AP®MathsCollege BoardCalculus BCExam QuestionsUnit 10: Infinite Sequences and SeriesUnit 10 Overview

Unit 10 Overview (College Board AP® Calculus BC): Exam Questions

2 hours12 questions
1a
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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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1b
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2 marks

Use Euler's method, starting at x = 1 with two steps of equal size, to approximate f(2). Show the work that leads to your answer.

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1c
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5 marks

Find the particular solution y = f(x) to the differential equation \frac{\text{d} y}{\text{d} x} = y \cdot (x \text{ln} x) with initial condition f(1) = 4.

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2a
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3 marks

Let y = f(x) be the particular solution to the differential equation \frac{\text{d} y}{\text{d} x} = (3 - x) y^{2} with initial condition f(1) = -1.

Find f ' ' (1), the value of \frac{\text{d}^{2} y}{\text{d} x^{2}} at the point (1, -1). Show the work that leads to your answer.

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2b
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2 marks

Write the second-degree Taylor polynomial for f about x = 1.

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2c
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2 marks

The second-degree Taylor polynomial for f about x = 1 is used to approximate f(1.1). Given that \left| f ' ' ' (x) \right| \leq 60 for all x in the interval 1 \leq x \leq 1.1, use the Lagrange error bound to show that this approximation differs from f(1.1) by at most 0.01.

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2d
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2 marks

Use Euler's method, starting at x = 1 with two steps of equal size, to approximate f(1.4). Show the work that leads to your answer.

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3a
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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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3b
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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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3c
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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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3d
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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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4a
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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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4b
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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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4c
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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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4d
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1 mark

Use the result from part (c) to find the value of f ' \left(\frac{1}{6}\right).

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5a
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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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5b
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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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5c
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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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6a
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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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6b
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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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6c
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1 mark

The Taylor series for f ' described in part (b) is a geometric series. For all x in the interval of convergence of the Taylor series for f ', show that f ' (x) = \frac{x - 4}{7 - x}.

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6d
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1 mark

It is known that the radius of convergence of the Taylor series for f about x = 4 is the same as the radius of convergence of the Taylor series for f ' about x = 4. Does the Taylor series for f ' described in part (b) converge to f ' (x) = \frac{x - 4}{7 - x} at x = 8? Give a reason for your answer.

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7a
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3 marks

The function g has derivatives of all orders for all real numbers. The Maclaurin series for g is given by g(x) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{n}}{2 e^{n} + 3} on its interval of convergence.

State the conditions necessary to use the integral test to determine convergence of the series \sum_{n = 0}^{\infty} \frac{1}{e^{n}}. Use the integral test to show that \sum_{n = 0}^{\infty} \frac{1}{e^{n}} converges.

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7b
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2 marks

Use the limit comparison test with the series \sum_{n = 0}^{\infty} \frac{1}{e^{n}} to show that the series g(1) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{2 e^{n} + 3} converges absolutely.

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7c
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3 marks

Determine the radius of convergence of the Maclaurin series for g.

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7d
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1 mark

The first two terms of the series g(1) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{2 e^{n} + 3} are used to approximate g(1). Use the alternating series error bound to determine an upper bound on the error of the approximation.

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8a
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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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8b
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2 marks

Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at x = 1. Explain your reasoning.

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8c
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3 marks

Write the first four nonzero terms and the general term of the Maclaurin series for g(x) = \int_{0}^{x} f(t) \text{d} t.

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8d
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1 mark

Let P_{n} \left( \frac{1}{2} \right) represent the nth-degree Taylor polynomial for g about x = 0 evaluated at x = \frac{1}{2}, where g is the function defined in part (c). Use the alternating series error bound to show that \left| P_{4} \left( \frac{1}{2} \right) - g \left( \frac{1}{2} \right) \right| < \frac{1}{500}.

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9a
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2 marks

Let f be the function defined by f(x) = \frac{3}{2 x^{2} - 7 x + 5}.

Find the slope of the line tangent to the graph of f at x = 3.

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9b
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2 marks

Find the x-coordinate of each critical point of f in the interval 1 < x < 2.5. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers.

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9c
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3 marks

Using the identity that \frac{3}{2 x^{2} - 7 x + 5} = \frac{2}{2 x - 5} - \frac{1}{x - 1}, evaluate \int_{5}^{\infty} f(x) \text{d} x or show that the integral diverges.

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9d
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2 marks

Determine whether the series \sum_{n = 5}^{\infty} \frac{3}{2 n^{2} - 7 n + 5} converges or diverges. State the conditions of the test used for determining convergence or divergence.

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10a
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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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10b
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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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10c
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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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10d
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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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11a
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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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11b
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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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11c
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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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12a
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2 marks

The Maclaurin series for a function f is given by \sum_{n = 1}^{\infty} \frac{(n + 1) x^{n}}{n^{2} 6^{n}} and converges to f(x) for all x in the interval of convergence. It can be shown that the Maclaurin series for f has a radius of convergence of 6.

Determine whether the Maclaurin series for f converges or diverges at x = 6. Give a reason for your answer.

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12b
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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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12c
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2 marks

Find the general term of the Maclaurin series for f ', the derivative of f. Find the radius of convergence of the Maclaurin series for f '.

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12d
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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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