Deck 10: Approximating Functions Using Series

Question 1

Use the first three nonzero terms of the Taylor polynomial to approximate $\int_{0.1}^{1.1} \frac{\sin x}{x} d x$ .Give your answer to 5 decimal places.

A)0.92868
B)0.92880
C)-0.92880
D)-0.92868
E)does not exist

Answer 1

0.92880

Question 2

Find the fourth term of the Taylor series for the function $f(x)=\ln (x+1)$ about x = 1.

A) $\frac{(x-1)^{3}}{3}$
B) $\frac{(x-1)^{3}}{6}$
C) $\frac{(x-1)^{3}}{12}$
D) $\frac{(x-1)^{3}}{24}$

Answer 2

$\frac{(x-1)^{3}}{24}$

Question 3

Recognize as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Answer 3

1.6094

Question 4

Approximate the function with a Taylor polynomial of degree 6.Use this to estimate the value of to 5 decimal places.

Question 5

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?

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 6

The hyperbolic cosine function is defined as follows: $f(x)=\cosh (x)=\frac{e^{x}+e^{-x}}{2}$ .Use the Taylor polynomial for $e^{x}$ near 0 to find the Taylor polynomial of degree 4 for $f(x)=4 \cosh (x)$ .

A) $4-\frac{4 x^{2}}{2 !}+\frac{4 x^{3}}{3 !}-\frac{4 x^{4}}{4 !}$
B) $4\left(1+\frac{x^{2}}{2}+\frac{x^{4}}{24}\right)$
C) $4+\frac{3 x}{1 !}-\frac{3 x^{2}}{2 !}+\frac{x^{4}}{24}$
D) $4+4 x+\frac{4 x^{2}}{2}+\frac{4 x^{3}}{3 !}+\frac{4 x^{4}}{4 !}$

Question 7

Find the fourth term of the Taylor series for the function $f(x)=\cos x$ about $x=\pi / 3$ .

A) $\frac{\left(x-\frac{\pi}{3}\right)^{3}}{12}$
B) $-\frac{\left(x-\frac{\pi}{3}\right)^{3}}{12}$
C) $\frac{\sqrt{3}\left(x-\frac{\pi}{3}\right)^{3}}{12}$
D) $-\frac{\sqrt{3}\left(x-\frac{\pi}{3}\right)^{3}}{12}$

Question 8

Find using a Taylor approximation for sin x.

Question 9

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, .Is b positive, negative, or zero?

Question 10

Find the Taylor polynomial of degree 3 around x = 0 for the function and use it to approximate .Give your answer to 4 decimal places.

Question 11

Approximate the function for values of x near 0 using the first three non-zero terms of its Taylor polynomial.

Question 12

Is the Taylor polynomial of degree 6 for for x near 0 given by ?

Question 13

Suppose you approximate $\sqrt{0.9}$ and $\sqrt{0.2}$ using the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ .Which approximation is more accurate?

A)the approximation of $\sqrt{0.9}$
B)the approximation of $\sqrt{0.2}$

Question 14

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

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 15

Solve for x.

Question 16

Construct the Taylor polynomial approximation of degree 3 to the function about the point x = 0.Use it to approximate the value to 5 decimal places.How does the approximation compare to the actual value?

Question 17

Recognize as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Question 18

The function g has the Taylor approximation and the graph given below: Is c₀ positive, negative, or zero?

Question 19

Estimate using a 4^th degree Taylor Polynomial for about t = 1.Round to 4 decimal places.

Question 20

What is the general term of the series $1-\frac{x^{5}}{5 !}+\frac{x^{10}}{10 !}-\frac{x^{15}}{15 !}+\cdots$ ?

A) $\frac{(-1)^{k} x^{5 k}}{(5 k) !}$ for $k \geq 0$
B) $\frac{(-1)^{k+1} x^{5 k}}{(5 k) !}$ for $k \geq 0$
C) $\frac{(-1)^{k} x^{5 k+5}}{(5 k+5) !}$ for $k \geq 0$
D) $\frac{(-1)^{k+1} x^{5 k+1}}{(5 k+1) !}$ for $k \geq 0$

Question 21

Find the Maclaurin series for $f(x)=\sin (5 x)$ .

A) $5 x-\frac{5^{3} x^{3}}{3 !}+\frac{5^{5} x^{5}}{5 !}-$
B) $1-\frac{5^{2} x^{2}}{2 !}+\frac{5^{4} x^{4}}{4 !}-\cdots$
C) $x-\frac{5^{2} x^{3}}{3 !}+\frac{5^{4} x^{5}}{5 !}-\cdots$
D) $5 x-\frac{5^{2} x^{2}}{2 !}+\frac{5^{3} x^{3}}{3 !}-\cdots$

Question 22

Find an expression for the general term of the Taylor series for $x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}$ .

A) $\frac{(-1)^{n-1} x^{2 n-1}}{(2 n+1) !}$
B) $(-1)^{2 n-1} \frac{x^{n+1}}{n !}$
C) $(-1)^{n} \frac{x^{2 n+2}}{(2 n+1) !}$
D) $\frac{x^{2 n}}{2 n-1}$

Question 23

The infinite series $x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots+\frac{(-1)^{n-1} x^{n}}{n}+\ldots$ does not converge for $x=-1$ .What behavior does it exhibit? It does converge for $x=1$ .To what number does it appear to converge?

A)At -1, the series diverges to $\infty$ , At 1, it appears to converge to ln(3).
B)At -1 the series diverges because the terms alternate back and forth between positive and negative.At 1, it appears to converge to ln(2).
C)At -1, the series diverges because some of the terms are undefined.At 1, it appears to converge to ln(3).
D)At -1, the series diverges to $-\infty$ .At 1, it appears to converge to ln(2).

Question 24

Find the Taylor series centered at $x=0$ (i.e.the Maclaurin series)for $\sin \left(x^{2}\right)$ .

A) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+3}}{(4 i+1) !}$
B) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+2}}{(2 i+1) !}$
C) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i}}{(2 i) !}$
D) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+1}}{(4 i) !}$

Question 25

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

Question 26

The graph of the function $f(x)=e^{-x^{2} / 5}$ is a bell-shaped curve similar to a normal probability density function.Is $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{5^{i}(i !)}$ the Maclaurin series for $f(x)$ ?

Question 27

Use the binomial series to find the coefficient of the term in the expansion of .

Question 28

Based on the Maclaurin series for the function , evaluate .

Question 29

Find the first four non-zero terms of the Taylor series about zero for the function .Leave coefficients in fraction form.

Question 30

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

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 31

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 , 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 .By what percentage do your two approximations differ?

Question 32

What is the interval of convergence of the Taylor series for the function $f(x)=9(1+x)^{1 / 2}$ about zero? (Exclude any possible endpoints.)

A) $-1$$<$$x$$<1$
B) $-9<$$x<$$9$
C) $-\frac{1}{9}$$<$$x<$$\frac{1}{9}$
D)none of the above

Question 33

Use the formula for the Taylor polynomial approximation to the function about to construct a polynomial approximation of degree 6 for .Use the first four nonzero terms of this approximation to estimate the value of .Give your answer to 5 decimal places.

Question 34

Suppose that you are told that the Taylor series of about x = 0 is .Find .

Question 35

Find the number to which the series converges.Round to 5 decimal places.

Question 36

Solve for x.Round to 2 decimal places.

Question 37

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.

A) $m c^{2}\left[1+\frac{1}{2} \frac{v^{2}}{c^{2}}+\cdots\right]$
B) $m c^{2}\left[\frac{1}{2} \frac{v}{c}+\frac{3}{4} \frac{v^{2}}{c^{2}}+\cdots\right]$
C) $m c^{2}\left[1+\frac{1}{2} \frac{v}{c}+\cdots\right]$
D) $m c^{2}\left[\frac{1}{2} \frac{v^{2}}{c^{2}}+\frac{3}{8} \frac{v^{4}}{c^{4}}+\cdots\right]$

Question 38

Use the derivative of the Taylor series about 0 for $\frac{1}{1-x}$ to find the Taylor series about 0 for $\frac{4 x}{(1-x)^{2}}$ .

A) $4+4 x^{3}+4 x^{5}+4 x^{7}+\cdots$
B) $4+4 x^{2}+4 x^{3}+4 x^{4}+\cdots$
C) $4 x+8 x^{2}+12 x^{3}+16 x^{4}+\cdots$
D) $4+8 x+12 x^{2}+16 x^{3}+\cdots$

Question 39

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.

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 40

Use the derivative of the Taylor series about 0 for to find the Taylor series about 0 for .Use this result to find the value of .Round to 3 decimal places.

Question 41

Find the first harmonic of 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}$ .

A) $-\sin x$
B) $\sin x$
C) $-2 \sin x$
D) $2 \sin x$
E)0

Question 42

Find the second harmonic of the function $f(x)=$$\left\{\begin{array}{c}-2 \\2\end{array}\right.$ $\begin{array}{c}-\pi< \\0<\end{array}$$\begin{array}{l}x \leq 0 \\x \leq \pi\end{array}$ .

A) $-2+\frac{8}{\pi} \sin x+\frac{4}{3 \pi} \sin 3 x$
B) $2-\frac{8}{\pi} \sin x-\frac{4}{3 \pi} \sin 3 x$
C) $\frac{8}{3 \pi} \sin 3 x$
D) $\frac{4}{3 \pi} \sin 3 x$
E)0

Question 43

Given the fact that the Taylor series about x = 0 for $e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots$ , is the Taylor series about x = 0 for $e^{x / 4}=1+\frac{x}{4}+\frac{x^{2}}{32}+\frac{x^{3}}{384}+\cdots$ ?

Question 44

Find the first harmonic of the function $f(x)=$$\left\{\begin{array}{c}-2 \\2\end{array}\right.$ $\begin{array}{c}-\pi< \\0<\end{array}$$\begin{array}{l}x \leq 0 \\x \leq \pi\end{array}$
.

A) $-2+\frac{8}{\pi} \sin x$
B) $2-\frac{4}{\pi} \sin x$
C) $\frac{8}{\pi} \sin x$
D) $\frac{4}{\pi} \sin x$
E)0

Question 45

Find the 12th-degree Taylor polynomial for $x \sin \left(x^{2}\right)$ centered at x = 0.Suppose you use the first two non-zero terms of the polynomial to approximate $x \sin \left(x^{2}\right)$ for 0 < x < 1.Is your approximation too big or too small?

A)Too small
B)Too big

Question 46

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

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

Question 47

Since and are inverse functions, we know that for x > -1.Find the Taylor series for using only up to the quadratic terms and show that the result is 1 + x.

Question 48

Find the first four terms of the Taylor series about x = -2 for .

Question 49

Use the Maclaurin series for $f(x)=\sin (4 x)$ to find the Maclaurin series for $f(x)=\cos (4 x)$ .

A) $4 x-\frac{4^{3} x^{3}}{3 !}+\frac{4^{5} x^{5}}{5 !}-\cdots$
B) $1-\frac{4^{2} x^{2}}{2 !}+\frac{4^{4} x^{4}}{4 !}-\cdots$
C) $x-\frac{4^{2} x^{3}}{3 !}+\frac{4^{4} x^{5}}{5 !}-\cdots$
D) $4 x-\frac{4^{2} x^{2}}{2 !}+\frac{4^{3} x^{3}}{3 !}-\cdots$

Question 50

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

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

Question 51

Approximate $\int_{0}^{0.3} \sqrt{1+x} d x$ using the first three terms of the Taylor series about zero for $(1+x)^{1 / 2}$ .

A)0.321
B)0.336
C)0.3
D)1.5

Question 52

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]?

Question 53

Find $a_{0}$ for the function $f(x)=$$\left\{\begin{array}{c}-1 \\1\end{array}\right.$$\begin{array}{c}-\pi< \\0<\end{array}$$\begin{array}{l}x \leq 0 \\x \leq \pi\end{array}$
.

A)1
B)-1
C)1/2
D)-1/2
E)0

Question 54

Show that the Taylor series about 0 for converges to for all values of x by showing that the error .

Question 55

Find the second harmonic of the function $h(x)=$$\left\{\begin{array}{c}2 \pi \\0\end{array}\right.$$\begin{aligned}-\pi< \\0<\end{aligned}$$\begin{array}{l}x \leq 0 \\x \leq \pi\end{array}$ .

A) $\frac{4}{3} \sin 3 x$
B) $-\frac{4}{3} \sin 3 x$
C) $\frac{2}{3} \sin 3 x$
D) $-\frac{2}{3} \sin 3 x$
E)0

Question 56

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}$ .

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

Question 57

Find the 12th-degree Taylor polynomial for $x \sin \left(x^{2}\right)$ centered at x = 0.Suppose you use the first two non-zero terms of the series to approximate $x \sin \left(x^{2}\right)$ for 0 < x < 1.Is the magnitude of the error always less than 0.011?

Question 58

a) is the Maclaurin series for what function?
b)What is its radius and interval of convergence (excluding possible endpoints)?
c)Use the Maclaurin series to determine .

Question 59

The function h(x)is a continuous differentiable function whose graph is drawn below.The accompanying table provides some information about h(x)and its derivatives. $$\begin{array} { c c c c c } \boldsymbol { x } & \boldsymbol { h } ( \boldsymbol { x } ) & \boldsymbol { h } ^ { \prime } ( \boldsymbol { x } ) & \boldsymbol { h } ^{ { \prime }{ \prime }} ( \boldsymbol { x } ) & \boldsymbol { h } ^ {{ \prime }{ \prime } { \prime } } ( \boldsymbol { x } ) \\0 & 2 & 1 & 0.50 & 0.25 \\1 & 3.29 & 1.64 & 0.82 & 0.41 \\2 & 5.43 & 2.71 & 1.35 & 0.67 \\3 & 8.96 & 4.48 & 2.24 & 1.12\end{array}$$ h(x), h'(x), h"(x)and h"'(x)are all increasing functions.Suppose we use a tangent line approximation at zero to approximate h(0.1).Find a good upper bound for the error.

A)0.0025
B)0.0820
C)0.1066
D)0.1558

Question 60

It can be shown that the Maclaurin series for , and converge for all values of z in the complex numbers, just as they do for all values of x in the real numbers.
a)Write down and simplify the Maclaurin series for .
b)Write down the Maclaurin series for and c)Use the series you found in parts a)and b)to show that .(This is one of several formulas called "Euler's Formula.")
d)Find the value of .

Question 61

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.

Question 62

a)Find the Taylor series for using a series for .
b)Use the series from part a)to find the Taylor series for .

Question 63

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

Question 64

Medicine balls are launched from the floor to a height of six feet.They bounce, reaching x/10 the height of the previous bounce each time.The heavier the medicine ball, the smaller the value of x.Write a power series that gives the total distance that a medicine ball bounces as a function of x.What is the function that gives this Taylor polynomial?

Question 65

Find the second degree Taylor polynomial approximation of $\frac{1}{1+x^{2}}$ about x = 1.

A) $\frac{1}{2}-\frac{x-1}{2}+\frac{(x-1)^{2}}{4}$
B) $-\frac{1}{2}+\frac{x-1}{3}-\frac{(x-1)^{2}}{4}$
C) $\frac{1}{2}-\frac{x-1}{4}+\frac{(x-1)^{2}}{6}$
D) $-\frac{1}{2}+\frac{x-1}{3}-\frac{(x-1)^{2}}{4}$ .

Question 66

Find the Taylor series for $\sin x-\cos x$ .

A) $1+x-\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}-\cdots$
B) $1-x-\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}-\frac{x^{5}}{5 !}-\cdots$
C) $-1+x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}+\ldots$
D) $-1-x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}-\frac{x^{5}}{5 !}+\ldots$

Question 67

Which gives the better approximation of $4 e^{03}$ , the Taylor polynomial about zero with three terms, or the Fourier polynomial with three terms?

A)Taylor
B)Fourier
C)Both give the same approximation.

Question 68

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 $2 \pi$ .

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 69

Fill in the blanks: Fourier polynomials give good __________ approximations to a function.Taylor polynomials give good _____________ approximations to a function.

Question 70

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.

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

Question 71

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.

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) $R(x)=\frac{120 x^{5}-24 x^{3}+x}{30 x^{4}-60 x^{2}+120}$