Math Problem Statement
Solution
Let's walk through this problem step-by-step!
Problem Summary
You are asked to approximate using a Taylor polynomial.
(a) Determine an appropriate power function .
- The function related to finding a cube root is , since we are interested in .
(b) What would be an appropriate base point ?
- In Taylor approximations, we pick a point close to the value we want to approximate (63), and one that makes calculations easier.
- A good choice is , because is easy to compute. This gives us a simple starting point for the approximation.
(c) Find .
This part asks for the first-order Taylor polynomial. The Taylor expansion of a function at is: For :
- Evaluate :
- Find the derivative :
- Evaluate at :
- Write the first-order Taylor polynomial:
(d) Use to estimate the value of .
Now, substitute into the Taylor polynomial :
Thus, the estimate for is approximately:
Conclusion
- The Taylor polynomial approximation suggests . This is close to the actual value, which is about 3.979057. So our approximation is quite good!
Do you have any questions, or would you like further details?
5 Related Questions:
- What is a second-order Taylor polynomial, and how would it improve the approximation?
- How does the error bound behave for Taylor polynomials? Can we calculate the error for this approximation?
- Can Taylor approximations work for other roots, such as ?
- How can we derive a general formula for the -th derivative of ?
- How do higher-order polynomials compare in accuracy to first-order approximations?
Tip:
When choosing the base point , try to select values that simplify calculations (like perfect squares or cubes). This reduces computational errors and simplifies derivatives.
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Math Problem Analysis
Mathematical Concepts
Taylor Polynomials
Approximations
Derivatives
Cube Roots
Formulas
First-order Taylor polynomial: T_1(x) = f(a) + f'(a) * (x - a)
Derivative of f(x) = x^(1/3): f'(x) = 1/3 * x^(-2/3)
Theorems
Taylor Polynomial Theorem
Suitable Grade Level
Grades 11-12, Early College