Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.) 1/((1 + 9x)^4) ≈ 1 − 36x

Part 1)

See Below.

Part 2)

Step-by-step explanation:

Part 1)

The linear approximation L for a function f at the point x = a is given by:

We want to verify that the expression:

Is the linear approximation for the function:

At x = 0.

So, find f'(x). We can use the chain rule:

Simplify. Hence:

Then the slope of the linear approximation at x = 0 will be:

And the value of the function at x = 0 is:

Thus, the linear approximation will be:

Hence verified.

Part B)

We want to determine the values of x for which the linear approximation L is accurate to within 0.1.

In other words:

By definition:

Therefore:

We can solve this by using a graphing calculator. Please refer to the graph shown below.

We can see that the inequality is true (i.e. the graph is between y = 0.1 and y = -0.1) for x values between -0.179 and -0.178 as well as -0.010 and 0.012.

In interval notation:

49/6 is going to equal 8 1/6