Question 1

Use the second Taylor polynomial at x = 1 to estimate $\int _ { 1 } ^ { 2 } \ln x ^ { 2 } d x$ .
Enter just a reduced fraction.

Accepted Answer

The answer of Use the second Taylor polynomial at x...

Question 2

Find the Taylor series of f(x) = $\frac { 1 } { 1 + x }$ for $| x |$ < 1 then multiply the series by x to obtain a series expansion of $\frac { x } { x + 1 }$ , and then use these two series to obtain a series expansion of $\frac { 1 - x } { 1 + x }$ for $| x |$ < 1.
Is $\frac { 1 - x } { 1 + x } = 1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 } + 2 x ^ { 4 } + \ldots$ the correct expansion?

Accepted Answer

A) True 
 B)False

Question 3

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \sqrt { 4 - x }$ .
Is $f ( x ) = 2 - \frac { x } { 4 } - \frac { x ^ { 2 } } { 32 \cdot 2 ! } - \frac { 3 x ^ { 3 } } { 256 \cdot 3 ! }$ correct?

Accepted Answer

A) True 
 B)False

Question 4

Determine the sum of the series $1 + \frac { 1 } { 1.01 } + \frac { 1 } { ( 1.01 ) ^ { 2 } } + \frac { 1 } { ( 1.01 ) ^ { 3 } } + \ldots$

Accepted Answer

A)  101 
B)  $\frac { 1 } { 1.01 }$ 
C)  10.1 
D)  none of these 
A)  101 
B)  $\frac { 1 } { 1.01 }$ 
C)  10.1 
D)  none of these

Question 5

Suppose $x 0$ is an initial approximation of a zero of the function f(x). Using the Newton-Raphson algorithm, a second approximation, $\mathrm { x } _ { 1 }$ is obtained. Which of the following must be true?

Accepted Answer

A)  $\mathrm { x } _ { 1 }$ is the x-coordinate of the x-intercept of the tangent line to f(x) at $\left( { }  { \mathrm { x } } _0 , \mathrm { f } \left( \mathrm { x } _ { 0 } \right) \right)$ 
B)  f( $\mathrm { x } _ { 1 }$ ) = 0 
C)  $\mathrm { x } _ { 1 }$ = $x _0$ - $\frac { f ^ { \prime } \left( x _ { 0 } \right) } { f \left( x _ { 0 } \right) }$ 
D)  $\mathrm { x } _ { 1 }$ is closer to the zero of f(x) than $x _0$ . 
E)  all of these 
A)  $\mathrm { x } _ { 1 }$ is the x-coordinate of the x-intercept of the tangent line to f(x) at $\left( { }  { \mathrm { x } } _0 , \mathrm { f } \left( \mathrm { x } _ { 0 } \right) \right)$ 
B)  f( $\mathrm { x } _ { 1 }$ ) = 0 
C)  $\mathrm { x } _ { 1 }$ = $x _0$ - $\frac { f ^ { \prime } \left( x _ { 0 } \right) } { f \left( x _ { 0 } \right) }$ 
D)  $\mathrm { x } _ { 1 }$ is closer to the zero of f(x) than $x _0$ . 
E)  all of these

Question 6

Find the Taylor series expansion at x = 0 of $\int \frac { x } { 1 + x ^ { 3 } }$ dx. Is $\left[ \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 5 } x ^ { 5 } + \frac { 1 } { 8 } x ^ { 8 } + \frac { 1 } { 11 } x ^ { 11 } \ldots \right] + C$ correct?

Accepted Answer

A) True 
 B)False

Question 7

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The fifth Taylor polynomial of f(x) at x = 1 is $\mathrm { P } 5 ( \mathrm { x } ) = x ^ { 5 } + x ^ { 4 } - 7 x ^ { 3 } + 2$ .

Accepted Answer

A) True 
 B)False

Question 8

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \sin 2 x$ .
Is $f ( x ) = 2 x - \frac { 8 x ^ { 3 } } { 3 ! } + \frac { 32 x ^ { 5 } } { 5 ! } - \frac { 128 x ^ { 7 } } { 7 ! }$ the correct answer?

Accepted Answer

A) True 
 B)False

Question 9

Suppose $\sum _ { \mathrm { k } = 0 } ^ { \infty } \left( \frac { 2 } { \mathrm {~m} } \right) ^ { \mathrm { k } }$ converges. What can you say about the value of m?
Enter your answer in standard interval notation (no variables).

Accepted Answer

The answer of Suppose \(\sum _ { \mathrm { k...

Question 10

Consider the following geometric series: $\frac { 25 } { 2 }$ - $\frac { 15 } { 2 }$ + $\frac { 9 } { 2 }$ - $\frac { 27 } { 10 }$ + ... . Which of the following statements is true?

Accepted Answer

A)  This series diverges. 
B)  The sum of this series is $\frac { 125 } { 4 }$ . 
C)  Another way writing this series is $\sum _ { k = 1 } ^ { \infty } \left( - \frac { 3 } { 5 } \right) ^ { k }$ 
D)  The ratio r of this series is $\frac { 3 } { 5 }$ . 
E)  none of these 
A)  This series diverges. 
B)  The sum of this series is $\frac { 125 } { 4 }$ . 
C)  Another way writing this series is $\sum _ { k = 1 } ^ { \infty } \left( - \frac { 3 } { 5 } \right) ^ { k }$ 
D)  The ratio r of this series is $\frac { 3 } { 5 }$ . 
E)  none of these

Question 11

The function f(x) = sin $x ^ { 2 }$ is approximated by its second Taylor polynomial $\mathrm { P } 2$ (x) at x = 0. Which of the following statements is NOT true?

Accepted Answer

A)  f'(0) = 0 
B)  $\mathrm { P } 2$ (x) = $\frac { 1 } { 3 }$ + $x ^ { 2 }$ 
C)  $\mathrm { P } 2$ (x) = $x ^ { 2 }$ 
D)  f''(0) = 2 
E)  none of these 
A)  f'(0) = 0 
B)  $\mathrm { P } 2$ (x) = $\frac { 1 } { 3 }$ + $x ^ { 2 }$ 
C)  $\mathrm { P } 2$ (x) = $x ^ { 2 }$ 
D)  f''(0) = 2 
E)  none of these

Question 12

Find the first four non-zero terms of the Taylor series at x = 0 for f(x) = 1 + $x e ^ { x }$ .
Is $1 + x + x ^ { 2 } + \frac { x ^ { 3 } } { 2 ! }$ the correct answer?

Accepted Answer

A) True 
 B)False

Question 13

Find the second Taylor polynomial for f(x) = $\frac { 1 } { \sqrt { x + 4 } }$ at x = 0 and use it to approximate $\frac { 1 } { \sqrt { 4.1 } }$ .
Enter just a real number rounded off to two decimal places.

Accepted Answer

The answer of Find the second Taylor polynomial for f(x)...

Question 14

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The third Taylor polynomial of f(x) at x = 1 is $\mathrm { P } 3 ( x ) = 2 - 7 x ^ { 3 }$ .

Accepted Answer

A) True 
 B)False

Question 15

If the following is a graph of f(x), which of the following could be the first Taylor polynomial of f at $x = 0$ ?

Accepted Answer

A)  $\mathrm { P } 1$ (x) = 2 + 2x 
B)  $\mathrm { P } 1$ (x) = 2 - 2x 
C)  $\mathrm { P } 1$ (x) = 2x - 2 
D)  $\mathrm { P } 1$ (x) = 2x + 5 
A)  $\mathrm { P } 1$ (x) = 2 + 2x 
B)  $\mathrm { P } 1$ (x) = 2 - 2x 
C)  $\mathrm { P } 1$ (x) = 2x - 2 
D)  $\mathrm { P } 1$ (x) = 2x + 5

Question 16

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

Accepted Answer

The answer of Use the comparison test to determine whether...

Question 17

Suppose f(0) = 1, f'(0) = 1, and f''(0) = -1. Use a Taylor polynomial of degree two to approximate f $\left( \frac { 1 } { 2 } \right)$ . Is $f\left( \frac { 1 } { 2 } \right) \approx \frac { 11 } { 8 }$ the solution?

Accepted Answer

A) True 
 B)False

Question 18

Determine the sum of the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { \pi } { e } \right) ^ { n }$ .

Accepted Answer

A)  $\frac { e } { e - \pi }$ 
B)  $\frac { \mathrm { e } } { \pi }$ 
C)  $\frac { \pi } { \mathrm { e } - \pi }$ 
D)  none of these 
A)  $\frac { e } { e - \pi }$ 
B)  $\frac { \mathrm { e } } { \pi }$ 
C)  $\frac { \pi } { \mathrm { e } - \pi }$ 
D)  none of these

Question 19

Use the Newton-Raphson algorithm with three repetitions to approximate the solution to $\mathrm { e } ^ { - \mathrm { x } }$ = 2 - x near x = 2.
Enter just a real number rounded off to two decimal places.

Accepted Answer

The answer of Use the Newton-Raphson algorithm with three repetitions...

Question 20

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = e ^ { - 2 x }$ .
Is $f ( x ) = 1 + 2 x + \frac { 4 x ^ { 2 } } { 2 ! } + \frac { 8 x ^ { 3 } } { 3 ! }$ correct?

Accepted Answer

A) True 
 B)False

Use the second Taylor polynomial at x = 1 to estimate $\int _ { 1 } ^ { 2 } \ln x ^ { 2 } d x$ . Enter just a reduced fraction.

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \sqrt { 4 - x }$ . Is $f ( x ) = 2 - \frac { x } { 4 } - \frac { x ^ { 2 } } { 32 \cdot 2 ! } - \frac { 3 x ^ { 3 } } { 256 \cdot 3 ! }$ correct?

Determine the sum of the series $1 + \frac { 1 } { 1.01 } + \frac { 1 } { ( 1.01 ) ^ { 2 } } + \frac { 1 } { ( 1.01 ) ^ { 3 } } + \ldots$

Suppose $x 0$ is an initial approximation of a zero of the function f(x). Using the Newton-Raphson algorithm, a second approximation, $\mathrm { x } _ { 1 }$ is obtained. Which of the following must be true?

Find the Taylor series expansion at x = 0 of $\int \frac { x } { 1 + x ^ { 3 } }$ dx. Is $\left[ \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 5 } x ^ { 5 } + \frac { 1 } { 8 } x ^ { 8 } + \frac { 1 } { 11 } x ^ { 11 } \ldots \right] + C$ correct?

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The fifth Taylor polynomial of f(x) at x = 1 is $\mathrm { P } 5 ( \mathrm { x } ) = x ^ { 5 } + x ^ { 4 } - 7 x ^ { 3 } + 2$ .

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \sin 2 x$ . Is $f ( x ) = 2 x - \frac { 8 x ^ { 3 } } { 3 ! } + \frac { 32 x ^ { 5 } } { 5 ! } - \frac { 128 x ^ { 7 } } { 7 ! }$ the correct answer?

Suppose $\sum _ { \mathrm { k } = 0 } ^ { \infty } \left( \frac { 2 } { \mathrm {~m} } \right) ^ { \mathrm { k } }$ converges. What can you say about the value of m? Enter your answer in standard interval notation (no variables).

Consider the following geometric series: $\frac { 25 } { 2 }$ - $\frac { 15 } { 2 }$ + $\frac { 9 } { 2 }$ - $\frac { 27 } { 10 }$ + ... . Which of the following statements is true?

The function f(x) = sin $x ^ { 2 }$ is approximated by its second Taylor polynomial $\mathrm { P } 2$ (x) at x = 0. Which of the following statements is NOT true?

Find the first four non-zero terms of the Taylor series at x = 0 for f(x) = 1 + $x e ^ { x }$ . Is $1 + x + x ^ { 2 } + \frac { x ^ { 3 } } { 2 ! }$ the correct answer?

Find the second Taylor polynomial for f(x) = $\frac { 1 } { \sqrt { x + 4 } }$ at x = 0 and use it to approximate $\frac { 1 } { \sqrt { 4.1 } }$ . Enter just a real number rounded off to two decimal places.

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The third Taylor polynomial of f(x) at x = 1 is $\mathrm { P } 3 ( x ) = 2 - 7 x ^ { 3 }$ .

If the following is a graph of f(x), which of the following could be the first Taylor polynomial of f at $x = 0$ ?

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

Suppose f(0) = 1, f'(0) = 1, and f''(0) = -1. Use a Taylor polynomial of degree two to approximate f $\left( \frac { 1 } { 2 } \right)$ . Is $f\left( \frac { 1 } { 2 } \right) \approx \frac { 11 } { 8 }$ the solution?

Determine the sum of the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { \pi } { e } \right) ^ { n }$ .

Use the Newton-Raphson algorithm with three repetitions to approximate the solution to $\mathrm { e } ^ { - \mathrm { x } }$ = 2 - x near x = 2. Enter just a real number rounded off to two decimal places.

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = e ^ { - 2 x }$ . Is $f ( x ) = 1 + 2 x + \frac { 4 x ^ { 2 } } { 2 ! } + \frac { 8 x ^ { 3 } } { 3 ! }$ correct?

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

Use the second Taylor polynomial at x = 1 to estimate ∫12ln⁡x2dx\int _ { 1 } ^ { 2 } \ln x ^ { 2 } d x∫12​lnx2dx . Enter just a reduced fraction.

Determine the sum of the series 1+11.01+1(1.01)2+1(1.01)3+…1 + \frac { 1 } { 1.01 } + \frac { 1 } { ( 1.01 ) ^ { 2 } } + \frac { 1 } { ( 1.01 ) ^ { 3 } } + \ldots1+1.011​+(1.01)21​+(1.01)31​+…

Suppose x0x 0x0 is an initial approximation of a zero of the function f(x). Using the Newton-Raphson algorithm, a second approximation, x1\mathrm { x } _ { 1 }x1​ is obtained. Which of the following must be true?

Suppose f(x) = x4x ^ { 4 }x4 - 7 x3x ^ { 3 }x3 + 2. The fifth Taylor polynomial of f(x) at x = 1 is P5(x)=x5+x4−7x3+2\mathrm { P } 5 ( \mathrm { x } ) = x ^ { 5 } + x ^ { 4 } - 7 x ^ { 3 } + 2P5(x)=x5+x4−7x3+2 .

Suppose ∑k=0∞(2 m)k\sum _ { \mathrm { k } = 0 } ^ { \infty } \left( \frac { 2 } { \mathrm {~m} } \right) ^ { \mathrm { k } }∑k=0∞​( m2​)k converges. What can you say about the value of m? Enter your answer in standard interval notation (no variables).

Consider the following geometric series: 252\frac { 25 } { 2 }225​ - 152\frac { 15 } { 2 }215​ + 92\frac { 9 } { 2 }29​ - 2710\frac { 27 } { 10 }1027​ + ... . Which of the following statements is true?

The function f(x) = sin x2x ^ { 2 }x2 is approximated by its second Taylor polynomial P2\mathrm { P } 2P2 (x) at x = 0. Which of the following statements is NOT true?

Find the first four non-zero terms of the Taylor series at x = 0 for f(x) = 1 + xexx e ^ { x }xex . Is 1+x+x2+x32!1 + x + x ^ { 2 } + \frac { x ^ { 3 } } { 2 ! }1+x+x2+2!x3​ the correct answer?

Find the second Taylor polynomial for f(x) = 1x+4\frac { 1 } { \sqrt { x + 4 } }x+4​1​ at x = 0 and use it to approximate 14.1\frac { 1 } { \sqrt { 4.1 } }4.1​1​ . Enter just a real number rounded off to two decimal places.

Suppose f(x) = x4x ^ { 4 }x4 - 7 x3x ^ { 3 }x3 + 2. The third Taylor polynomial of f(x) at x = 1 is P3(x)=2−7x3\mathrm { P } 3 ( x ) = 2 - 7 x ^ { 3 }P3(x)=2−7x3 .

If the following is a graph of f(x), which of the following could be the first Taylor polynomial of f at x=0x = 0x=0 ?

Use the comparison test to determine whether the infinite series ∑k=1∞1k+1\sum _ { k = 1 } ^ { \infty } \frac { 1 } { \sqrt { k } + 1 }∑k=1∞​k​+11​ is convergent or divergent. Enter just "convergent" or "divergent".

Suppose f(0) = 1, f'(0) = 1, and f''(0) = -1. Use a Taylor polynomial of degree two to approximate f (12)\left( \frac { 1 } { 2 } \right)(21​) . Is f(12)≈118f\left( \frac { 1 } { 2 } \right) \approx \frac { 11 } { 8 }f(21​)≈811​ the solution?

Determine the sum of the series ∑n=1∞(πe)n\sum _ { n = 1 } ^ { \infty } \left( \frac { \pi } { e } \right) ^ { n }∑n=1∞​(eπ​)n .

Use the Newton-Raphson algorithm with three repetitions to approximate the solution to e−x\mathrm { e } ^ { - \mathrm { x } }e−x = 2 - x near x = 2. Enter just a real number rounded off to two decimal places.

Find the first four non-zero terms of the Taylor series at x = 0 of f(x)=e−2xf ( x ) = e ^ { - 2 x }f(x)=e−2x . Is f(x)=1+2x+4x22!+8x33!f ( x ) = 1 + 2 x + \frac { 4 x ^ { 2 } } { 2 ! } + \frac { 8 x ^ { 3 } } { 3 ! }f(x)=1+2x+2!4x2​+3!8x3​ correct?

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

Use the second Taylor polynomial at x = 1 to estimate $\int _ { 1 } ^ { 2 } \ln x ^ { 2 } d x$ . Enter just a reduced fraction.

Determine the sum of the series $1 + \frac { 1 } { 1.01 } + \frac { 1 } { ( 1.01 ) ^ { 2 } } + \frac { 1 } { ( 1.01 ) ^ { 3 } } + \ldots$

Suppose $x 0$ is an initial approximation of a zero of the function f(x). Using the Newton-Raphson algorithm, a second approximation, $\mathrm { x } _ { 1 }$ is obtained. Which of the following must be true?

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The fifth Taylor polynomial of f(x) at x = 1 is $\mathrm { P } 5 ( \mathrm { x } ) = x ^ { 5 } + x ^ { 4 } - 7 x ^ { 3 } + 2$ .

Suppose $\sum _ { \mathrm { k } = 0 } ^ { \infty } \left( \frac { 2 } { \mathrm {~m} } \right) ^ { \mathrm { k } }$ converges. What can you say about the value of m? Enter your answer in standard interval notation (no variables).

Consider the following geometric series: $\frac { 25 } { 2 }$ - $\frac { 15 } { 2 }$ + $\frac { 9 } { 2 }$ - $\frac { 27 } { 10 }$ + ... . Which of the following statements is true?

The function f(x) = sin $x ^ { 2 }$ is approximated by its second Taylor polynomial $\mathrm { P } 2$ (x) at x = 0. Which of the following statements is NOT true?

Find the first four non-zero terms of the Taylor series at x = 0 for f(x) = 1 + $x e ^ { x }$ . Is $1 + x + x ^ { 2 } + \frac { x ^ { 3 } } { 2 ! }$ the correct answer?

Find the second Taylor polynomial for f(x) = $\frac { 1 } { \sqrt { x + 4 } }$ at x = 0 and use it to approximate $\frac { 1 } { \sqrt { 4.1 } }$ . Enter just a real number rounded off to two decimal places.

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The third Taylor polynomial of f(x) at x = 1 is $\mathrm { P } 3 ( x ) = 2 - 7 x ^ { 3 }$ .

If the following is a graph of f(x), which of the following could be the first Taylor polynomial of f at $x = 0$ ?

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

Suppose f(0) = 1, f'(0) = 1, and f''(0) = -1. Use a Taylor polynomial of degree two to approximate f $\left( \frac { 1 } { 2 } \right)$ . Is $f\left( \frac { 1 } { 2 } \right) \approx \frac { 11 } { 8 }$ the solution?

Determine the sum of the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { \pi } { e } \right) ^ { n }$ .

Use the Newton-Raphson algorithm with three repetitions to approximate the solution to $\mathrm { e } ^ { - \mathrm { x } }$ = 2 - x near x = 2. Enter just a real number rounded off to two decimal places.

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = e ^ { - 2 x }$ . Is $f ( x ) = 1 + 2 x + \frac { 4 x ^ { 2 } } { 2 ! } + \frac { 8 x ^ { 3 } } { 3 ! }$ correct?