Chapter 11 Review – Calculus: Taylor Series, Approximations, and Power Series

Q1. Find the linear approximating polynomial for $f(x) = \tan x$ centered at $a = \pi$.

Background

Topic: Linear Approximation (Taylor Polynomial of Degree 1)

This question tests your understanding of how to construct a linear (first-degree) Taylor polynomial for a function at a given point.

Key Terms and Formulas:

• Linear approximation: $p_1(x) = f(a) + f'(a)(x - a)$

• $f(x) = \tan x$

• $a = \pi$

Step-by-Step Guidance

1. Compute $f(a)$: Evaluate $\tan(\pi)$.

2. Find $f'(x)$: Differentiate $f(x) = \tan x$ to get $f'(x)$.

3. Evaluate $f'(a)$: Plug $a = \pi$ into $f'(x)$.

4. Write the linear approximation formula: $p_1(x) = f(a) + f'(a)(x - a)$.

Try solving on your own before revealing the answer!

Q2. Find the quadratic approximation of $f(x) = 64 - x^2$ at $x = 0$.

Background

Topic: Quadratic Approximation (Taylor Polynomial of Degree 2)

This question asks you to find the second-degree Taylor polynomial (quadratic approximation) for a function at a specific point.

Key Terms and Formulas:

• Quadratic approximation: $p_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2$

• $f(x) = 64 - x^2$

• $a = 0$

Step-by-Step Guidance

1. Compute $f(0)$: Plug $x = 0$ into $f(x)$.

2. Find $f'(x)$ and $f''(x)$: Differentiate $f(x)$ once and twice.

3. Evaluate $f'(0)$ and $f''(0)$.

4. Plug these values into the quadratic approximation formula.

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Q3. Find the Taylor polynomial of order 3 generated by $f(x) = \ln(x + 1)$ at $a = 4$.

Background

Topic: Taylor Polynomial (Degree 3)

This question tests your ability to construct a third-degree Taylor polynomial for a function at a specified point.

Key Terms and Formulas:

• Taylor polynomial of degree 3: $P_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 + \frac{f'''(a)}{6}(x - a)^3$

• $f(x) = \ln(x + 1)$

• $a = 4$

Step-by-Step Guidance

1. Compute $f(4)$: Evaluate $\ln(4 + 1)$.

2. Find $f'(x)$, $f''(x)$, and $f'''(x)$: Differentiate $f(x)$ up to the third derivative.

3. Evaluate $f'(4)$, $f''(4)$, and $f'''(4)$.

4. Plug these values into the Taylor polynomial formula for degree 3.

Try solving on your own before revealing the answer!

Q4. Use the appropriate Taylor polynomial of degree 3 at $x = 16$ to approximate $16.07$.

Background

Topic: Taylor Polynomial Approximation

This question asks you to use a third-degree Taylor polynomial centered at $x = 16$ to approximate a nearby value.

Key Terms and Formulas:

• Taylor polynomial: $P_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 + \frac{f'''(a)}{6}(x - a)^3$

• Approximate $f(16.07)$ using $a = 16$

Step-by-Step Guidance

1. Identify the function $f(x)$ you are approximating (not specified, but likely $f(x) = \sqrt{x}$ or similar).

2. Compute $f(16)$, $f'(16)$, $f''(16)$, and $f'''(16)$.

3. Set up the Taylor polynomial formula centered at $x = 16$.

4. Plug $x = 16.07$ into the polynomial and simplify up to the last step.

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Q5. Find the series' radius of convergence and interval of convergence for $\sum_{n=0}^{\infty} (x - 4)^n$.

Background

Topic: Radius and Interval of Convergence

This question tests your ability to determine where a power series converges.

Key Terms and Formulas:

• Radius of convergence: Use the ratio test or root test.

• Interval of convergence: Find all $x$ for which the series converges.

• General form: $\sum_{n=0}^{\infty} (x - 4)^n$

Step-by-Step Guidance

1. Apply the ratio test to $a_n = (x - 4)^n$.

2. Set up the limit $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.

3. Solve for $x$ such that the limit is less than 1.

4. Express the interval of convergence based on the radius found.

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Q6. Find the series' radius of convergence and interval of convergence for $\sum_{n=0}^{\infty} \frac{(x - 7)^n}{n56^n}$.

Background

Topic: Radius and Interval of Convergence

This question tests your ability to use the ratio test to find the radius and interval of convergence for a more complex power series.

Key Terms and Formulas:

• Ratio test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

• General term: $a_n = \frac{(x - 7)^n}{n56^n}$

Step-by-Step Guidance

1. Write the ratio $\left| \frac{a_{n+1}}{a_n} \right|$ for the given series.

2. Simplify the ratio and take the limit as $n \to \infty$.

3. Set the limit less than 1 to solve for $x$.

4. Determine the radius and interval of convergence.

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Q7. Find the series' interval of convergence for $\sum_{n=1}^{\infty} \frac{(x - 3)^n}{(5n)!}$.

Background

Topic: Interval of Convergence

This question tests your ability to use the ratio test for a series with factorials in the denominator.

Key Terms and Formulas:

• Ratio test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

• General term: $a_n = \frac{(x - 3)^n}{(5n)!}$

Step-by-Step Guidance

1. Set up the ratio $\left| \frac{a_{n+1}}{a_n} \right|$.

2. Simplify the factorial expression in the denominator.

3. Take the limit as $n \to \infty$ and analyze for which $x$ the series converges.

4. Express the interval of convergence.

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Q8. Use the geometric series $f(x) = \frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k$ for $|x| < 1$ to find the power series representation for $f(10x) = \frac{1}{1 - 10x}$ centered at 0.

Background

Topic: Power Series Representation (Geometric Series)

This question tests your ability to manipulate geometric series to represent related functions.

Key Terms and Formulas:

• Geometric series: $\frac{1}{1 - r} = \sum_{k=0}^{\infty} r^k$ for $|r| < 1$

• Here, $r = 10x$

Step-by-Step Guidance

1. Substitute $r = 10x$ into the geometric series formula.

2. Write the power series representation for $f(10x)$.

3. State the interval of convergence for $|10x| < 1$.

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Q9. Use $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$ for $-\infty < x < \infty$ to find the power series representation for $f(x) = e^{9x}$.

Background

Topic: Power Series Representation (Exponential Function)

This question tests your ability to use the power series for $e^x$ to represent $e^{9x}$.

Key Terms and Formulas:

• $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$

• Substitute $x$ with $9x$

Step-by-Step Guidance

1. Replace $x$ with $9x$ in the series for $e^x$.

2. Write the resulting power series for $e^{9x}$.

3. State the interval of convergence.

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Q10. Show that by differentiating term by term the expansion of $f(x) = \frac{1}{(1 + x)^2}$, the result is the expansion for $f(x) = -\frac{2}{(1 + x)^3}$.

Background

Topic: Differentiation of Power Series

This question tests your ability to differentiate a power series term by term and relate it to another function's series expansion.

Key Terms and Formulas:

• Power series for $\frac{1}{(1 + x)^2}$

• Term-by-term differentiation: $\frac{d}{dx} \sum a_n x^n = \sum a_n n x^{n-1}$

Step-by-Step Guidance

1. Write the power series expansion for $\frac{1}{(1 + x)^2}$.

2. Differentiate each term in the series with respect to $x$.

3. Compare the resulting series to the expansion for $-\frac{2}{(1 + x)^3}$.

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Q11. Find the first four nonzero terms in the Maclaurin series for $f(x) = e^{-5x}$.

Background

Topic: Maclaurin Series (Taylor Series at $x = 0$)

This question tests your ability to expand $e^{-5x}$ as a Maclaurin series and find the first four nonzero terms.

Key Terms and Formulas:

• Maclaurin series for $e^{ax}$: $\sum_{n=0}^{\infty} \frac{(a x)^n}{n!}$

• Here, $a = -5$

Step-by-Step Guidance

1. Write the general Maclaurin series for $e^{-5x}$.

2. Expand the first four terms: $n = 0, 1, 2, 3$.

3. Simplify each term.

4. Write the series up to the fourth term.

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Q12. Find the Taylor series generated by $f(x) = e^x$ at $x = a = 8$.

Background

Topic: Taylor Series (Centered at $a$)

This question tests your ability to write the Taylor series for $e^x$ centered at $x = 8$.

Key Terms and Formulas:

• Taylor series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

• For $e^x$, $f^{(n)}(a) = e^a$ for all $n$

Step-by-Step Guidance

1. Write the general Taylor series formula for $e^x$ centered at $a = 8$.

2. Express $f^{(n)}(8)$ for $e^x$.

3. Write the series in terms of $e^8$ and $(x - 8)^n$.

4. State the interval of convergence.

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Q13. Use power series operations to find the Taylor series at $x = 0$ for $f(x) = x^6 e^x$.

Background

Topic: Power Series Operations

This question tests your ability to combine power series (multiplication) to find the Taylor series for a product of functions.

Key Terms and Formulas:

• $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

• Multiply $x^6$ by the series for $e^x$

Step-by-Step Guidance

1. Write the power series for $e^x$.

2. Multiply each term in the series by $x^6$.

3. Express the resulting series as $\sum_{n=0}^{\infty} \frac{x^{n+6}}{n!}$.

4. State the interval of convergence.

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Q14. Find the first four terms of the binomial series for $\sqrt[3]{1 + x}$.

Background

Topic: Binomial Series Expansion

This question tests your ability to use the binomial theorem for fractional exponents to expand a function as a power series.

Key Terms and Formulas:

• Binomial series: $(1 + x)^r = 1 + r x + \frac{r(r-1)}{2!} x^2 + \frac{r(r-1)(r-2)}{3!} x^3 + \ldots$

• Here, $r = \frac{1}{3}$

Step-by-Step Guidance

1. Identify $r = \frac{1}{3}$ and write the binomial series formula.

2. Compute the coefficients for $x$, $x^2$, and $x^3$ using the formula.

3. Write out the first four terms of the series.

4. State the interval of convergence for the binomial series.

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