Deck 11: Taylor Polynomials and Infinite Series

Question 1

Is this the graph of y = and are its first two Taylor polynomials at x = 0 on the same axis?

Answer 1

True

Question 2

Write down the fourth Taylor polynomial of f(x) = at x = 0.
Enter your answer an an unlabeled polynomial in x in standard Taylor polynomial form (i.e., constant first, highest power last ).

Answer 2

1 - +

Question 3

Suppose f(x) = - 7 + 2. The fifth Taylor polynomial of f(x) at x = 0 is .

Answer 3

True

Question 4

Let f(x) = . Determine the second Taylor polynomial (x) of f(x) at x = 0.

A) 1 - x
B) 1 - 2x + 2
C) 1 - x -
D) 1 - x +
E) none of these

Question 5

Find the third Taylor polynomial of f(x) = at x = 0 and use it to approximate e.
Enter just a reduced fraction of form .

Question 6

Suppose f(x) = - 7 + 2. The third Taylor polynomial of f(x) at x = 0 is .

Question 7

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

A) x + +
B) x + +
C) x - +
D) x + +

Question 8

Find the third Taylor polynomial of f(x) = sin x at x = 0 and use it to approximate .
Enter just a real number rounded off to two decimal places.

Question 9

Find the second Taylor polynomial for f(x) = at x = 0 and use it to approximate .
Enter just a real number rounded off to two decimal places.

Question 10

Estimate by using the second Taylor polynomial for f(x) = . Is the solution?

Question 11

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

Question 12

The function f(x) = sin is approximated by its second Taylor polynomial (x) at x = 0. Which of the following statements is NOT true?

A) f'(0) = 0
B) (x) = +
C) (x) =
D) f''(0) = 2
E) none of these

Question 13

Let f(x) = . Determine the fourth Taylor polynomial at x = 0.

A) 1 + x + + +
B) 1 + x + 2 + +
C) 1 - x + - +
D) 1 - x + 2 - +

Question 14

Determine the second Taylor polynomial of sin at x = 0.
Enter an unlabeled polynomial in x in standard form (i.e., highest powers first).

Question 15

Find the third Taylor polynomial of f(x) = + sin x at x = 0.
Enter an unlabeled polynomial in x in standard form (i.e., highest powers first).

Question 16

Find the second Taylor polynomial of f(x) = sin at x = 0 and use it to approximate the area under the curve f(x) between 0 and .
Enter an unlabeled polynomial in x in standard form followed by a comma and then just a quotient representing the area (π in the numerator).

Question 17

Determine the third Taylor polynomial of f(x) = - 3x at x = 0.
Enter an unlabeled polynomial in x in standard form (i.e., highest powers first).

Question 18

The area of a circle with radius 1 is π. If f(x) = gives the top half of this circle, as illustrated below, use the second Taylor polynomial of f(x) at x = 0 to find an approximate value for π. Is the following correct?

Question 19

Let f(x) = - 4x - 1. Which of the following statements is true? (All Taylor polynomials are at x = 0.)

A) (-1) = 0
B) (1) = 7
C) = f(x) for all n ≥ 3
D) (3) = -11
E) none of these

Question 20

Determine the third Taylor polynomial of f(x) = at x = 0.
Enter your answer as an unlabeled polynomial in x in standard form (i.e., highest powers first).

Question 21

Suppose f(x) = - 7 + 2. The fifth Taylor polynomial of f(x) at x = 1 is .

Question 22

Find the third Taylor polynomial of f(x) = cos x at x = .
Enter your answer as an unlabeled polynomial in x - in standard form (i.e., highest powers first).

Question 23

Suppose f(0) = 1, f'(0) = 1, and f''(0) = -1. Use a Taylor polynomial of degree two to approximate f . Is the solution?

Question 24

The Newton-Raphson algorithm is applied to estimate . If = 3, find .

A)
B)
C)
D)
E) none of these

Question 25

Suppose the second Taylor polynomial for f(x) at x = 3 is . Find f''(3).
Enter just a reduced fraction.

Question 26

Suppose f(x) = - 7 + 2. The third Taylor polynomial of f(x) at x = 1 is .

Question 27

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

A) (x) = 2 + 2x
B) (x) = 2 - 2x
C) (x) = 2x - 2
D) (x) = 2x + 5

Question 28

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

Question 29

Use the second Taylor polynomial at x = 1 to estimate .
Enter just a reduced fraction.

Question 30

Suppose that the first Taylor polynomial of a function f(x) at x = 0 is (x) = 2 - 3x. Which of the following could be a graph of f(x) ?

A)
B)
C)
D)

Question 31

Determine the third Taylor polynomial of f(x) = ln(2 - x) at x = 1 and use it to estimate [Hint: .]
Enter your answer as an unlabeled polynomial in in standard form :

Question 32

Below is a graph of the function f(x). Which of the following could be the first Taylor polynomial of f(x) at

A)
B) (x) = -2 - 3x
C)
D) (x) = 3 + 4x
E) none of these

Question 33

The Newton-Raphson algorithm is used to approximate the zero of f(x) = + x - 5 between x = 1 and x=2 . If x₀ = 1, find x₁.

A) 7/4
B) 1/4
C) 3/4
D) 7/3
E) none of these

Question 34

If f(x) = 1 - 3(x - 2) + 4 + 6 , then what is f''(2)? Enter just an integer.

Question 35

Find the second Taylor polynomial of f(x) = at x = 9 and use it to approximate .
Enter just a real number rounded off to two decimal places.

Question 36

Below is a graph of function f(x). Which of the following could be the second Taylor polynomial of f(x) at

A) (x) = + (x - a)
B) (x) = +
C) (x) = -
D) (x) = - (x - a) +
E) (x) = (x - a) -

Question 37

Let f(x) = . Determine the second Taylor polynomial of f(x) at x = 2.

A) 1 + (x - 2) +
B) 1 - (x - 2) +
C) 1 + (x - 2) +
D) e + (x - 2) +

Question 38

The Newton-Raphson algorithm is applied to estimate a zero of f(x) with = 3. Which of the following statements is true?

A) = 3 -
B) = 3 +
C) = 3 -
D) =
E) none of these

Question 39

Let f(x) = ln x. Find the second Taylor polynomial of f(x) at x = 2.

A) 1 + (x - 2) +
B) ln 2 - (x - 2) -
C) 1 + (x - 2) -
D) ln 2 + (x - 2) -

Question 40

A polynomial f(x) of degree 3 for which f(1) = -1, f'(1) = 2, f''(1) = -1, and f'''(1) = -2 is given by

A) f(x) = -1 + 2(x - 1) - -
B) f(x) = -1 + 2(x - 1) - 1 - 2
C) f(x) = -1 + 2x - - 2
D) f(x) = - 2 - 3x + 2
E) none of these

Question 41

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

Question 42

Below is a graph of the functions h(x) and g(x). In using the Newton-Raphson algorithm to find where , which of the following statements is false?

A) = 3 could be used as the initial approximation.
B) Use the Newton-Raphson algorithm to find the zeroes of f(x) = h(x) - g(x).
C) Use the Newton-Raphson algorithm to find the zeroes of f(x) = h(x) + g(x).
D) = 4 could be used as the initial approximation.
E) Use the Newton-Raphson algorithm to find the zeroes of f(x) = g(x) - h(x).

Question 43

Use two repetitions of the Newton-Raphson algorithm to approximate the zero of f(x) = sin x - cos x near x = 0.
Enter just a real number rounded off to two decimal places.

Question 44

Determine the sum of the series .

A) -
B) 3
C)
D) none of these

Question 45

Use three repetitions of the Newton-Raphson algorithm to approximate . Let .
Enter just a real number rounded off to two decimal places.

Question 46

Use the Newton-Raphson algorithm with three repetitions to approximate the solution to = 2 - x near x = 2.
Enter just a real number rounded off to two decimal places.

Question 47

Use the Newton-Raphson algorithm with three repetitions to approximate the zero of f(x) = - 2 near Enter just a real number rounded off to two decimal places.

Question 48

Use the Newton-Raphson algorithm with three repetitions to approximate the zero of near Enter just a real number rounded off to two decimal places.

Question 49

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

Question 50

Below is a graph of the function f(x). If is taken as the initial approximation of the zero of f(x), then which of the following points, A, B, C, or D could be given by the Newton-Raphson algorithm as the next approximation?

A) A
B) B
C) C
D) D

Question 51

Let = 2. Use three repetitions of the Newton-Raphson algorithm to approximate .
Enter just a real number rounded off to two decimal places (no label).

Question 52

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

Question 53

Determine the sum of the series .

A)
B)
C)
D) none of these

Question 54

Determine the sum of the series + + + ... if it converges.

A) diverges
B)
C)
D) e

Question 55

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?

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 56

Determine the sum of the series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 2 } { e } \right) ^ { n }$ .

A) $\frac { 2 } { \mathrm { e } }$
B) $\frac { \mathrm { e } } { \mathrm { e } + 2 }$
C) 1 - $\frac { 2 } { \mathrm { e } }$
D) none of these

Question 57

Use two repetitions of the Newton-Raphson algorithm to approximate .
Enter just a real number rounded off to two decimal places.

Question 58

Determine the sum of the series

A) 101
B)
C) 10.1
D) none of these

Question 59

Use two repetitions of the Newton-Raphson algorithm to approximate the value of x for which = 3x. Use as the first approximation.
Enter just a real number rounded off to two decimal places.

Question 60

Use the Newton-Raphson algorithm with two repetitions to estimate the positive solution of sin x = x. Use = 2.
Enter just a real number rounded off to two decimal places.

Question 61

Sum an appropriate infinite series to find the rational number whose decimal expansion is: Enter just a reduced fraction of form .

Question 62

Determine the sum of the following geometric series: 1 - + - + - ... .
Enter just a reduced fraction of form .

Question 63

Sum an appropriate infinite series to find the rational number whose decimal expansion is: Enter just a reduced fraction of form .

Question 64

Determine the sum of the series

A)
B)
C)
D) none of these

Question 65

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

A) 2
B) 1
C) $\frac { 1 } { 2 }$
D) none of these

Question 66

Determine the sum of the infinite geometric series Enter your answer exactly in the reduced form .

Question 67

Determine the sum of the following infinite series: Enter just a reduced fraction of form .

Question 68

Sum an appropriate infinite series to find the rational number whose decimal expansion is: Enter just a reduced fraction of form .

Question 69

Determine the sum of the following geometric series: 2 + + + ... .
Enter just an integer.

Question 70

Determine the sum of the following infinite series: .
Enter just a reduced fraction of form .

Question 71

Determine the sum of the following infinite series: .
Enter your answer exactly in the reduced form .

Question 72

Determine the sum of the following geometric series: 1 + + + ... .
Enter your answer exactly in the form .

Question 73

Determine the sum of the following geometric series: + + + + + ... .
Enter just a reduced fraction of form .

Question 74

Determine the sum of the following geometric series: + + + + ... .
Enter a reduced fraction of form .

Question 75

Determine the sum of the following infinite series: .
Enter just an integer.

Question 76

Determine the sum of the following infinite series: .
Enter just an integer.

Question 77

Determine the sum of the following geometric series: 3 - 1.8 + 1.08 + .648 - ... .
Enter just a real number rounded off to three decimal places.

Question 78

Sum an appropriate infinite series to find the rational number whose decimal expansion is: .
Enter just a reduced fraction of form .

Question 79

Determine the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { 4 ^ { n - 1 } }$ .

A) $\frac { 3 } { 4 }$
B) 3
C) $\frac { 1 } { 4 }$
D) 12

Question 80

Determine the sum of the following geometric series: .
Enter a reduced fraction of form .