Deck 10: Taylor Polynomials and Infinite Series

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)

Answer 1

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)

Answer 2

Question 3

Find the indicated Taylor polynomial at 0.
)

A)
B)
C)
D)

Answer 3

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)

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)

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)

Question 7

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

A)
B)
C)
D) -3 < x < -1

Question 8

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)

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)

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)

Question 12

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

A)
B)
C)
D)

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

Question 14

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

A)
B)
C)
D)

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)

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)

Question 17

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

A)
B)
C)
D)

Question 18

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

A)
B)
C)
D)

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)

Question 20

Find f(3)(x).

A) 4x
B) 24x
C) 4x - 3
D)

Question 21

Find the Taylor series at 0. State the interval of convergence for the series.
ln (1 - x) + ln (1 + x)

A)
B)
C)
D)

Question 22

Use the substitution t = x - a to find the Taylor series at the indicated value of a. State the interval of convergence for the
series.

A)
B)
C)
D)

Question 23

Solve the problem.
Find the Taylor polynomial of degree 2n at 0 for g(x) = cos x.

A)
B)
C)
D)

Question 24

Find the Taylor series at 0. State the interval of convergence for the series.
The Taylor series at 0 for the sine and cosine functions are given below. F(x) satisfies = sin x and F(0) = -3

A)
B)
C)
D)

Question 25

Find the nth-degree Taylor polynomial at the indicated value of a for f, find the Taylor series at a for f and determine the
values of x for which
f(x) =

A)
B)
C)
D)

Question 26

Find the Taylor series at 0. State the interval of convergence for the series.
f(x) satisfies

A)
B)
C)
D)

Question 27

Use a Taylor polynomial at 0 to approximate the expression with an error of no more than 0.000005. Select the polynomial
of lowest degree that can be used to obtain this accuracy and state the degree of this polynomial. Round to six decimal
places.

A) 0.740818; n = 6
B) 0.740838; n = 5
C) 0.740817; n = 4
D) 0.740817; n = 5

Question 28

Find the Taylor series at 0. State the interval of convergence for the series.

A)
B)
C)
D)

Question 29

Find the Taylor series at 0. State the interval of convergence for the series.
f(x) satisfies

A)
B)
C)
D)

Question 30

Find the Taylor series at 0. State the interval of convergence for the series.
The Taylor series at 0 for the sine and cosine functions are given below.

A)
B)
C)
D)

Question 31

Solve the problem.
Evaluating the Taylor series at 0 for f(x) = ln (1 + x) at x = 0.6 produces the following series. Use four terms in this series to approximate ln 1.6, and then estimate the error in this approximation.

A)
B)
C)
D)

Question 32

Use the substitution t = x - a to find the Taylor series at the indicated value of a. State the interval of convergence for the
series.

A)
B)
C)
D)

Question 33

Find the nth-degree Taylor polynomial at the indicated value of a for f, find the Taylor series at a for f and determine the
values of x for which
f(x) = ln (5 - x); a = 4

A)
B)
C)
D)

Question 34

Use the third-degree Taylor polynomial at 0 for the given function to approximate the expression, and then estimate the
error in the approximation. Round to six decimal places as needed.

A)
B)
C)
D)

Question 35

Use the third-degree Taylor polynomial at 0 for the given function to approximate the expression, and then estimate the
error in the approximation. Round to six decimal places as needed.

A)
B)
C)
D)

Question 36

Find the Taylor series at 0. State the interval of convergence for the series.
use the Taylor series at 0 for and repeated applications of the differentiation property of Taylor series

A)
B)
C)
D)

Question 37

Solve the problem.
Evaluating the Taylor series at 0 for produces the following series. Use three terms in this series to approximate and then estimate the error in this approximation.

A)
B)
C)
D)

Question 38

Find the Taylor series at 0. State the interval of convergence for the series.

A)
B)
C)
D)

Question 39

Solve the problem.
Find the Taylor series at 0 for g(x) = cos x (while evaluating cannot be directly applied, other techniques can be used to show that the Taylor series at 0 for g(x) = cos x converges for all values of x).

A)
B)
C)
D)

Question 40

Find the Taylor series at 0. State the interval of convergence for the series.

A)
B)
C)
D)

Question 41

Solve the problem.
Assume that f(x) is a function such that for all n and all x and let be the nth-degree Taylor polynomial for f at 0. Use Taylor's formula for the remainder to find the smallest value of n such that the error
In the approximation of f(6) by pn(6) is guaranteed to be less than 0.001.

A) n = 20
B) n = 25
C) n = 6
D) n = 13

Question 42

Solve the problem. Use the theorem for error estimation for alternating series to perform the indicated error estimation.
Given the demand equation approximate the average price (in dollars) over the demand interval 0, 20 to within ± 0.005.

A) $13.924
B) $9.284
C) $0.716
D) $13.926

Question 43

Solve the problem. Use the theorem for error estimation for alternating series to perform the indicated error estimation.
The rate of healing for a skin wound (in square centimeters per day) is given by A The initial wound has an area of 12 square centimeters. Use the second-degree Taylor polynomial at 0 for A‛(t) to
Approximate the area of the would after 3 days, and estimate the error in this approximation.

A)
B)
C)
D)

Question 44

Solve the problem. Use the theorem for error estimation for alternating series to perform the indicated error estimation.
The income distribution for a certain country is represented by the Lorenz curve with the equation f Approximate the index of income concentration to within ± 0.005.

A) 0.740
B) 0.731
C) 0.733
D) 0.833

Question 45

Use a Taylor series at 0 to approximate the integral with an error of no more than 0.0005.

A) 0.2980
B) 0.3
C) 0.0020
D) 0.2979

Question 46

Use a Taylor polynomial at 0 to approximate the expression with an error of no more than 0.000005. Select the polynomial
of lowest degree that can be used to obtain this accuracy and state the degree of this polynomial. Round to six decimal
places.

A) 0.048792; n = 3
B) 0.04875; n = 2
C) 0.048790; n = 3
D) 0.048790; n = 4

Question 47

Solve the problem.
Use the second-degree Taylor polynomial at 0 for to approximate e0.04. Use Taylor's formula for the remainder to estimate the error in the approximation.

A)
B)
C)
D)

Question 48

Use a Taylor series at 0 to approximate the integral with an error of no more than 0.0005.

A) 0.0284
B) 0.0286
C) 0.0287
D) 0.0324