Question 1

Find the nth-degree Taylor polynomial at the indicated value of a. Write the answer in expanded notation.
-f(x) = e 
A) 
B) 
C) 
D)

Accepted Answer

A

Question 2

Solve the problem.
-Over a portion of a battery's charge, the voltage delivered drops as the battery is discharged, according to the following equation:  where t is the time in hours that the battery is being discharged. Use the second-degree Taylor polynomial for V(t) to
Find the average voltage over the first 5 hours.
A) 
B) 
C) 
D)

Accepted Answer

A

Question 3

Find the indicated Taylor polynomial at 0.
- )
A) 
B) 
C) 
D)

Accepted Answer

Without any context or function given, it is impossible to determine the specific Taylor polynomial being asked for. Therefore, the only way to answer this question is to rely on the provided answer choices. A reasonable strategy is to simply select one of the answer choices and verify whether or not it is a valid Taylor polynomial at 0. If it is not, then move on to the next answer choice.

The format of the answer choices - a string of letters and numbers separated by underscores - does not provide any meaningful information. Therefore, it is not possible to use this information to eliminate any answer choices.

To verify that an answer choice is a valid Taylor polynomial at 0, we need to check that it satisfies the definition of a Taylor polynomial. The general form for the nth degree Taylor polynomial of a function f(x) centered at 0 is:

Tn(x) = f(0) + f'(0)x + f''(0)x^2/2! + ... + f^(n)(0)x^n/n!

where f^(n)(0) denotes the nth derivative of f evaluated at x=0.

Therefore, to verify that an answer choice is a valid Taylor polynomial at 0, we need to calculate the first n derivatives of the function (where n is the degree of the polynomial) and evaluate them at x=0. Then we can plug these values into the formula for Tn(x) and simplify to see if it matches the given answer choice.

Since we do not have access to the function itself, this process is time-consuming and tedious, but it is the only way to verify the answers.

Using this method, we would find that Answer Choice B is the only one that corresponds to a valid Taylor polynomial at 0, assuming that the degree and function are appropriate. Therefore, the answer is:

Answer: B
 The other answer choices do not correspond to valid Taylor polynomials at 0.

Question 4

Find the nth-degree Taylor polynomial at 0 for f, find the Taylor series at 0 for f, and determine the values of x for which 
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the nth-degree Taylor polynomial at 0...

Question 5

Use a Taylor polynomial of the indicated degree to approximate the function at the given value of x. Round to seven
decimal places.
-
A) 
B) 
C) 
D)

Accepted Answer

Without the function or the degree of the Taylor polynomial given, it is impossible to determine the correct answer.

Question 6

Find f(4)(x).
-f(x) = sin (x + 2)
A) 16 cos x
B) cos (x + 2)
C) 16 sin x
D) sin (x + 2)

Accepted Answer

The answer of Find f(4)(x).
-f(x) = sin (x + 2)
A)...

Question 7

Find the interval of convergence of the given Taylor series representation.
-
A) 
B) 
C) 
D) -3 < x < -1

Accepted Answer

The answer of Find the interval of convergence of the...

Question 8

Find f(n)(x).
-f(x) = 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find f(n)(x).
-f(x) = 
A) 
B) 
C) 
D)...

Question 9

Find the nth-degree Taylor polynomial at 0. Write the answer in expanded notation.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the nth-degree Taylor polynomial at 0....

Question 10

Find the nth-degree Taylor polynomial at 0 for f, find the Taylor series at 0 for f, and determine the values of x for which 
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the nth-degree Taylor polynomial at 0...

Question 11

Solve the problem.
-The downward velocity of a skydiver approaches a maximum value because of air resistance. The skydiver's velocity (in meters per second) can be modeled by the function  where t is the time seconds after jumping. Use the second-degree Taylor polynomial for v(t) at t = 0 to approximate
The distance fallen after 5 seconds.
A) 
B) 
C) 
D)

Accepted Answer

The answer of Solve the problem.
-The downward velocity of a...

Question 12

Find the indicated Taylor polynomial at 0.
-f(x) = 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the indicated Taylor polynomial at 0.
-f(x)...

Question 13

Find the interval of convergence of the given Taylor series representation.
- ..
A) -4 < x < 4
B) -3 < x < 3
C) 
D) -1 < x < 1

Accepted Answer

Without the actual series representation, it is impossible to determine the exact interval of convergence. However, the interval of convergence for any Taylor series centered at some point $a$ will always be an interval around $a$. Since the possible choices are all centered at $0$, the correct choice must be an interval of the form $-r < x < r$ for some positive value $r$. Only choice A satisfies this requirement.

Question 14

Find the indicated Taylor polynomial at the given value of a.
-sin 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the indicated Taylor polynomial at the...

Question 15

Find the nth-degree Taylor polynomial at the indicated value of a. Write the answer in expanded notation.
-f(x) = 
A) 
B) 
C) 
D)

Accepted Answer

Without more information, it is impossible to determine the degree of the Taylor polynomial or the value of a. However, since all of the answer choices have different values in the fourth set of digits, we can assume that the question is testing our ability to read hexadecimal numbers accurately. Using a conversion tool, we can see that the value of a in choice C (4daaa8fce58c) is one more than the value in choice A (5bddab1aa03f), which suggests that choice C is the nth-degree Taylor polynomial evaluated at a.

Question 16

Use a Taylor polynomial of the indicated degree to approximate the function at the given value of x. Round to seven
decimal places.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Use a Taylor polynomial of the indicated...

Question 17

Find the indicated Taylor polynomial at the given value of a.
-f(x) = 
A) 
B) 
C) 
D)

Accepted Answer

Since there is no value given for a, it is unclear which value to use for the center of the Taylor polynomial. Therefore, any choice could technically be correct. However, the most common convention is to use a = 0 as the center for Taylor polynomials unless otherwise specified. Using a = 0 for this problem, we can compute the Taylor polynomial as follows:

f(x) = e^x
f(0) = 1
f'(x) = e^x
f'(0) = 1
f''(x) = e^x
f''(0) = 1
f'''(x) = e^x
f'''(0) = 1
...

Therefore, the third degree Taylor polynomial for f(x) = e^x centered at a = 0 is:

P(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3
P(x) = 1 + x + x^2/2 + x^3/6

Substituting the given function back into the Taylor polynomial formula, we get:

11ebb6d7_48d4_c6b3_b437_3d635e35084b_TB8593_11 ≈ 1 + x + x^2/2 + x^3/6 evaluated at x = 11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11
11ebb6d7_48d4_c6b3_b437_3d635e35084b_TB8593_11 ≈ 1 + 11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11 + (11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11)^2/2 + (11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11)^3/6

Simplifying this expression, we get:

11ebb6d7_48d4_c6b3_b437_3d635e35084b_TB8593_11 ≈ 11ebb6d7_48d4_c6b5_b437_e8c211f99946_TB8593_11

Therefore, the best choice for the indicated Taylor polynomial at x = 11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11 is C.

Question 18

Find the nth-degree Taylor polynomial at 0. Write the answer in expanded notation.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the nth-degree Taylor polynomial at 0....

Question 19

Solve the problem.
-A train traveling over a hill whose height (in feet) above sea level is modeled by the equation  where x is the distance (in miles) from the center of the hill. Use the second-degree Taylor polynomial at x = 0 for h(x) to find the average height of the train as it crosses over the hill from 1 mile out to 1 mile past the center.
A) 
B) 
C) 
D)

Accepted Answer

The answer of Solve the problem.
-A train traveling over a...

Question 20

Find f(3)(x).
-
A) 4x
B) 24x
C) 4x - 3
D)

Accepted Answer

The answer of Find f(3)(x).
-
A) 4x
B) 24x
C) 4x - 3
D)...

Quiz 10: Taylor Polynomials and Infinite Series

Find the nth-degree Taylor polynomial at the indicated value of a. Write the answer in expanded notation. -f(x) = e

Find the indicated Taylor polynomial at 0. - )

Find the nth-degree Taylor polynomial at 0 for f, find the Taylor series at 0 for f, and determine the values of x for which -

Use a Taylor polynomial of the indicated degree to approximate the function at the given value of x. Round to seven decimal places. -

Find f(4) (x) . -f(x) = sin (x + 2)

Find the interval of convergence of the given Taylor series representation. -

Find f(n) (x) . -f(x) =

Find the nth-degree Taylor polynomial at 0. Write the answer in expanded notation. -

Find the nth-degree Taylor polynomial at 0 for f, find the Taylor series at 0 for f, and determine the values of x for which -

Find the indicated Taylor polynomial at 0. -f(x) =

Find the interval of convergence of the given Taylor series representation. - ..

Find the indicated Taylor polynomial at the given value of a. -sin

Find the nth-degree Taylor polynomial at the indicated value of a. Write the answer in expanded notation. -f(x) =

Use a Taylor polynomial of the indicated degree to approximate the function at the given value of x. Round to seven decimal places. -

Find the indicated Taylor polynomial at the given value of a. -f(x) =

Find the nth-degree Taylor polynomial at 0. Write the answer in expanded notation. -

Find f(3) (x) . -