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Let's walk through this problem step-by-step!

### Problem Summary
You are asked to **approximate \( 63^{1/3} \)** using a **Taylor polynomial**.

### (a) Determine an appropriate power function \( f(x) \).

- The function related to finding a cube root is \( f(x) = x^{1/3} \), since we are interested in \( 63^{1/3} \).

### (b) What would be an appropriate base point \( a \)?

- In Taylor approximations, we pick a point \( a \) close to the value we want to approximate (63), and one that makes calculations easier.
- A good choice is \( a = 64 \), because \( 64^{1/3} = 4 \) is easy to compute. This gives us a simple starting point for the approximation.

### (c) Find \( T_1(x) \).

This part asks for the **first-order Taylor polynomial**. The Taylor expansion of a function \( f(x) \) at \( a \) is:
\[
T_1(x) = f(a) + f'(a) (x - a)
\]
For \( f(x) = x^{1/3} \):
1. **Evaluate** \( f(a) \):
   \[
   f(64) = 64^{1/3} = 4
   \]
2. **Find the derivative** \( f'(x) \):
   \[
   f'(x) = \frac{1}{3} x^{-2/3}
   \]
3. **Evaluate** \( f'(a) \) at \( a = 64 \):
   \[
   f'(64) = \frac{1}{3} \cdot 64^{-2/3} = \frac{1}{3} \cdot \frac{1}{16} = \frac{1}{48}
   \]
4. **Write the first-order Taylor polynomial**:
   \[
   T_1(x) = 4 + \frac{1}{48} (x - 64)
   \]

### (d) Use \( T_1(x) \) to estimate the value of \( 63^{1/3} \).

Now, substitute \( x = 63 \) into the Taylor polynomial \( T_1(x) \):
\[
T_1(63) = 4 + \frac{1}{48} (63 - 64) = 4 + \frac{1}{48} (-1)
\]
\[
T_1(63) = 4 - \frac{1}{48} = 4 - 0.0208 \approx 3.9792
\]

Thus, the estimate for \( 63^{1/3} \) is approximately:
\[
63^{1/3} \approx 3.9792
\]

---

### Conclusion

- The Taylor polynomial approximation suggests \( 63^{1/3} \approx 3.9792 \). This is close to the actual value, which is about 3.979057. So our approximation is quite good!

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Do you have any questions, or would you like further details?

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### 5 Related Questions:
1. What is a second-order Taylor polynomial, and how would it improve the approximation?
2. How does the error bound behave for Taylor polynomials? Can we calculate the error for this approximation?
3. Can Taylor approximations work for other roots, such as \( x^{1/5} \)?
4. How can we derive a general formula for the \( n \)-th derivative of \( f(x) = x^{1/3} \)?
5. How do higher-order polynomials compare in accuracy to first-order approximations?

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### Tip:
When choosing the base point \( a \), try to select values that simplify calculations (like perfect squares or cubes). This reduces computational errors and simplifies derivatives.

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 T_1(x). (d) Use T_1(x) to estimate the value of 63^(1/3).

Learn how to approximate 63^(1/3) using a first-order Taylor polynomial. This step-by-step solution explains the process of selecting a power function, choosing an appropriate basepoint, and calculating the first-order polynomial to estimate the cube root of 63. Suitable for students in Grades 11-12 and early college.

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