Math Problem Statement
Find the third order Taylor polynomial of f(x)= index 3 sqrt x at x=64.
Solution
We are tasked with finding the third-order Taylor polynomial of the function around .
The general form of the third-order Taylor polynomial for a function around is given by:
Here, and . We'll need to compute , , , and .
Step 1: Compute
Since :
Step 2: Compute
Using the power rule, the derivative of is:
Now, evaluate at :
Step 3: Compute
Now, find the second derivative of :
Evaluate at :
Step 4: Compute
Finally, compute the third derivative of :
Evaluate at :
Step 5: Construct the Third-Order Taylor Polynomial
Now, we substitute the values of , , , and into the Taylor polynomial formula:
Simplifying:
This is the third-order Taylor polynomial for at .
Would you like further details or any clarification?
Here are 5 questions for further exploration:
- How does a Taylor polynomial approximate a function?
- What are the conditions for a Taylor series to converge to the function it represents?
- Can higher-order terms in the Taylor series improve the accuracy of approximation?
- How would you find the fourth-order Taylor polynomial for the same function?
- What are the limitations of using a Taylor polynomial for approximation near singular points?
Tip: The accuracy of a Taylor polynomial depends on how far you are from the expansion point (here, ). Near the point, lower-order polynomials work well, but as you move further, higher-order terms become crucial for precision.
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Math Problem Analysis
Mathematical Concepts
Taylor Series
Derivatives
Polynomial Approximation
Power Functions
Formulas
Taylor polynomial: T_n(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + (f'''(a)/3!)(x - a)^3
Derivative of a power function: f'(x) = nx^(n-1)
Theorems
Taylor's Theorem
Suitable Grade Level
College Calculus or Advanced High School Calculus