1. Section 9.7 – Taylor Theorem

2. Taylor’s Theorem Like all of the “Value Theorems,” this is an existence theorem.

3. Taylor’s Theorem Re-Worded The amount needed for the approximate to be equal to the actual. The approximate value. The actual value.

4. Example 1 Find the third degree Taylor polynomial: Use the Taylor Theorem Equation: We can calculate this value since we know the function. But what if we did not know the function?

5. The Remainder in Taylor’s Theorem According to Taylor’s Theorem, the remainder (or error) is one degree higher than the Taylor Polynomial used to approximate the actual value. Remember, this is the same result from power zooming!

6. Example 1 Continued What would happen if we could find the Taylor polynomial but didn’t know the function and could not calculate this value?

7. Example 1 Continued Seek Comfort in the Familiar: Taylor’s Theorem, like the Intermediate Value, Extreme Value, and Mean Value Theorem is an existence theorem.

8. Example 1 Continued Instead of actually calculating the remainder (or error). The best we can usually do is find a bound for the error. In other words, something the error is always less than. The worst case scenario for your approximation.

9. The Lagrange Error Bound One useful consequence of Taylor’s Theorem is that: The Error Use the absolute value to keep things positive when calculating error.

10. Example 2 nMax Error 00.82436 10.20609 20.03435 30.00429 40.00043 50.00004 5 th Degree Polynomial

11. Example 3

12. Extension: Taylor Theorem and The Mean Value Theorem Taylor Series is 1 term long Remainder Mean Value Theorem is a special case of Taylors Theorem. Solve for the derivative