help? [linear approximations]

Question

Answer 1

1)

Step 1:

The function is and the value of x is .

Linearization of the function is at is  .

Consider .

Differentiate on each side with respect to x.

.

Derivative of the cosine function: .

.

.

Substitute in above function.

.

.

Substitute in above function.

.

Step 2:

Linearization of the function at is .

Substitute corresponding values in linear approximation equation.

.

Solution:

Linear approximation to the function at the point is .

Answer 2

2)

Step 1:

The function is  and the value of x is .

Linearization of the function is at a is  .

Consider .

Differentiate on each side with respect to x.

.

Derivative of the logarithmic function: .

.

Substitute in above function.

.

.

Substitute in the function.

.

Step 2:

Linearization of the function is at is .

Substitute corresponding values in linear approximation equation.

.

Solution:

Linear approximation to the function at the point is .

Answer 3

3)

Step 1:

The function is  and the value of x is .

Linearization of the function is at a is  .

Consider .

Differentiate on each side with respect to x.

.

Power rule of derivatives: .

.

Substitute in above function.

.

.

Substitute in the function.

.

Step 2:

Linearization of the function is at is .

Substitute corresponding values in linear approximation equation.

.

Solution:

Linear approximation to the function at the point is .