AP®MathsCollege BoardCalculus BCExam QuestionsUnit 10: Infinite Sequences and SeriesSeries Representations of Functions

Series Representations of Functions (College Board AP® Calculus BC): Exam Questions

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Series Representations of Functions

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1
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2 marks

The function f has derivatives of all orders for all real numbers. It is known that f open parentheses 0 close parentheses equals 6, f apostrophe open parentheses 0 close parentheses equals 1, f to the power of apostrophe apostrophe end exponent open parentheses 0 close parentheses equals negative 18, and f to the power of apostrophe apostrophe apostrophe end exponent open parentheses 0 close parentheses equals 3. Write the third-degree Taylor polynomial for f about x equals 0 and use it to approximate the value of f open parentheses 1 close parentheses.

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2
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1 mark

The Maclaurin series for the exponential function e to the power of x is

e to the power of x space equals space 1 plus x plus fraction numerator x squared over denominator 2 factorial end fraction plus fraction numerator x cubed over denominator 3 factorial end fraction plus... plus fraction numerator x to the power of n over denominator n factorial end fraction plus...

Let f be the function defined by f open parentheses x close parentheses equals e to the power of negative 4 x end exponent. Write the third-degree Taylor polynomial for f about x equals 0.

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3a
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3 marks

A function f is given in power series form as f open parentheses x close parentheses equals sum from n equals 1 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n times x to the power of n over denominator 4 to the power of n times n end fraction.

Use the ratio test to determine the radius of convergence of the series.

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3b
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2 marks

By testing the endpoints, determine the full interval of convergence for f open parentheses x close parentheses. Show the work that leads to your answer.

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4
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3 marks

A function f is given in power series form as f open parentheses x close parentheses equals sum from n equals 1 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n times x to the power of n over denominator 5 to the power of n end fraction.

Show that the power series is a geometric series, and hence determine the interval of convergence for f open parentheses x close parentheses.

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5a
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2 marks

The function f is defined by f open parentheses x close parentheses equals fraction numerator 1 over denominator 1 plus x squared end fraction. Find f apostrophe open parentheses x close parentheses.

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5b
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2 marks

The Maclaurin series for f is given by 1 minus x squared plus x to the power of 4 minus x to the power of 6 plus... plus open parentheses negative 1 close parentheses to the power of n x to the power of 2 n end exponent+..., which converges to f open parentheses x close parentheses for negative 1 less than x less than 1.

Find the first three nonzero terms and the general term for the Maclaurin series for f to the power of apostrophe open parentheses x close parentheses.

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5c
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1 mark

Use your results from parts (a) and (b) to find the sum of the infinite series negative 2 over 3 plus 4 over 3 cubed minus 6 over 3 to the power of 5 plus... plus open parentheses negative 1 close parentheses to the power of n fraction numerator 2 n over denominator 3 to the power of 2 n minus 1 end exponent end fraction plus....

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1a
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2 marks
A graph of function f on axes labelled from -3 to 5 on the x-axis and -4 to 5 on the y-axis. The tangent line to f at the point (0, 2) is also shown, which also goes through the point (1, -2)

The function f has derivatives of all orders for all real numbers x. A portion of the graph of f is shown above, including the tangent line to the graph of f at x equals 0. Selected derivatives for f at x equals 0 are given in the table below.

bold italic n

Error converting from MathML to accessible text.

2

negative 15

3

147

4

negative 9274 over 5

Write the third-degree Taylor polynomial for f about x equals 0.

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1b
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2 marks

Write the first three nonzero terms of the Taylor series for e to the power of x about x equals 0. Write the second-degree Taylor polynomial for e to the power of x f open parentheses x close parentheses about x equals 0.

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2
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3 marks

The function f has derivatives of all orders for all real numbers. It is known that f open parentheses 0 close parentheses equals 4 and f apostrophe open parentheses 0 close parentheses equals negative 3.

Let g be the function such that g open parentheses 0 close parentheses equals 5 and g to the power of apostrophe open parentheses x close parentheses equals e to the power of 2 x end exponent times f open parentheses x close parentheses. Write the second-degree Taylor polynomial for g about x equals 0.

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3a
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2 marks

The Taylor series for ln open parentheses 1 plus x close parentheses about x equals 0 is given by

x minus x squared over 2 plus x cubed over 3 minus x to the power of 4 over 4 plus... plus fraction numerator open parentheses negative 1 close parentheses to the power of n plus 1 end exponent x to the power of n over denominator n end fraction plus...,

which converges to ln open parentheses 1 plus x close parentheses on its interval of convergence.

Let f be the function defined by f open parentheses x close parentheses equals x squared ln open parentheses 1 plus x over 2 close parentheses space.

Write the first four nonzero terms and the general term of the Taylor series for f about x equals 0.

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3b
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5 marks

Determine the interval of convergence of the Taylor series for f about x equals 0. Show the work that leads to your answer.

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4
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3 marks

The Taylor series for a function f about x equals 2 is given by sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent fraction numerator 3 to the power of n over denominator 2 n end fraction open parentheses x minus 2 close parentheses to the power of n. Find the first three nonzero terms and the general term of the Taylor series for f apostrophe, the derivative of f, about x equals 2.

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5
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2 marks

The Maclaurin series for a function f is given by f open parentheses x close parentheses equals sum from n equals 0 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n open parentheses 2 x close parentheses to the power of 2 n end exponent over denominator open parentheses 2 n close parentheses factorial end fraction equals 1 minus 2 x squared plus fraction numerator 2 x to the power of 4 over denominator 3 end fraction minus fraction numerator 4 x to the power of 6 over denominator 45 end fraction plus..., which converges for all real numbers x. If the first three nonzero terms of the series are used to approximate f open parentheses x close parentheses on the interval space minus 1 half less or equal than x less or equal than 1 third, use the alternating series error bound to determine an upper bound for the error of the approximations.

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6
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2 marks

A function f has derivatives of all orders for all real numbers x. The fourth-degree Taylor polynomial for f about x equals 2 is used to approximate f open parentheses 1.9 close parentheses. Given that open vertical bar f to the power of open parentheses 5 close parentheses end exponent open parentheses x close parentheses close vertical bar less or equal than 9 for 1.5 less or equal than x less or equal than 2, use the Lagrange error bound to show that this approximation is within 1 over 10 to the power of 6 of the exact value of f open parentheses 1.9 close parentheses.

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7
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2 marks

Let y = f(x) be the particular solution to the differential equation \frac{\text{d} y}{\text{d} x} = y \cdot (x \text{ln} x) with initial condition f(1) = 4. It can be shown that f ' ' (1) = 4.

Write the second-degree Taylor polynomial for f about x = 1. Use the Taylor polynomial to approximate f(2).

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8
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2 marks

It can be shown that f(- 3) = \sum_{n = 1}^{\infty} \frac{(n + 1)(- 3)^{n}}{n^{2} 6^{n}} = \sum_{n = 1}^{\infty} \frac{n + 1}{n^{2}} \left( - \frac{1}{2} \right)^{n} and that the first three terms of this series sum to S_{3} = - \frac{125}{144}. Show that \left| f(- 3) - S_{3} \right| < \frac{1}{50}.

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9
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3 marks

Let g(x) = \sum_{n = 1}^{\infty} \frac{(n + 1) x^{2 n}}{n^{2} 3^{n}}. Use the ratio test to determine the radius of convergence of the Maclaurin series for g.

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1
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2 marks

Let y equals f open parentheses x close parentheses be the particular solution to the differential equation fraction numerator d y over denominator d x end fraction equals 2 x squared y times ln x with the initial condition f open parentheses 1 close parentheses equals 5. It can be shown that f to the power of apostrophe apostrophe end exponent open parentheses 1 close parentheses equals 10.

Write the second-degree Taylor polynomial for f about x equals 1, and use the polynomial to approximate f open parentheses 2 close parentheses.

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2
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5 marks

The function f has derivatives of all orders for all real numbers. It is known that f open parentheses 0 close parentheses equals 3, f apostrophe open parentheses 0 close parentheses equals negative 2, and f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals f open parentheses 2 x squared close parentheses. Write the fourth-degree Taylor polynomial for f about x equals 0. Show the work that leads to your answer.

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3
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4 marks

A function f is such that f open parentheses 3 close parentheses equals 2 and the Taylor series of its derivative f to the power of apostrophe is given by space f to the power of apostrophe open parentheses x close parentheses equals sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent 5 to the power of n open parentheses x minus 3 close parentheses to the power of n minus 1 end exponent. Use this function to determine f explicitly within the radius of convergence of the series.

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4a
4 marks

The Maclaurin series for a function f is given by f open parentheses x close parentheses equals x squared minus x to the power of 4 over 2 plus x to the power of 6 over 3 minus x to the power of 8 over 4 plus... plus open parentheses negative 1 close parentheses to the power of n plus 1 end exponent times x to the power of 2 n end exponent over n plus..., which converges on open square brackets negative 1 comma space 1 close square brackets.

Write the first four nonzero terms of the Maclaurin series for f to the power of apostrophe open parentheses t to the power of 4 close parentheses. Given that g open parentheses x close parentheses equals integral subscript 0 superscript x f to the power of apostrophe open parentheses t to the power of 4 close parentheses space d t, use the first two nonzero terms of the Maclaurin series for g to approximate g open parentheses 1 close parentheses.

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4b
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3 marks

Show that your approximation in part (a) must differ from g open parentheses 1 close parentheses by less than 1 over 10. Justify your answer.

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5
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3 marks

Let the function f be defined by f open parentheses x close parentheses equals ln open parentheses 1 plus 2 x squared close parentheses. The Taylor series for f about x equals 0 is given by sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent fraction numerator 2 to the power of n x to the power of 2 n end exponent over denominator n end fraction, which converges on the interval open square brackets negative fraction numerator 1 over denominator square root of 2 end fraction comma space fraction numerator 1 over denominator square root of 2 end fraction close square brackets. Use the Taylor series to find a rational number A such that open vertical bar A minus ln open parentheses 3 over 2 close parentheses close vertical bar less than 1 over 50. Justify your answer.

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6
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4 marks

Let the function f be defined by f open parentheses x close parentheses equals sin open parentheses 3 x close parentheses. The fourth-degree Taylor polynomial for f about x equals 1 is used to approximate f open parentheses x close parentheses. Use the Lagrange error bound to show that this approximation will be within 7 over 10 to the power of 4 of the exact value of f open parentheses x close parentheses for all x in the interval 0.8 less or equal than x less or equal than 1.2.

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7
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4 marks

For a function f, the Maclaurin series is given by f open parentheses x close parentheses equals sum from n equals 4 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over denominator n minus 1 end fraction times open parentheses x over 3 close parentheses to the power of n. For open vertical bar x close vertical bar less than R, where R is the radius of convergence of the series, show that y equals f open parentheses x close parentheses is a solution to the differential equation y minus x y apostrophe equals fraction numerator x to the power of 4 over denominator 27 open parentheses x plus 3 close parentheses end fraction.

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8a
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4 marks

The function f has a Taylor series about x = 1 that converges to f(x) for all x in the interval of convergence. It is known that f(1) = 1, f'(1) = - \frac{1}{2}, and the nth derivative of f at x = 1 is given by f^{(n)}(1) = (- 1)^{n} \frac{(n - 1) !}{2^{n}} for n \geq 2.

Write the first four nonzero terms and the general term of the Taylor series for f about x = 1.

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8b
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2 marks

The Taylor series for f about x = 1 has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer.

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8c
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1 mark

The Taylor series for f about x = 1 can be used to represent f(1.2) as an alternating series. Use the first three nonzero terms of the alternating series to approximate f(1.2).

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8d
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2 marks

Show that the approximation found in part (c) is within 0.001 of the exact value of f(1.2).

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9a
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4 marks

The function f is defined by the power series f(x) = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots + \frac{(- 1)^{n} x^{2 n + 1}}{2 n + 1} + \cdots for all real numbers x for which the series converges.

Using the ratio test, find the interval of convergence of the power series for f. Justify your answer.

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9b
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2 marks

Show that \left| f\left(\frac{1}{2}\right) - \frac{1}{2} \right| < \frac{1}{10}. Justify your answer.

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9c
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2 marks

Write the first four nonzero terms and the general term for an infinite series that represents f ' (x).

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10a
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2 marks

The Maclaurin series for \text{ln}(1 + x) is given by

x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \cdots + (- 1)^{n + 1} \frac{x^{n}}{n} + \cdots

On its interval of convergence, this series converges to \text{ln}(1 + x). Let f be the function defined by f(x) = x \text{ln} \left(1 + \frac{x}{3}\right).

Write the first four nonzero terms and the general term of the Maclaurin series for f.

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10b
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5 marks

Determine the interval of convergence of the Maclaurin series for f. Show the work that leads to your answer.

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10c
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2 marks

Let P_{4}(x) be the fourth-degree Taylor polynomial for f about x = 0. Use the alternating series error bound to find an upper bound for \left| P_{4}(2) - f(2) \right|.

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11a
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5 marks

The Taylor series for a function f about x = 4 is given by

\sum_{n = 1}^{\infty} \frac{(x - 4)^{n + 1}}{(n + 1) 3^{n}} = \frac{(x - 4)^{2}}{2 \cdot 3} + \frac{(x - 4)^{3}}{3 \cdot 3^{2}} + \frac{(x - 4)^{4}}{4 \cdot 3^{3}} + \cdots + \frac{(x - 4)^{n + 1}}{(n + 1) 3^{n}} + \cdots

and converges to f(x) on its interval of convergence.

Using the ratio test, find the interval of convergence of the Taylor series for f about x = 4. Justify your answer.

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11b
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2 marks

Find the first three nonzero terms and the general term of the Taylor series for f ', the derivative of f, about x = 4.

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12
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3 marks

A function f has derivatives of all orders for - 1 < x < 1. The derivatives of f satisfy the following.

f(0) = 0

f'(0) = 1

f^{(n + 1)}(0) = - n f^{(n)}(0) for all n \geq 1

The Maclaurin series for f converges to f(x) for open vertical bar x close vertical bar less than 1.

Show that the first four nonzero terms of the Maclaurin series for f are x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4}, and write the general term of the Maclaurin series for f.

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13a
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2 marks

A function f has derivatives of all orders for all real numbers x. A portion of the graph of f is shown below, along with the line tangent to the graph of f at x = 0. Selected derivatives of f at x = 0 are given in the table below.

Graph of y = f(x) on an xy-grid. The curve comes down from the upper left, crosses the y-axis at y = 3, decreases to a minimum just to the right of x = 1 (a little below y = 2), then rises again. The straight line tangent to f at x = 0 is also drawn; it passes through (0, 3) and (1, 1)

n

f^{(n)}(0)

2

3

3

- \frac{23}{2}

4

54

Write the third-degree Taylor polynomial for f about x = 0.

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13b
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2 marks

Write the first three nonzero terms of the Maclaurin series for e^{x}. Write the second-degree Taylor polynomial for e^{x} f(x) about x = 0.

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13c
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2 marks

Let h be the function defined by h(x) = \int_{0}^{x} f(t) \text{d} t. Use the Taylor polynomial found in part (a) to find an approximation for h(1).

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13d
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3 marks

It is known that the Maclaurin series for h converges to h(x) for all real numbers x. It is also known that the individual terms of the series for h(1) alternate in sign and decrease in absolute value to 0. Use the alternating series error bound to show that the approximation found in part (c) differs from h(1) by at most 0.45.

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14a
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4 marks

The function f has derivatives of all orders for all real numbers. It is known that f(0) = 2, f ' (0) = 3, f ' ' (x) = - f(x^{2}), and f ' ' ' (x) = - 2 x \cdot f ' (x^{2}).

Find f^{(4)}(x), the fourth derivative of f with respect to x. Write the fourth-degree Taylor polynomial for f about x = 0. Show the work that leads to your answer.

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14b
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2 marks

The fourth-degree Taylor polynomial for f about x = 0 is used to approximate f(0.1). Given that \left| f^{(5)}(x) \right| \leq 15 for 0 \leq x \leq 0.5, use the Lagrange error bound to show that this approximation is within \frac{1}{10^{5}} of the exact value of f(0.1).

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14c
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3 marks

Let g be the function such that g(0) = 4 and g ' (x) = e^{x} f(x). Write the second-degree Taylor polynomial for g about x = 0.

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