1. Using a tangent line approximation of the function, find an approximate value for

2. Tangent line (9,3) Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point.

3. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). Tangent line (9,3)

4. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). We can rewrite the function as and use the shortcut rule for the derivative of a power function. Tangent line (9,3)

5. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). We can rewrite the function as and use the shortcut rule for the derivative of a power function. Bringing the power down, and then subtracting 1, we get Evaluate this derivative at x=9 to getThis can be simplified to Tangent line (9,3)

6. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). We can rewrite the function as and use the shortcut rule for the derivative of a power function. Bringing the power down, and then subtracting 1, we get Evaluate this derivative at x=9 to getThis can be simplified to Now that we have the slope, we can find the approximate value for The idea is that we start from the known value of, then take 2 steps forward, corresponding to up. Tangent line (9,3)

7. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). We can rewrite the function as and use the shortcut rule for the derivative of a power function. Bringing the power down, and then subtracting 1, we get Evaluate this derivative at x=9 to getThis can be simplified to Now that we have the slope, we can find the approximate value for The idea is that we start from the known value of, then take 2 steps forward, corresponding to up. Tangent line (9,3) The calculation should look like this:

8. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). We can rewrite the function as and use the shortcut rule for the derivative of a power function. Bringing the power down, and then subtracting 1, we get Evaluate this derivative at x=9 to getThis can be simplified to Now that we have the slope, we can find the approximate value for The idea is that we start from the known value of, then take 2 steps forward, corresponding to up. Tangent line (9,3) The calculation should look like this: This value is a bit too high, as expected. The actual value is close to 3.3166.

9. Using a tangent line approximation of the function, find an approximate value for The first step is to find some exact value of the function near x=11. We know so we will use x=9 as the starting point. Next we need the slope of the tangent line to f(x) at x=9. So we need to find the derivative f’(9). We can rewrite the function as and use the shortcut rule for the derivative of a power function. Bringing the power down, and then subtracting 1, we get Evaluate this derivative at x=9 to getThis can be simplified to Now that we have the slope, we can find the approximate value for The idea is that we start from the known value of, then take 2 steps forward, corresponding to up. Tangent line (9,3) The calculation should look like this: This value is a bit too high, as expected. The actual value is close to 3.3166. The percent error is: