Estimating Maximum Error Using Taylor Series Approximation of e^x

Math Problem Statement

Homework:HW SECTION 11.1 Question 8, 11.1.55 HW Score: 72.22%, 6.5 of 9 points Points: 0 of 1

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question content area top Part 1 Use the remainder term to estimate the maximum error in the following approximation on the given interval. e Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction ; left bracket negative one tenth comma one tenth right bracket Question content area bottom Part 1 Select the correct choice below and fill in the answer box to complete your choice. (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to two decimal places as needed.) A. The maximum error is approximately enter your response here for Mequals2. B. The maximum error is approximately enter your response here for Mequalsone tenth . C. The maximum error is approximately enter your response here for Mequals1. D. The maximum error is approximately enter your response here for MequalsStartFraction 1 Over e EndFraction . e Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative seven eighths comma seven eighths right brackete Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative seven eighths comma seven eighths right brackete Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative seven eighths comma seven eighths right brackete Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative seven eighths comma seven eighths right bracket)e Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative one tenth comma one tenth right brackete Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative one tenth comma one tenth right brackete Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative one tenth comma one tenth right brackete Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction; left bracket negative one tenth comma one tenth right bracket) Get more help

Solution

To estimate the maximum error in the given approximation of $e^x$ on the interval $[ -\frac{1}{10}, \frac{1}{10} ]$ , we need to apply the Lagrange remainder term for the Taylor series approximation.

The given approximation is:

$e^x \approx 1 + x + \frac{x^2}{2}$

This is the second-degree Taylor polynomial for $e^x$ around $x = 0$ . The remainder (error) after the second-degree term is given by:

$R_2(x) = \frac{f^{(3)}(c)}{3!} x^3$

Where:

$f^{(3)}(x) = e^x$ is the third derivative of $e^x$ ,
$c$ is some number in the interval $[-0.1, 0.1]$ ,
$x^3$ comes from the third-degree term.

Since $e^x$ is increasing, the maximum value of $e^c$ for $c \in [-0.1, 0.1]$ is approximately $e^{0.1} \approx 1.105$ .

Thus, the maximum error is:

$R_2(x) = \frac{1.105}{6} \cdot (0.1)^3 = \frac{1.105}{6} \cdot 0.001 = 0.000184$

Rounding to scientific notation and two decimal places:

$R_2(x) \approx 1.84 \times 10^{-4}$

So, the maximum error is approximately $1.84 \times 10^{-4}$ .

Let me know if you need more details or have any questions!

Here are 5 related questions to expand your understanding:

How is the Taylor series of $e^x$ derived?
Why does the remainder term for Taylor series represent the error in approximation?
How does the interval of approximation affect the maximum error in a Taylor series?
Can you generalize the error estimation for higher-degree Taylor polynomials of $e^x$ ?
What is the relationship between the size of the interval and the accuracy of the Taylor approximation?

Tip: Always check how the remainder term changes with the degree of the polynomial. The more terms you include, the smaller the error in approximation!

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Math Problem Analysis

Mathematical Concepts

Taylor Series
Error Estimation
Exponential Function

Formulas

e^x ≈ 1 + x + x^2/2
R_2(x) = f^(3)(c)/3! * x^3

Theorems

Taylor's Theorem
Lagrange Remainder Term

Suitable Grade Level

College Level (Calculus)

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