f(x)=6{{x}^{4}}+2{{x}^{3}}**What is the coefficient for the term containing \[(x+1)^2\] in the Taylor polynomial, centered at \[x=-1\], of \[f\]?**

Choose 1 answer:

Choose 1 answer:

(Choice A)   

\[30\]

A

\[30\]

(Choice B)   

\[60\]

B

\[60\]

(Choice C)   

\[42\]

C

\[42\]

(Choice D)   

\[20\]

D

\[20\]

To find the coefficient of the term containing \((x+1)^2\) in the Taylor polynomial of \(f(x) = 6x^4 + 2x^3\) centered at \(x = -1\), we need to compute the second derivative of \(f(x)\) and evaluate it at \(x = -1\).

### Step 1: Compute the first and second derivatives of \(f(x)\)

The function is:
\[
f(x) = 6x^4 + 2x^3
\]

**First derivative:**
\[
f'(x) = \frac{d}{dx}\left(6x^4 + 2x^3\right) = 24x^3 + 6x^2
\]

**Second derivative:**
\[
f''(x) = \frac{d}{dx}\left(24x^3 + 6x^2\right) = 72x^2 + 12x
\]

### Step 2: Evaluate the second derivative at \(x = -1\)

Substitute \(x = -1\) into \(f''(x)\):
\[
f''(-1) = 72(-1)^2 + 12(-1) = 72(1) - 12 = 72 - 12 = 60
\]

### Step 3: Use the Taylor series formula

The coefficient of the \((x + 1)^2\) term in the Taylor polynomial is given by:
\[
\frac{f''(-1)}{2!}
\]

Since \(f''(-1) = 60\) and \(2! = 2\), the coefficient is:
\[
\frac{60}{2} = 30
\]

### Final Answer

The coefficient for the term containing \((x+1)^2\) is:

**(Choice A) 30**

Would you like more details or have any questions? Here are 5 related questions that could further your understanding:

1. How do you find the general formula for the nth term in a Taylor series?
2. What is the significance of centering a Taylor series at a point \(x = a\)?
3. How do higher-order derivatives affect the accuracy of a Taylor polynomial?
4. Can you calculate the Taylor series of \(f(x)\) centered at a different point, such as \(x = 0\)?
5. What happens if we center the Taylor series at a point where the function or its derivatives are undefined?

**Tip:** Always ensure that you evaluate derivatives carefully at the point of expansion when calculating Taylor series coefficients.

Discover how to find the coefficient of the term containing (x+1)^2 in the Taylor polynomial of f(x) = 6x^4 + 2x^3 centered at x = -1. This detailed solution involves computing derivatives, applying the Taylor series formula, and evaluating at the center point. Suitable for advanced high school students and learners interested in calculus.

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