Study GuidesExam QuestionsPast Papers
AP®MathsCollege BoardCalculus BCExam QuestionsUnit 10: Infinite Sequences and SeriesSeries Representations of Functions

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

19 mins19 questions

Select a download format for Series Representations of Functions

Scroll for more
Easy
Medium
Hard
1
Sme Calculator1 mark

What is the approximate value of the exponential constant e found by using the third-degree Taylor polynomial for e to the power of x about x equals 0?

  • 1 minus 1 half plus 1 over 24

  • 1 minus 1 over 6 plus 1 over 120

  • 1 plus 1 plus 1 half

  • 2 plus 1 half plus 1 over 6

Choose your answer

Did this page help you?

2
Sme Calculator1 mark

The value of a function f and its first three derivatives at x equals 2 are f open parentheses 2 close parentheses equals 3, f to the power of apostrophe open parentheses 2 close parentheses equals negative 2, f to the power of apostrophe apostrophe end exponent open parentheses 2 close parentheses equals negative 4, and f to the power of apostrophe apostrophe apostrophe end exponent open parentheses 2 close parentheses equals 18. Which of the following is the third-degree Taylor polynomial for f about x equals 2?

  • 3 minus 2 x minus 4 x squared plus 18 x cubed

  • 3 minus 2 x minus 2 x squared plus 3 x cubed

  • 3 minus 2 open parentheses x minus 2 close parentheses minus 4 open parentheses x minus 2 close parentheses squared plus 18 open parentheses x minus 2 close parentheses cubed

  • 3 minus 2 open parentheses x minus 2 close parentheses minus 2 open parentheses x minus 2 close parentheses squared plus 3 open parentheses x minus 2 close parentheses cubed

Choose your answer

Did this page help you?

3
Sme Calculator1 mark

The coefficient of x to the power of 4 in the Taylor series for negative cos x about x equals 0 is

  • negative 1 fourth

  • negative 1 over 24

  • 1 over 24

  • 1 fourth

Choose your answer

Did this page help you?

4
Sme Calculator1 mark

A power series is given by sum from n equals 0 to infinity of a subscript n open parentheses x minus 5 close parentheses to the power of n , where open curly brackets a subscript n close curly brackets subscript n equals 0 end subscript superscript infinity is the sequence of coefficients. Given that the series converges at x equals 2, which of the following must be true?

  • The series diverges at x equals 0

  • The series converges at x equals 6

  • The series converges at x equals 8

  • The series diverges at x equals 8

Choose your answer

Did this page help you?

5
Sme Calculator1 mark

What is the radius of convergence of the series sum from n equals 0 to infinity of x to the power of n over 2 to the power of n?

  • 0

  • 1

  • square root of 2

  • 2

Choose your answer

Did this page help you?

6
Sme Calculator1 mark

The Taylor series for e to the power of x about x equals 2 is sum from n equals 0 to infinity of fraction numerator e squared open parentheses x minus 2 close parentheses to the power of n over denominator n factorial end fraction. Let function f be the second-degree Taylor polynomial for e to the power of x about x equals 2. The maximum value of open vertical bar e to the power of x minus f open parentheses x close parentheses close vertical bar for 1.2 less or equal than x less or equal than 2.8 is

  • 0.522

  • 0.780

  • 1.842

  • 3.144

Choose your answer

Did this page help you?

1
Sme Calculator1 mark

What is the approximation of the value of cos open parentheses negative 1 close parentheses found by using the fourth-degree Taylor polynomial for cos x about x equals 0?

  • 1 minus 1 half plus 1 over 24

  • 1 minus 1 half plus 1 fourth

  • negative 1 plus 1 third minus 1 fifth

  • negative 1 plus 1 over 6 minus 1 over 120

Choose your answer

Did this page help you?

2
Sme Calculator1 mark

x

f open parentheses x close parentheses

f to the power of apostrophe open parentheses x close parentheses

f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses

f to the power of apostrophe apostrophe apostrophe end exponent open parentheses x close parentheses

3

4

2

negative 3

1

4

3

negative 1

negative 4

2

5

2

negative 4

1

6

The table above gives selected values for a function f and its first three derivatives. Which of the following is the third-degree Taylor polynomial for f about x equals 4?

  • 3 minus x minus 2 x squared plus 1 third x cubed

  • 3 minus open parentheses x minus 4 close parentheses minus 2 open parentheses x minus 4 close parentheses squared plus 1 third open parentheses x minus 4 close parentheses cubed

  • 3 minus open parentheses x minus 4 close parentheses minus 2 open parentheses x minus 4 close parentheses squared plus 2 over 3 open parentheses x minus 4 close parentheses cubed

  • 3 minus open parentheses x minus 4 close parentheses minus 4 open parentheses x minus 4 close parentheses squared plus 2 open parentheses x minus 4 close parentheses cubed

Choose your answer

Did this page help you?

3
Sme Calculator1 mark

The graph of the function represented by the Maclaurin series 1 minus x to the power of 4 over 2 plus x to the power of 8 over 24 minus x to the power of 12 over 720 plus... plus fraction numerator open parentheses negative 1 close parentheses to the power of n x to the power of 4 n end exponent over denominator open parentheses 2 n close parentheses factorial end fraction plus... intersects the graph of y equals x cubed at x equals

  • 0.865

  • 0.889

  • 0.896

  • 0.929

Choose your answer

Did this page help you?

4
Sme Calculator1 mark

The coefficient of x cubed in the Taylor series for e to the power of negative 4 x end exponent about x equals 0 is

  • negative 64 over 3

  • negative 32 over 3

  • negative 2 over 3

  • 1 over 6

Choose your answer

Did this page help you?

5
Sme Calculator1 mark

A function f is given in power series form as f open parentheses x close parentheses equals sum from n equals 0 to infinity of a subscript n x to the power of n, which is known to converge for all real values of x. space f to the power of apostrophe open parentheses 2 close parentheses equals

  • a subscript 1

  • sum from n equals 0 to infinity of 2 to the power of n a subscript n

  • sum from n equals 1 to infinity of 2 to the power of n n a subscript n

  • sum from n equals 1 to infinity of 2 to the power of n minus 1 end exponent n a subscript n

Choose your answer

Did this page help you?

6
Sme Calculator1 mark

What are all values of x for which the series sum from n equals 1 to infinity of fraction numerator 2 x to the power of n over denominator n end fraction converges?

  • negative 1 half less than x less than 1 half

  • negative 1 less than x less than 1

  • negative 1 less or equal than x less than 1

  • All real x

Choose your answer

Did this page help you?

1
Sme Calculator1 mark

Which of the following is an approximation for the value of pi that can be found by using the third-degree Taylor polynomial for arctan x about x equals 0?

  • 2 over 3

  • 8 over 3

  • 10 over 3

  • 14 over 3

Choose your answer

Did this page help you?

2
Sme Calculator1 mark

Let f be the function defined by f open parentheses x close parentheses equals ln open parentheses 5 minus x close parentheses. The third-degree Taylor polynomial for f about x equals 4 is

  • table row blank blank cell open parentheses x minus 4 close parentheses plus open parentheses x minus 4 close parentheses squared over 2 plus open parentheses x minus 4 close parentheses cubed over 3 end cell end table

  • table row blank blank cell open parentheses x minus 4 close parentheses minus open parentheses x minus 4 close parentheses squared over 2 plus open parentheses x minus 4 close parentheses cubed over 3 end cell end table

  • table row blank blank cell negative open parentheses x minus 4 close parentheses plus open parentheses x minus 4 close parentheses squared over 2 minus open parentheses x minus 4 close parentheses cubed over 3 end cell end table

  • table row blank blank cell negative open parentheses x minus 4 close parentheses minus open parentheses x minus 4 close parentheses squared over 2 minus open parentheses x minus 4 close parentheses cubed over 3 end cell end table

Choose your answer

Did this page help you?

3
Sme Calculator1 mark

The function f is defined for x greater than 0 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 x to the power of 2 n end exponent over denominator open parentheses 2 n plus 1 close parentheses factorial end fraction. What is limit as x rightwards arrow 2 of f open parentheses x close parentheses?

  • -0.416

  • -0.208

  • 0.455

  • 0.909

Choose your answer

Did this page help you?

4
Sme Calculator1 mark

The coefficient of x cubed in the Taylor series for fraction numerator cos open parentheses 3 x close parentheses over denominator e to the power of x end fraction about x equals 0 is

  • negative 13 over 3

  • negative 1 third

  • 1 third

  • 13 over 3

Choose your answer

Did this page help you?

5
Sme Calculator1 mark

Let f be a function such that f to the power of apostrophe open parentheses x close parentheses equals cos open parentheses 2 x squared close parentheses. The coefficient of x to the power of 9 in the Taylor series for f open parentheses x close parentheses about x equals 0 is

  • negative fraction numerator 1 over denominator 9 factorial end fraction

  • negative 2 over 3

  • 2 over 27

  • fraction numerator 1 over denominator 9 factorial end fraction

Choose your answer

Did this page help you?

6
Sme Calculator1 mark

What are all values of x for which the series sum from n equals 1 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n open parentheses x plus 3 close parentheses to the power of n over denominator n times 4 to the power of n end fraction converges?

  • negative 7 less or equal than x less than 1

  • negative 7 less than x less or equal than 1

  • negative 1 less or equal than x less than 7

  • negative 1 less than x less or equal than 7

Choose your answer

Did this page help you?

7
Sme Calculator1 mark

The Taylor series for a function f about x equals 3 is given by 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 open parentheses x minus 3 close parentheses to the power of n over denominator 2 to the power of n times open parentheses n squared minus n plus 3 close parentheses end fraction, which converges for 1 less or equal than x less or equal than 5. Let P subscript n open parentheses x close parentheses be the nth-degree Taylor polynomial for f about x equals 3. Of the following, which is the smallest number M for which the alternating series error bound guarantees that open vertical bar f open parentheses x close parentheses minus P subscript 4 open parentheses x close parentheses close vertical bar less or equal than M for all x in the interval 7 over 2 less or equal than x less or equal than 9 over 2?

  • table row blank blank cell fraction numerator 1 over denominator 2 to the power of 10 times 23 end fraction end cell end table

  • table row blank blank cell fraction numerator 243 over denominator 2 to the power of 10 times 23 end fraction end cell end table

  • table row blank blank cell fraction numerator 1 over denominator 2 to the power of 8 times 15 end fraction end cell end table

  • table row blank blank cell fraction numerator 81 over denominator 2 to the power of 8 times 15 end fraction end cell end table

Choose your answer

Did this page help you?