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The problem provided asks to approximate \( 63^{1/3} \) using a Taylor polynomial.

Let's break it down step-by-step:

### (a) Determine an appropriate power function \( f(x) \):
To approximate \( 63^{1/3} \), we can choose the function \( f(x) = x^{1/3} \). This function gives the cube root of \( x \), which directly applies to the given expression.

### (b) What would be an appropriate basepoint \( a \)?
To find a basepoint \( a \), we want to select a value near 63 for which we can easily compute the cube root. A good choice is \( a = 64 \), because \( 64^{1/3} = 4 \). This makes calculations simpler and keeps the approximation close to the target value.

### (c) Find \( T_1(x) \):
The first-degree Taylor polynomial \( T_1(x) \) around \( x = a \) is given by:
\[
T_1(x) = f(a) + f'(a)(x - a)
\]
For \( f(x) = x^{1/3} \), we first compute \( f'(x) \):
\[
f'(x) = \frac{1}{3}x^{-2/3}
\]
Now, using \( a = 64 \):
\[
f(64) = 64^{1/3} = 4
\]
\[
f'(64) = \frac{1}{3} \cdot 64^{-2/3} = \frac{1}{3} \cdot \frac{1}{16} = \frac{1}{48}
\]
Thus, the first-degree Taylor polynomial is:
\[
T_1(x) = 4 + \frac{1}{48}(x - 64)
\]

### (d) Use \( T_1(x) \) to estimate the value of \( 63^{1/3} \):
Now, substitute \( x = 63 \) into the Taylor polynomial \( T_1(x) \):
\[
T_1(63) = 4 + \frac{1}{48}(63 - 64) = 4 + \frac{1}{48}(-1) = 4 - \frac{1}{48} = 4 - 0.02083 = 3.97917
\]
Thus, the estimated value of \( 63^{1/3} \) is approximately \( 3.97917 \).

Would you like more details on any of the steps or further clarifications?

Here are 5 questions that extend the discussion:
1. How can the accuracy of this approximation be improved with higher-order terms?
2. What is the general formula for the \( n \)-th derivative of \( f(x) = x^{1/3} \)?
3. Why is \( a = 64 \) a better choice than \( a = 60 \) or \( a = 65 \) for this approximation?
4. How would the approximation differ if you used a second-degree Taylor polynomial?
5. Can this method be used to approximate other roots, like \( 63^{1/5} \)?

**Tip:** When using Taylor polynomials, selecting a basepoint \( a \) close to the value you're approximating improves the accuracy of the approximation!

Suppose we want to approximate the value of 63^(1/3) using a Taylor polynomial.

Learn how to approximate the cube root of 63 using a first-degree Taylor polynomial. This example includes step-by-step instructions, starting with determining an appropriate function, selecting a basepoint, and calculating the first derivative for an accurate approximation.

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