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

Accepted Answer

B

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

Accepted Answer

D

Question 3

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

1.6094

Question 4

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 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) $10+25 x-\frac{35}{2} x^{2}+6 x^{3}$

Accepted Answer

The third degree Taylor polynomial for a function $f$ about $x = 5$ is given by $f(5) + f'(5)(x-5) + \frac{f''(5)}{2}(x-5)^2 + \frac{f'''(5)}{6}(x-5)^3$. Substituting the given values, we get $2 + 5(x-5) - \frac{7}{2}(x-5)^2 + 2(x-5)^3$.

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) $4+4 x+\frac{4 x^{2}}{2}+\frac{4 x^{3}}{3 !}+\frac{4 x^{4}}{4 !}$

Accepted Answer

The Taylor polynomial of $e^x$ near 0 up to degree 4 is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}$. For $e^{-x}$, it is $1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}$. Adding these and dividing by 2 gives $\cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!}$. Multiplying by 4 gives $4\left(1 + \frac{x^2}{2!} + \frac{x^4}{4!}\right)$, which simplifies to $4\left(1 + \frac{x^2}{2} + \frac{x^4}{24}\right)$.

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

Accepted Answer

The answer of Find the fourth term of the Taylor...

Question 8

Find $\lim _{x \rightarrow 0} \frac{\sin x}{4 x}$ using a Taylor approximation for sin x.

Accepted Answer

The answer of Find \(\lim _{x \rightarrow 0} \frac{\sin x}{4...

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, $a+b(x-17)+c(x-17)^{2}$ .Is b positive, negative, or zero?

Accepted Answer

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

Question 10

Find the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ and use it to approximate $\sqrt{0.7}$ .Give your answer to 4 decimal places.

Accepted Answer

The answer of Find the Taylor polynomial of degree 3...

Question 11

Approximate the function $f(x)=x^{3} e^{-x^{2}}$ for values of x near 0 using the first three non-zero terms of its Taylor polynomial.

Accepted Answer

The answer of Approximate the function $f(x)=x^{3} e^{-x^{2}}$ for values...

Question 12

Is the Taylor polynomial of degree 6 for $e^{-x^{2}}$ for x near 0 given by $1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}$ ?

Accepted Answer

The answer of Is the Taylor polynomial of degree 6...

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 $\sqrt{0.2}$

Accepted Answer

The error in the Taylor approximation is $R_3(x) = \frac{f^{(4)}(c)}{4!} x^4$, where $c$ is some number between 0 and $x$. Since $f^{(4)}(x) = \frac{1}{8} (1-x)^{-5/2}$, the error is largest when $x$ is close to 1. Therefore, the approximation of $\sqrt{0.9}$ is more accurate than 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) $x+\frac{3}{2} x^{2}-\frac{9}{6} x^{3}+\frac{27}{24} x^{4}$

Accepted Answer

The fourth degree Taylor polynomial for a function $f(x)$ about $x = 0$ includes terms up to $x^4$. For $\cos(3x^2)$, the Taylor series expansion is $1 - \frac{(3x^2)^2}{2!} + \cdots$, which simplifies to $1 - \frac{9x^4}{2}$ for the terms up to $x^4$, matching option A.

Question 15

Solve $1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots=e^{4}$ for x.

Accepted Answer

The answer of Solve $1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots=e^{4}$ for x....

Question 16

Construct the Taylor polynomial approximation of degree 3 to the function $f(x)=\arctan x$ about the point x = 0.Use it to approximate the value $f(0.35)$ to 5 decimal places.How does the approximation compare to the actual value?

Accepted Answer

The answer of Construct the Taylor polynomial approximation of degree...

Question 17

Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ 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 $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a...

Question 18

The function g has the Taylor approximation $g(x)=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}$ and the graph given below: Is c₀ positive, negative, or zero?

Accepted Answer

The answer of The function g has the Taylor approximation...

Question 19

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.

Accepted Answer

The answer of Estimate $\int_{1}^{1.5} \ln t d t$ using...

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 $k \geq 0$

Accepted Answer

The general term of the series is $\frac{(-1)^{k} x^{5 k}}{(5 k) !}$ for $k \geq 0$, which correctly represents the pattern of alternating signs, powers of $x$ increasing by multiples of 5, and factorial denominators also increasing by multiples of 5.

Quiz 10: Approximating Functions Using Series

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.

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

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?

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 $f(x) =4 \cosh (x)$ .

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

Find $\lim _{x \rightarrow 0} \frac{\sin x}{4 x}$ using a Taylor approximation for sin x.

Find the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ and use it to approximate $\sqrt{0.7}$ .Give your answer to 4 decimal places.

Approximate the function $f(x)=x^{3} e^{-x^{2}}$ for values of x near 0 using the first three non-zero terms of its Taylor polynomial.

Is the Taylor polynomial of degree 6 for $e^{-x^{2}}$ for x near 0 given by $1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}$ ?

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?

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

Solve $1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots=e^{4}$ for x.

Construct the Taylor polynomial approximation of degree 3 to the function $f(x)=\arctan x$ about the point x = 0.Use it to approximate the value $f(0.35)$ to 5 decimal places.How does the approximation compare to the actual value?

Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

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.

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

Quiz 10: Approximating Functions Using Series

Use the first three nonzero terms of the Taylor polynomial to approximate ∫0.11.1sin⁡xxdx\int_{0.1}^{1.1} \frac{\sin x}{x} d x∫0.11.1​xsinx​dx .Give your answer to 5 decimal places.

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

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?

The hyperbolic cosine function is defined as follows: f(x) =cosh⁡(x) =ex+e−x2f(x) =\cosh (x) =\frac{e^{x}+e^{-x}}{2}f(x) =cosh(x) =2ex+e−x​ .Use the Taylor polynomial for exe^{x}ex near 0 to find the Taylor polynomial of degree 4 for f(x) =4cosh⁡(x) f(x) =4 \cosh (x) f(x) =4cosh(x) .

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

Find lim⁡x→0sin⁡x4x\lim _{x \rightarrow 0} \frac{\sin x}{4 x}limx→0​4xsinx​ using a Taylor approximation for sin x.

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 Taylor polynomial of degree 3 around x = 0 for the function f(x)=1−xf(x)=\sqrt{1-x}f(x)=1−x​ and use it to approximate 0.7\sqrt{0.7}0.7​ .Give your answer to 4 decimal places.

Approximate the function f(x)=x3e−x2f(x)=x^{3} e^{-x^{2}}f(x)=x3e−x2 for values of x near 0 using the first three non-zero terms of its Taylor polynomial.

Is the Taylor polynomial of degree 6 for e−x2e^{-x^{2}}e−x2 for x near 0 given by 1+x22!+x44!+x66!1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}1+2!x2​+4!x4​+6!x6​ ?

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

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

Solve 1+x+x22!+x33!+⋯=e41+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots=e^{4}1+x+2!x2​+3!x3​+⋯=e4 for x.

Construct the Taylor polynomial approximation of degree 3 to the function f(x)=arctan⁡xf(x)=\arctan xf(x)=arctanx about the point x = 0.Use it to approximate the value f(0.35)f(0.35)f(0.35) to 5 decimal places.How does the approximation compare to the actual value?

Recognize 5−533!+555!−577!+⋯5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots5−3!53​+5!55​−7!57​+⋯ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

The function g has the Taylor approximation g(x)=c0+c1(x−a)+c2(x−a)2g(x)=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}g(x)=c0​+c1​(x−a)+c2​(x−a)2 and the graph given below: Is c0 positive, negative, or zero?

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.

What is the general term of the series 1−x55!+x1010!−x1515!+⋯1-\frac{x^{5}}{5 !}+\frac{x^{10}}{10 !}-\frac{x^{15}}{15 !}+\cdots1−5!x5​+10!x10​−15!x15​+⋯ ?

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.

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

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?

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 $f(x) =4 \cosh (x)$ .

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

Find $\lim _{x \rightarrow 0} \frac{\sin x}{4 x}$ using a Taylor approximation for sin x.

Find the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ and use it to approximate $\sqrt{0.7}$ .Give your answer to 4 decimal places.

Approximate the function $f(x)=x^{3} e^{-x^{2}}$ for values of x near 0 using the first three non-zero terms of its Taylor polynomial.

Is the Taylor polynomial of degree 6 for $e^{-x^{2}}$ for x near 0 given by $1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}$ ?

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?

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

Solve $1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots=e^{4}$ for x.

Construct the Taylor polynomial approximation of degree 3 to the function $f(x)=\arctan x$ about the point x = 0.Use it to approximate the value $f(0.35)$ to 5 decimal places.How does the approximation compare to the actual value?

Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

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.

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