the function.

.

Linear approximation at  is .

Substitute all the corresponding values in linear approximation equation.

.

The corresponding linear approximation is .

Comments

Continued....

Step 2:

Approximate value for . 

Here, .

.

Approximate value for .

Exact value of the .

The approximation is not good since the difference between approximation and exact value is more.

Approximate value for . 

Here, .

Approximate value for .

Exact value of the .

The approximation is not good since the difference between approximation and exact value is large.

Solution:

 Linear approximation of the function is  .

Approximate value for .

Approximate value for .

The approximation is not good since the approximation values are differ from the exact values.