Exam 11: Taylor Polynomials and Infinite Series

f(x) = $x ^ { 5 }$ + x - 3 has a zero between 1 and 2 . Use two repetitions of the Newton-Raphson algorithm to approximate this zero with $x _ { 0 } = 1$ Enter just a real number rounded off to two decimal places.

(Short Answer)

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Question 61

The Taylor series at x = 0 for f(x) = tan x is x + $\frac { 1 } { 3 }$ $x ^ { 3 }$ + $\frac { 2 } { 15 }$ $x ^ { 5 }$ + $\frac { 17 } { 315 }$ $x ^ { 7 }$ + ... . Find $f ( 5 )$ (0). Enter just an integer.

(Short Answer)

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Question 62

Find the Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) by computing three or four derivatives and using the definition of the Taylor series.

(Multiple Choice)

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Question 63

Let f(x) = ln(1 + x). Determine the third Taylor polynomial of f(x) at x = 0.

(Multiple Choice)

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Question 64

Determine the sum of the following geometric series: 2 + $\frac { 4 } { 3 }$ + $\frac { 8 } { 9 }$ + ... . Enter just an integer.

(Short Answer)

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Question 65

The Bessel function of order 1 has the Taylor Series at x = 0 given by $f ( x ) = \frac { x } { 2 } - \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 5 } } { 384 } - \ldots$ , find $f ( 5 ) ( 0 )$ .

(Multiple Choice)

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Question 66

Determine the third Taylor polynomial of f(x) = $x ^ { 3 }$ - 2x + 4 at x = 1. Enter an unlabeled polynomial in $x - 1$ in standard form (i.e., highest powers first).

(Short Answer)

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Question 67

Use three repetitions of the Newton-Raphson algorithm to approximate $\sqrt { 3 }$ . Let $x _ { 0 } = 4$ . Enter just a real number rounded off to two decimal places.

(Short Answer)

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Question 68

Determine the sum of the following infinite series: $\sum _ { n = 1 } ^ { \infty } 2 ^ { n } \left( \frac { 1 } { 3 } \right) ^ { n - 1 }$ . Enter just an integer.

(Short Answer)

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Question 69

Sum an appropriate infinite series to find the rational number whose decimal expansion is: $0 . \overline { 37 }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

(Short Answer)

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Question 70

A patient receives M milligrams of a certain drug every hour. Each hour the body eliminates a fraction p of the amount of drug in the body. After an extended period of time, which of the following series approximates the amount of drugs present in the patient's body immediately before receiving an hourly dose?

(Multiple Choice)

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Question 71

Which of the following series converge?

(Multiple Choice)

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Question 72

Use two repetitions of the Newton-Raphson algorithm to find the value of x near zero for which $\cos x = x$ . Enter just a real number rounded off to two decimal places.

(Short Answer)

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Question 73

If f(x) = 2 + 3x - 2 $x ^ { 2 }$ + 2 $x ^ { 3 }$ , then what i f'''(0)? Enter just an integer.

(Short Answer)

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Question 74

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { 1 } { 2 + e ^ { k } }$ is convergent or divergent. Enter just "convergent" or "divergent".

(Short Answer)

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Question 75

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 2 } } { k + k ^ { 2 } + k ^ { 5 / 2 } }$ is convergent or divergent. Enter just "convergent" or "divergent".

(Short Answer)

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Question 76

Find an infinite series that converges to the value of $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } }$ dx. Is $\frac { 1 } { 3 } - \frac { 1 } { 6 } + \frac { 1 } { 18 } - \frac { 1 } { 72 } \ldots$ correct?

(True/False)

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Question 77

Find the third Taylor polynomial of f(x) = $e ^ { x }$ at x = 0 and use it to approximate e. Enter just a reduced fraction of form $\frac { a } { b }$ .

(Short Answer)

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Question 78

Let f(x) = $\frac { 1 } { x + 1 }$ . Determine the second Taylor polynomial $\mathrm { P } 2$ (x) of f(x) at x = 0.

(Multiple Choice)

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Question 79

Use the integral test to determine whether the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { \ln k } { k }$ is convergent or divergent. Enter just "convergent" or "divergent".

(Short Answer)

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Question 80

f(x) = $x ^ { 5 }$ + x - 3 has a zero between 1 and 2 . Use two repetitions of the Newton-Raphson algorithm to approximate this zero with $x _ { 0 } = 1$ Enter just a real number rounded off to two decimal places.

The Taylor series at x = 0 for f(x) = tan x is x + $\frac { 1 } { 3 }$ $x ^ { 3 }$ + $\frac { 2 } { 15 }$ $x ^ { 5 }$ + $\frac { 17 } { 315 }$ $x ^ { 7 }$ + ... . Find $f ( 5 )$ (0). Enter just an integer.

Find the Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) by computing three or four derivatives and using the definition of the Taylor series.

Let f(x) = ln(1 + x). Determine the third Taylor polynomial of f(x) at x = 0.

Determine the sum of the following geometric series: 2 + $\frac { 4 } { 3 }$ + $\frac { 8 } { 9 }$ + ... . Enter just an integer.

The Bessel function of order 1 has the Taylor Series at x = 0 given by $f ( x ) = \frac { x } { 2 } - \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 5 } } { 384 } - \ldots$ , find $f ( 5 ) ( 0 )$ .

Determine the third Taylor polynomial of f(x) = $x ^ { 3 }$ - 2x + 4 at x = 1. Enter an unlabeled polynomial in $x - 1$ in standard form (i.e., highest powers first).

Use three repetitions of the Newton-Raphson algorithm to approximate $\sqrt { 3 }$ . Let $x _ { 0 } = 4$ . Enter just a real number rounded off to two decimal places.

Determine the sum of the following infinite series: $\sum _ { n = 1 } ^ { \infty } 2 ^ { n } \left( \frac { 1 } { 3 } \right) ^ { n - 1 }$ . Enter just an integer.

Sum an appropriate infinite series to find the rational number whose decimal expansion is: $0 . \overline { 37 }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Which of the following series converge?

Use two repetitions of the Newton-Raphson algorithm to find the value of x near zero for which $\cos x = x$ . Enter just a real number rounded off to two decimal places.

If f(x) = 2 + 3x - 2 $x ^ { 2 }$ + 2 $x ^ { 3 }$ , then what i f'''(0)? Enter just an integer.

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { 1 } { 2 + e ^ { k } }$ is convergent or divergent. Enter just "convergent" or "divergent".

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 2 } } { k + k ^ { 2 } + k ^ { 5 / 2 } }$ is convergent or divergent. Enter just "convergent" or "divergent".

Find an infinite series that converges to the value of $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } }$ dx. Is $\frac { 1 } { 3 } - \frac { 1 } { 6 } + \frac { 1 } { 18 } - \frac { 1 } { 72 } \ldots$ correct?

Find the third Taylor polynomial of f(x) = $e ^ { x }$ at x = 0 and use it to approximate e. Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x) = $\frac { 1 } { x + 1 }$ . Determine the second Taylor polynomial $\mathrm { P } 2$ (x) of f(x) at x = 0.

Use the integral test to determine whether the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { \ln k } { k }$ is convergent or divergent. Enter just "convergent" or "divergent".

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Probability and Calculus

Filters

Exam 11: Taylor Polynomials and Infinite Series

f(x) = x5x ^ { 5 }x5 + x - 3 has a zero between 1 and 2 . Use two repetitions of the Newton-Raphson algorithm to approximate this zero with x0=1x _ { 0 } = 1x0​=1 Enter just a real number rounded off to two decimal places.

The Taylor series at x = 0 for f(x) = tan x is x + 13\frac { 1 } { 3 }31​ x3x ^ { 3 }x3 + 215\frac { 2 } { 15 }152​ x5x ^ { 5 }x5 + 17315\frac { 17 } { 315 }31517​ x7x ^ { 7 }x7 + ... . Find f(5)f ( 5 )f(5) (0). Enter just an integer.

Find the Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) by computing three or four derivatives and using the definition of the Taylor series.

Let f(x) = ln(1 + x). Determine the third Taylor polynomial of f(x) at x = 0.

Determine the sum of the following geometric series: 2 + 43\frac { 4 } { 3 }34​ + 89\frac { 8 } { 9 }98​ + ... . Enter just an integer.

The Bessel function of order 1 has the Taylor Series at x = 0 given by f(x)=x2−x316+x5384−…f ( x ) = \frac { x } { 2 } - \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 5 } } { 384 } - \ldotsf(x)=2x​−16x3​+384x5​−… , find f(5)(0)f ( 5 ) ( 0 )f(5)(0) .

Determine the third Taylor polynomial of f(x) = x3x ^ { 3 }x3 - 2x + 4 at x = 1. Enter an unlabeled polynomial in x−1x - 1x−1 in standard form (i.e., highest powers first).

Use three repetitions of the Newton-Raphson algorithm to approximate 3\sqrt { 3 }3​ . Let x0=4x _ { 0 } = 4x0​=4 . Enter just a real number rounded off to two decimal places.

Determine the sum of the following infinite series: ∑n=1∞2n(13)n−1\sum _ { n = 1 } ^ { \infty } 2 ^ { n } \left( \frac { 1 } { 3 } \right) ^ { n - 1 }∑n=1∞​2n(31​)n−1 . Enter just an integer.

Sum an appropriate infinite series to find the rational number whose decimal expansion is: 0.37‾0 . \overline { 37 }0.37 Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Which of the following series converge?

Use two repetitions of the Newton-Raphson algorithm to find the value of x near zero for which cos⁡x=x\cos x = xcosx=x . Enter just a real number rounded off to two decimal places.

If f(x) = 2 + 3x - 2 x2x ^ { 2 }x2 + 2 x3x ^ { 3 }x3 , then what i f'''(0)? Enter just an integer.

Use the comparison test to determine whether the infinite series ∑k=1∞12+ek\sum _ { k = 1 } ^ { \infty } \frac { 1 } { 2 + e ^ { k } }∑k=1∞​2+ek1​ is convergent or divergent. Enter just "convergent" or "divergent".

Use the comparison test to determine whether the infinite series ∑k=1∞k2k+k2+k5/2\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 2 } } { k + k ^ { 2 } + k ^ { 5 / 2 } }∑k=1∞​k+k2+k5/2k2​ is convergent or divergent. Enter just "convergent" or "divergent".

Find an infinite series that converges to the value of ∫01x2e−x3\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } }∫01​x2e−x3 dx. Is 13−16+118−172…\frac { 1 } { 3 } - \frac { 1 } { 6 } + \frac { 1 } { 18 } - \frac { 1 } { 72 } \ldots31​−61​+181​−721​… correct?

Find the third Taylor polynomial of f(x) = exe ^ { x }ex at x = 0 and use it to approximate e. Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Let f(x) = 1x+1\frac { 1 } { x + 1 }x+11​ . Determine the second Taylor polynomial P2\mathrm { P } 2P2 (x) of f(x) at x = 0.

Use the integral test to determine whether the infinite series ∑k=2∞ln⁡kk\sum _ { k = 2 } ^ { \infty } \frac { \ln k } { k }∑k=2∞​klnk​ is convergent or divergent. Enter just "convergent" or "divergent".

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Probability and Calculus

Filters

f(x) = $x ^ { 5 }$ + x - 3 has a zero between 1 and 2 . Use two repetitions of the Newton-Raphson algorithm to approximate this zero with $x _ { 0 } = 1$ Enter just a real number rounded off to two decimal places.

The Taylor series at x = 0 for f(x) = tan x is x + $\frac { 1 } { 3 }$ $x ^ { 3 }$ + $\frac { 2 } { 15 }$ $x ^ { 5 }$ + $\frac { 17 } { 315 }$ $x ^ { 7 }$ + ... . Find $f ( 5 )$ (0). Enter just an integer.

Determine the sum of the following geometric series: 2 + $\frac { 4 } { 3 }$ + $\frac { 8 } { 9 }$ + ... . Enter just an integer.

The Bessel function of order 1 has the Taylor Series at x = 0 given by $f ( x ) = \frac { x } { 2 } - \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 5 } } { 384 } - \ldots$ , find $f ( 5 ) ( 0 )$ .

Determine the third Taylor polynomial of f(x) = $x ^ { 3 }$ - 2x + 4 at x = 1. Enter an unlabeled polynomial in $x - 1$ in standard form (i.e., highest powers first).

Use three repetitions of the Newton-Raphson algorithm to approximate $\sqrt { 3 }$ . Let $x _ { 0 } = 4$ . Enter just a real number rounded off to two decimal places.

Determine the sum of the following infinite series: $\sum _ { n = 1 } ^ { \infty } 2 ^ { n } \left( \frac { 1 } { 3 } \right) ^ { n - 1 }$ . Enter just an integer.

Sum an appropriate infinite series to find the rational number whose decimal expansion is: $0 . \overline { 37 }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Use two repetitions of the Newton-Raphson algorithm to find the value of x near zero for which $\cos x = x$ . Enter just a real number rounded off to two decimal places.

If f(x) = 2 + 3x - 2 $x ^ { 2 }$ + 2 $x ^ { 3 }$ , then what i f'''(0)? Enter just an integer.

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { 1 } { 2 + e ^ { k } }$ is convergent or divergent. Enter just "convergent" or "divergent".

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 2 } } { k + k ^ { 2 } + k ^ { 5 / 2 } }$ is convergent or divergent. Enter just "convergent" or "divergent".

Find an infinite series that converges to the value of $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } }$ dx. Is $\frac { 1 } { 3 } - \frac { 1 } { 6 } + \frac { 1 } { 18 } - \frac { 1 } { 72 } \ldots$ correct?

Find the third Taylor polynomial of f(x) = $e ^ { x }$ at x = 0 and use it to approximate e. Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x) = $\frac { 1 } { x + 1 }$ . Determine the second Taylor polynomial $\mathrm { P } 2$ (x) of f(x) at x = 0.

Use the integral test to determine whether the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { \ln k } { k }$ is convergent or divergent. Enter just "convergent" or "divergent".