Question 1

Estimate $\int_{1}^{1.5} \ln t d t$ using a 4th degree Taylor Polynomial for $\ln t$ about t = 1.Round to 4 decimal places.

Accepted Answer

Question 2

A Taylor polynomial of degree six always has six non-zero terms.

Accepted Answer

A) True 
 B)False

Question 3

Consider the function $f(x)=1-\cos x$ .Is the Maclaurin series for $f(x)$ given by $\sum_{i=1}^{\infty} \frac{(-1)^{i+1} x^{2 i}}{(2 i) !}$ ?

Accepted Answer

A) True 
 B)False

Question 4

Suppose that g is the pulse train of width 0.5.What percent of the energy of g is contained in the constant term of its Fourier series? Round to one decimal place.

Accepted Answer

The answer of Suppose that g is the pulse train...

Question 5

The function $f(x)=e^{-x^{2} / 2}$ is part of the normal probability density function (or bell-shaped curve).Find the Maclaurin series for $\int e^{-x^{2} / 2} d x$ by first finding the Maclaurin series for $f(x)$ and then integrating it term by term.

Accepted Answer

A)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i+1}}{(2 i+1) 2^{i}(i !)}+C$ 
B)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{(2 i) 2^{i}(i !)}+C$ 
C)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i+1}}{(2 i+1) 2^{i}(2 i !)}+C$ 
D)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{(2 i) 2^{i}(2 i !)}+C$ 
A)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i+1}}{(2 i+1) 2^{i}(i !)}+C$ 
B)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{(2 i) 2^{i}(i !)}+C$ 
C)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i+1}}{(2 i+1) 2^{i}(2 i !)}+C$ 
D)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{(2 i) 2^{i}(2 i !)}+C$

Question 6

According to the theory of relativity, the energy, E, of a body of mass m is given as a function of its speed, v, by $E=m c^{2}\left(\frac{1}{\sqrt{1-v^{2} / c^{2}}}-1\right)$ , where c is a constant, the speed of light.Assuming v < c, expand E as a series in v/c, as far as the second non-zero term.If v = 0.05c, approximate E using your expansion.Also, approximate E by the formula $E=\frac{1}{2} m v^{2}$ .By what percentage do your two approximations differ?

Accepted Answer

The answer of According to the theory of relativity, the...

Question 7

What is the fourth degree Taylor polynomial for $\cos \left(3 x^{2}\right)$ about x = 0?

Accepted Answer

A)  $1-\frac{9}{2} x^{4}$ 
B)  $1-\frac{3}{2} x^{4}$ 
C)  $1-\frac{9}{2} x^{2}+\frac{27}{6} x^{3}-\frac{81}{24} x^{4}$ 
D)  $x+\frac{3}{2} x^{2}-\frac{9}{6} x^{3}+\frac{27}{24} x^{4}$ 
A)  $1-\frac{9}{2} x^{4}$ 
B)  $1-\frac{3}{2} x^{4}$ 
C)  $1-\frac{9}{2} x^{2}+\frac{27}{6} x^{3}-\frac{81}{24} x^{4}$ 
D)  $x+\frac{3}{2} x^{2}-\frac{9}{6} x^{3}+\frac{27}{24} x^{4}$

Question 8

The graph of y = f(x)is given below.   Suppose we approximate f(x)near x = 17 by the second degree Taylor polynomial centered about 17, $a+b(x-17)+c(x-17)^{2}$ .Is b positive, negative, or zero?

Accepted Answer

The answer of The graph of y = f(x)is given...

Question 9

Find the third-degree Fourier polynomial for $f(t)=$$\left\{\begin{array}{l}
0 \
c
\end{array}\right.$$\begin{array}{c}
-2 <\
0<
\end{array}$$\begin{array}{l}
x \leq 0 \
x \leq 2
\end{array}$ , where c is a constant, by writing a new function, $g(x)=f(t)$ , with period $2 \pi$ .

Accepted Answer

A)  $\frac{c}{2}-\frac{2 c}{\pi} \sin \left(\frac{\pi t}{2}\right)-\frac{2 c}{3 \pi} \sin \left(\frac{3 \pi t}{2}\right)$ 
B)  $\frac{c}{2}+\frac{2 c}{\pi} \sin \left(\frac{\pi t}{2}\right)+\frac{2 c}{3 \pi} \sin \left(\frac{3 \pi t}{2}\right)$ 
C)  $\frac{c}{2}+\frac{2 c}{\pi} \sin (t)+\frac{2 c}{3 \pi} \sin (3 t)$ 
D)  $\frac{c}{2}-\frac{2 c}{\pi} \sin (t)-\frac{2 c}{3 \pi} \sin (3 t)$ 
A)  $\frac{c}{2}-\frac{2 c}{\pi} \sin \left(\frac{\pi t}{2}\right)-\frac{2 c}{3 \pi} \sin \left(\frac{3 \pi t}{2}\right)$ 
B)  $\frac{c}{2}+\frac{2 c}{\pi} \sin \left(\frac{\pi t}{2}\right)+\frac{2 c}{3 \pi} \sin \left(\frac{3 \pi t}{2}\right)$ 
C)  $\frac{c}{2}+\frac{2 c}{\pi} \sin (t)+\frac{2 c}{3 \pi} \sin (3 t)$ 
D)  $\frac{c}{2}-\frac{2 c}{\pi} \sin (t)-\frac{2 c}{3 \pi} \sin (3 t)$

Question 10

Approximate the function $f(x)=e^{-x^{2}}$ with a Taylor polynomial of degree 6.Use this to estimate the value of $\int_{0}^{0.6} e^{-x^{2}} d x$ to 5 decimal places.

Accepted Answer

The answer of Approximate the function $f(x)=e^{-x^{2}}$ with a Taylor...

Question 11

Suppose a function satisfies $f(5)=2$ , $f^{\prime}(5)=5$ , $f^{\prime \prime}(5)=-7$ , and $f^{\prime \prime \prime}$$(5)=12$ .What is the third degree Taylor polynomial for f about x = 5?

Accepted Answer

A)  $2+5(x-5)-7(x-5)^{2}+12(x-5)^{3}$ 
B)  $2+5(x-5)-\frac{7}{2}(x-5)^{2}+2(x-5)^{3}$ 
C)  $2+5 x-\frac{7}{2} x^{2}+2 x^{3}$ 
D)  $10+25 x-\frac{35}{2} x^{2}+6 x^{3}$ 
A)  $2+5(x-5)-7(x-5)^{2}+12(x-5)^{3}$ 
B)  $2+5(x-5)-\frac{7}{2}(x-5)^{2}+2(x-5)^{3}$ 
C)  $2+5 x-\frac{7}{2} x^{2}+2 x^{3}$ 
D)  $10+25 x-\frac{35}{2} x^{2}+6 x^{3}$

Question 12

Suppose that you are told that the Taylor series of $f(x)=e^{-x^{2}}$ about x = 0 is $1-\frac{x^{2}}{1 !}+\frac{x^{4}}{2 !}-\frac{x^{6}}{3 !}+\cdots$ .Find $\left.\frac{d^{2}}{d^{2}}\left(e^{-x^{2}}\right)\right|_{x=0}$ .

Accepted Answer

The answer of Suppose that you are told that the...

Question 13

Estimate the magnitude of the error in approximating $\sin (1)$ using a third degree Taylor polynomial about x = 0.

Accepted Answer

A) 0.0125 
B) 0.0333 
C) 0.0417 
D) 0.0625 
A) 0.0125 
B) 0.0333 
C) 0.0417 
D) 0.0625

Question 14

Use the Taylor polynomials for the sine and cosine functions to find a rational function with a degree 5 numerator and no fractional coefficients that approximates the tangent function near 0.

Accepted Answer

A)  $R(x)=\frac{x^{5}+24 x^{4}-120 x^{3}}{x^{4}-6 x^{2}+120}$ 
B)  $R(x)=\frac{x^{5}-20 x^{3}+120 x}{5 x^{4}-60 x^{2}+120}$ 
C)  $R(x)=\frac{30 x^{4}-6 x^{2}+120}{-120 x^{3}+x^{5}}$ 
D)  $R(x)=\frac{120 x^{5}-24 x^{3}+x}{30 x^{4}-60 x^{2}+120}$ 
A)  $R(x)=\frac{x^{5}+24 x^{4}-120 x^{3}}{x^{4}-6 x^{2}+120}$ 
B)  $R(x)=\frac{x^{5}-20 x^{3}+120 x}{5 x^{4}-60 x^{2}+120}$ 
C)  $R(x)=\frac{30 x^{4}-6 x^{2}+120}{-120 x^{3}+x^{5}}$ 
D)  $R(x)=\frac{120 x^{5}-24 x^{3}+x}{30 x^{4}-60 x^{2}+120}$

Question 15

Use a Taylor polynomial of degree 3 for $f(x)=e^{4 x}$ to approximate the value of $e^{0.8}$ .Give your answer to five decimal places.

Accepted Answer

A) 23.66667 
B) 1.22133 
C) 2.22554 
D) 2.20533 
A) 23.66667 
B) 1.22133 
C) 2.22554 
D) 2.20533

Question 16

Is $\frac{1}{(n) !}$ a good bound for the maximum possible error for the nth degree Taylor polynomial about x = 0 approximating $\sin \left(\frac{x}{2}\right)$ on the interval [0, 1]?

Accepted Answer

A) True 
 B)False

Question 17

Recognize $4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Accepted Answer

The answer of Recognize $4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots$ as a Taylor series evaluated...

Question 18

Estimate $\int_{-2}^{-19} \ln (1-x) d x$ using the first two terms of the Taylor series about x = -2 for $\ln (1-x)$ .

Accepted Answer

A)  $1.098612$ 
B)  $0.108195$ 
C)  $-2.08903$ 
D) 20.085537 
A)  $1.098612$ 
B)  $0.108195$ 
C)  $-2.08903$ 
D) 20.085537

Question 19

Use the Taylor series for $f(x)=\cos (x)$ at x = 0 to find the Taylor series for $\cos (\sqrt{x})$ at x = 0.

Accepted Answer

A)  $1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\cdots$ 
B)  $1-\frac{x}{2 !}+\frac{x^{2}}{4 !}-\frac{x^{3}}{6 !}+\cdots$ 
C)  $\sqrt{x}-\frac{x}{2 !}+\frac{x^{3 / 2}}{4 !}-\frac{x^{2}}{6 !}+\cdots$ 
D)  $\sqrt{x}-x+\frac{x^{3 / 2}}{2 !}-\frac{x^{2}}{3 !}+\cdots$ 
A)  $1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\cdots$ 
B)  $1-\frac{x}{2 !}+\frac{x^{2}}{4 !}-\frac{x^{3}}{6 !}+\cdots$ 
C)  $\sqrt{x}-\frac{x}{2 !}+\frac{x^{3 / 2}}{4 !}-\frac{x^{2}}{6 !}+\cdots$ 
D)  $\sqrt{x}-x+\frac{x^{3 / 2}}{2 !}-\frac{x^{2}}{3 !}+\cdots$

Question 20

Find $a_{0}$ for the function $h(x)=$$\left\{\begin{array}{l}
\pi \
0
\end{array}\right.$$\begin{aligned}
-\pi &<\
0<
\end{aligned}$$\begin{array}{l}
x \leq 0 \
x \leq \pi
\end{array}$ .

Accepted Answer

A)  $-\frac{\pi}{4}$ 
B)  $\frac{\pi}{4}$ 
C)  $-\frac{\pi}{2}$ 
D)  $\frac{\pi}{2}$ 
E) 0 
A)  $-\frac{\pi}{4}$ 
B)  $\frac{\pi}{4}$ 
C)  $-\frac{\pi}{2}$ 
D)  $\frac{\pi}{2}$ 
E) 0

Estimate $\int_{1}^{1.5} \ln t d t$ using a 4^th degree Taylor Polynomial for $\ln t$ about t = 1.Round to 4 decimal places.

A Taylor polynomial of degree six always has six non-zero terms.

Consider the function $f(x)=1-\cos x$ .Is the Maclaurin series for $f(x)$ given by $\sum_{i=1}^{\infty} \frac{(-1)^{i+1} x^{2 i}}{(2 i) !}$ ?

Suppose that g is the pulse train of width 0.5.What percent of the energy of g is contained in the constant term of its Fourier series? Round to one decimal place.

The function $f(x)=e^{-x^{2} / 2}$ is part of the normal probability density function (or bell-shaped curve).Find the Maclaurin series for $\int e^{-x^{2} / 2} d x$ by first finding the Maclaurin series for $f(x)$ and then integrating it term by term.

What is the fourth degree Taylor polynomial for $\cos \left(3 x^{2}\right)$ about x = 0?

Find the third-degree Fourier polynomial for $f(t)=$ $\left\{\begin{array}{l}0 \\c\end{array}\right.$ -2< 0< x\leq0 x\leq2 , where c is a constant, by writing a new function, $g(x)=f(t)$ , with period $2 \pi$ .

Approximate the function $f(x)=e^{-x^{2}}$ with a Taylor polynomial of degree 6.Use this to estimate the value of $\int_{0}^{0.6} e^{-x^{2}} d x$ to 5 decimal places.

Suppose a function satisfies $f(5)=2$ , $f^{\prime}(5)=5$ , $f^{\prime \prime}(5)=-7$ , and $f^{\prime \prime \prime}$ $(5)=12$ .What is the third degree Taylor polynomial for f about x = 5?

Suppose that you are told that the Taylor series of $f(x)=e^{-x^{2}}$ about x = 0 is $1-\frac{x^{2}}{1 !}+\frac{x^{4}}{2 !}-\frac{x^{6}}{3 !}+\cdots$ .Find $\left.\frac{d^{2}}{d^{2}}\left(e^{-x^{2}}\right)\right|_{x=0}$ .

Estimate the magnitude of the error in approximating $\sin (1)$ using a third degree Taylor polynomial about x = 0.

Use the Taylor polynomials for the sine and cosine functions to find a rational function with a degree 5 numerator and no fractional coefficients that approximates the tangent function near 0.

Use a Taylor polynomial of degree 3 for $f(x)=e^{4 x}$ to approximate the value of $e^{0.8}$ .Give your answer to five decimal places.

Is $\frac{1}{(n) !}$ a good bound for the maximum possible error for the n^th degree Taylor polynomial about x = 0 approximating $\sin \left(\frac{x}{2}\right)$ on the interval [0, 1]?

Recognize $4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Estimate $\int_{-2}^{-19} \ln (1-x) d x$ using the first two terms of the Taylor series about x = -2 for $\ln (1-x)$ .

Use the Taylor series for $f(x)=\cos (x)$ at x = 0 to find the Taylor series for $\cos (\sqrt{x})$ at x = 0.

Find $a_{0}$ for the function $h(x)=$ $\left\{\begin{array}{l}\pi \\0\end{array}\right.$ -\pi < 0< x\leq0 x\leq\pi .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 10: Approximating Functions Using Series

Estimate ∫11.5ln⁡tdt\int_{1}^{1.5} \ln t d t∫11.5​lntdt using a 4th degree Taylor Polynomial for ln⁡t\ln tlnt about t = 1.Round to 4 decimal places.

A Taylor polynomial of degree six always has six non-zero terms.

Consider the function f(x)=1−cos⁡xf(x)=1-\cos xf(x)=1−cosx .Is the Maclaurin series for f(x)f(x)f(x) given by ∑i=1∞(−1)i+1x2i(2i)!\sum_{i=1}^{\infty} \frac{(-1)^{i+1} x^{2 i}}{(2 i) !}∑i=1∞​(2i)!(−1)i+1x2i​ ?

Suppose that g is the pulse train of width 0.5.What percent of the energy of g is contained in the constant term of its Fourier series? Round to one decimal place.

What is the fourth degree Taylor polynomial for cos⁡(3x2)\cos \left(3 x^{2}\right)cos(3x2) about x = 0?

The graph of y = f(x)is given below. Suppose we approximate f(x)near x = 17 by the second degree Taylor polynomial centered about 17, a+b(x−17)+c(x−17)2a+b(x-17)+c(x-17)^{2}a+b(x−17)+c(x−17)2 .Is b positive, negative, or zero?

Find the third-degree Fourier polynomial for f(t)=f(t)=f(t)={0c\left\{\begin{array}{l}0 \\c\end{array}\right.{0c​ -2< 0< x\leq0 x\leq2 , where c is a constant, by writing a new function, g(x)=f(t)g(x)=f(t)g(x)=f(t) , with period 2π2 \pi2π .

Approximate the function f(x)=e−x2f(x)=e^{-x^{2}}f(x)=e−x2 with a Taylor polynomial of degree 6.Use this to estimate the value of ∫00.6e−x2dx\int_{0}^{0.6} e^{-x^{2}} d x∫00.6​e−x2dx to 5 decimal places.

Suppose a function satisfies f(5)=2f(5)=2f(5)=2 , f′(5)=5f^{\prime}(5)=5f′(5)=5 , f′′(5)=−7f^{\prime \prime}(5)=-7f′′(5)=−7 , and f′′′f^{\prime \prime \prime}f′′′(5)=12(5)=12(5)=12 .What is the third degree Taylor polynomial for f about x = 5?

Estimate the magnitude of the error in approximating sin⁡(1)\sin (1)sin(1) using a third degree Taylor polynomial about x = 0.

Use the Taylor polynomials for the sine and cosine functions to find a rational function with a degree 5 numerator and no fractional coefficients that approximates the tangent function near 0.

Use a Taylor polynomial of degree 3 for f(x)=e4xf(x)=e^{4 x}f(x)=e4x to approximate the value of e0.8e^{0.8}e0.8 .Give your answer to five decimal places.

Is 1(n)!\frac{1}{(n) !}(n)!1​ a good bound for the maximum possible error for the nth degree Taylor polynomial about x = 0 approximating sin⁡(x2)\sin \left(\frac{x}{2}\right)sin(2x​) on the interval [0, 1]?

Recognize 4−422+433−444+⋯4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots4−242​+343​−444​+⋯ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Estimate ∫−2−19ln⁡(1−x)dx\int_{-2}^{-19} \ln (1-x) d x∫−2−19​ln(1−x)dx using the first two terms of the Taylor series about x = -2 for ln⁡(1−x)\ln (1-x)ln(1−x) .

Use the Taylor series for f(x)=cos⁡(x)f(x)=\cos (x)f(x)=cos(x) at x = 0 to find the Taylor series for cos⁡(x)\cos (\sqrt{x})cos(x​) at x = 0.

Find a0a_{0}a0​ for the function h(x)=h(x)=h(x)={π0\left\{\begin{array}{l}\pi \\0\end{array}\right.{π0​ -\pi < 0< x\leq0 x\leq\pi .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Estimate $\int_{1}^{1.5} \ln t d t$ using a 4^th degree Taylor Polynomial for $\ln t$ about t = 1.Round to 4 decimal places.

Consider the function $f(x)=1-\cos x$ .Is the Maclaurin series for $f(x)$ given by $\sum_{i=1}^{\infty} \frac{(-1)^{i+1} x^{2 i}}{(2 i) !}$ ?

What is the fourth degree Taylor polynomial for $\cos \left(3 x^{2}\right)$ about x = 0?

Find the third-degree Fourier polynomial for $f(t)=$ $\left\{\begin{array}{l}0 \\c\end{array}\right.$ -2< 0< x\leq0 x\leq2 , where c is a constant, by writing a new function, $g(x)=f(t)$ , with period $2 \pi$ .

Approximate the function $f(x)=e^{-x^{2}}$ with a Taylor polynomial of degree 6.Use this to estimate the value of $\int_{0}^{0.6} e^{-x^{2}} d x$ to 5 decimal places.

Suppose a function satisfies $f(5)=2$ , $f^{\prime}(5)=5$ , $f^{\prime \prime}(5)=-7$ , and $f^{\prime \prime \prime}$ $(5)=12$ .What is the third degree Taylor polynomial for f about x = 5?

Estimate the magnitude of the error in approximating $\sin (1)$ using a third degree Taylor polynomial about x = 0.

Use a Taylor polynomial of degree 3 for $f(x)=e^{4 x}$ to approximate the value of $e^{0.8}$ .Give your answer to five decimal places.

Is $\frac{1}{(n) !}$ a good bound for the maximum possible error for the n^th degree Taylor polynomial about x = 0 approximating $\sin \left(\frac{x}{2}\right)$ on the interval [0, 1]?

Recognize $4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Estimate $\int_{-2}^{-19} \ln (1-x) d x$ using the first two terms of the Taylor series about x = -2 for $\ln (1-x)$ .

Use the Taylor series for $f(x)=\cos (x)$ at x = 0 to find the Taylor series for $\cos (\sqrt{x})$ at x = 0.

Find $a_{0}$ for the function $h(x)=$ $\left\{\begin{array}{l}\pi \\0\end{array}\right.$ -\pi < 0< x\leq0 x\leq\pi .