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 answer to two decimal places.)
tan x ≈ x

did you get an answerr?

12

To verify the given linear approximation at a = 0, we need to compare the linear approximation with the actual function at that point.

The linear approximation of a function f(x) at a point x = a is given by the formula:
L(x) = f(a) + f'(a)(x - a)

In this case, the function is f(x) = tan(x), and we want to verify the linear approximation at a = 0. Therefore, we need to find L(x) using the formula above.

First, let's find f(a) and f'(a):

f(a) = tan(0) = 0 (since tan(0) = 0)

To find f'(a), we differentiate f(x) = tan(x) using the derivative rules:

f'(x) = sec^2(x) (since the derivative of tan(x) is sec^2(x))

Now, evaluate f'(a) at a = 0:

f'(a) = f'(0) = sec^2(0) = 1 (since sec(0) = 1)

Now we can find L(x):

L(x) = f(a) + f'(a)(x - a)
= 0 + 1(x - 0)
= x

Therefore, the linear approximation of tan(x) at a = 0 is given by L(x) = x.

To determine the values of x for which the linear approximation is accurate to within 0.1, we need to find the interval of x values where the difference between the linear approximation and the actual function is less than or equal to 0.1.

|tan(x) - L(x)| ≤ 0.1

Since L(x) = x, we have:

|tan(x) - x| ≤ 0.1

We can solve this inequality to find the interval of x values:

-0.1 ≤ tan(x) - x ≤ 0.1

Next, we need to solve the inequalities:

tan(x) - x ≥ -0.1
tan(x) - x ≤ 0.1

To solve these inequalities, we'll need to use a graphical or numerical approach. The simplest way to do this is by using a graphing calculator or an online graphing tool.

Plot the graphs of y = tan(x) and y = x, and observe the x-values where the functions are within ±0.1 of each other. Alternatively, you can use a numerical method to find the specific x-values.

By solving these inequalities, we can determine the interval of x-values for which the linear approximation is accurate to within 0.1. Round your answer to two decimal places and express it using interval notation.