https://file.aiquickdraw.com/mathbot/file/1729051948905q9kdffil.jpg

Let's address each part of the problem using a Taylor polynomial approximation for \( 63^{1/3} \).

### (a) Determine an appropriate power function \( f(x) \).
Since we want to approximate \( 63^{1/3} \), a suitable choice for \( f(x) \) would be:
\[
f(x) = x^{1/3}
\]

### (b) What would be an appropriate basepoint \( a \)?
The basepoint \( a \) should be a value close to 63 that makes the calculations simpler and for which we can easily find the cube root. A good choice here is \( a = 64 \), since \( 64^{1/3} = 4 \).

### (c) Find \( T_1(x) \).
The Taylor polynomial of degree 1 (linear approximation) centered at \( a = 64 \) for the function \( f(x) = x^{1/3} \) is given by:
\[
T_1(x) = f(a) + f'(a)(x - a)
\]

1. Calculate \( f(a) \):
   \[
   f(64) = 64^{1/3} = 4
   \]

2. Calculate \( f'(x) \):
   The derivative of \( f(x) = x^{1/3} \) is:
   \[
   f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \cdot x^{2/3}}
   \]

3. Evaluate \( f'(64) \):
   \[
   f'(64) = \frac{1}{3 \cdot 64^{2/3}} = \frac{1}{3 \cdot 16} = \frac{1}{48}
   \]

4. Write \( T_1(x) \):
   \[
   T_1(x) = 4 + \frac{1}{48}(x - 64)
   \]

### (d) Use \( T_1(x) \) to estimate the value of \( 63^{1/3} \).
Substitute \( x = 63 \) into \( T_1(x) \):
\[
T_1(63) = 4 + \frac{1}{48}(63 - 64) = 4 - \frac{1}{48} = 4 - 0.02083 \approx 3.9792
\]

So, the estimate for \( 63^{1/3} \) using the Taylor polynomial \( T_1(x) \) is approximately \( 3.9792 \).

Would you like more details on any of the steps or have any questions?

### Related Questions:
1. How would the approximation improve if we used a second-degree Taylor polynomial \( T_2(x) \)?
2. What is the actual value of \( 63^{1/3} \), and how close is the Taylor polynomial approximation?
3. How do higher-order derivatives affect the accuracy of a Taylor polynomial?
4. Can you use a different basepoint \( a \) for the Taylor expansion, like \( a = 60 \), and how would that change the result?
5. What other methods can be used to approximate \( 63^{1/3} \) besides Taylor series?

### Tip:
For better approximations, using a basepoint \( a \) that is very close to the desired value \( x \) often results in a more accurate Taylor polynomial approximation, especially when using only a few terms.

Suppose we want to approximate the value of 63^(1/3) using a Taylor polynomial.
(a) Determine an appropriate power function f(x).
(b) What would be an appropriate basepoint a?
(c) Find T1(x).
(d) Use T1(x) to estimate the value of 63^(1/3).

Learn how to approximate the cube root of 63 using a Taylor polynomial. This example walks through the steps of determining an appropriate power function, basepoint, and using the first-degree Taylor polynomial for estimation. Suitable for advanced high school or early college math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation