Find the third order Taylor polynomial of f(x)= index 3 sqrt x at x=64.

We are tasked with finding the third-order Taylor polynomial of the function \( f(x) = \sqrt[3]{x} \) around \( x = 64 \).

The general form of the third-order Taylor polynomial for a function \( f(x) \) around \( x = a \) is given by:

\[
T_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3
\]

Here, \( a = 64 \) and \( f(x) = x^{1/3} \). We'll need to compute \( f(a) \), \( f'(a) \), \( f''(a) \), and \( f'''(a) \).

### Step 1: Compute \( f(a) \)
Since \( f(x) = x^{1/3} \):

\[
f(64) = 64^{1/3} = 4
\]

### Step 2: Compute \( f'(x) \)
Using the power rule, the derivative of \( f(x) = x^{1/3} \) is:

\[
f'(x) = \frac{1}{3}x^{-2/3}
\]

Now, evaluate \( f'(x) \) at \( x = 64 \):

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

### Step 3: Compute \( f''(x) \)
Now, find the second derivative of \( f(x) = \frac{1}{3}x^{-2/3} \):

\[
f''(x) = \frac{d}{dx} \left( \frac{1}{3}x^{-2/3} \right) = -\frac{2}{9}x^{-5/3}
\]

Evaluate \( f''(x) \) at \( x = 64 \):

\[
f''(64) = -\frac{2}{9} \cdot 64^{-5/3} = -\frac{2}{9} \cdot \frac{1}{1024} = -\frac{2}{9216}
\]

### Step 4: Compute \( f'''(x) \)
Finally, compute the third derivative of \( f(x) = -\frac{2}{9}x^{-5/3} \):

\[
f'''(x) = \frac{10}{27}x^{-8/3}
\]

Evaluate \( f'''(x) \) at \( x = 64 \):

\[
f'''(64) = \frac{10}{27} \cdot 64^{-8/3} = \frac{10}{27} \cdot \frac{1}{4096} = \frac{10}{110592}
\]

### Step 5: Construct the Third-Order Taylor Polynomial
Now, we substitute the values of \( f(64) \), \( f'(64) \), \( f''(64) \), and \( f'''(64) \) into the Taylor polynomial formula:

\[
T_3(x) = 4 + \frac{1}{48}(x - 64) + \frac{-2}{9216} \frac{(x - 64)^2}{2!} + \frac{10}{110592} \frac{(x - 64)^3}{3!}
\]

Simplifying:

\[
T_3(x) = 4 + \frac{1}{48}(x - 64) - \frac{1}{9216}(x - 64)^2 + \frac{5}{110592}(x - 64)^3
\]

This is the third-order Taylor polynomial for \( f(x) = \sqrt[3]{x} \) at \( x = 64 \).

Would you like further details or any clarification? 

Here are 5 questions for further exploration:
1. How does a Taylor polynomial approximate a function?
2. What are the conditions for a Taylor series to converge to the function it represents?
3. Can higher-order terms in the Taylor series improve the accuracy of approximation?
4. How would you find the fourth-order Taylor polynomial for the same function?
5. 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, \( x = 64 \)). Near the point, lower-order polynomials work well, but as you move further, higher-order terms become crucial for precision.

Find the third order Taylor polynomial of f(x) = ³√x at x = 64.

Explore the process of finding the third-order Taylor polynomial for the function f(x) = ³√x around x = 64. This detailed solution involves the use of derivatives and Taylor's theorem to create an approximation for the cube root function.

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