To solve the problem, we need to correctly interpret the fifth Maclaurin polynomial for $e^{2x}$ given in a compact form and then approximate $e^{-0.98}$ using $\frac{1}{P(0.49)}$. Finally, we will calculate the absolute and relative errors.

Fifth Maclaurin Polynomial for $e^{2x}$

Given the polynomial: $P(x) = (((x+3)x + )x + 2)x + 2)x + 1$

It seems the given expression is incorrect or typographically unclear. Let's start with the correct Maclaurin series expansion for $e^{2x}$:

$e^{2x} = \sum_{n=0}^{\infty} \frac{(2x)^n}{n!} = 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \frac{(2x)^4}{4!} + \frac{(2x)^5}{5!} + \cdots$

The fifth Maclaurin polynomial is: $P_5(x) = 1 + 2x + 2x^2 + \frac{8x^3}{6} + \frac{16x^4}{24} + \frac{32x^5}{120}$ $P_5(x) = 1 + 2x + 2x^2 + \frac{4x^3}{3} + \frac{2x^4}{3} + \frac{4x^5}{15}$

Calculation of $P(0.49)$

$P_5(0.49) = 1 + 2(0.49) + 2(0.49)^2 + \frac{4(0.49)^3}{3} + \frac{2(0.49)^4}{3} + \frac{4(0.49)^5}{15}$

Calculating each term: $1$ $+ 2(0.49) = 0.98$ $+ 2(0.49)^2 = 2(0.2401) = 0.4802$ $+ \frac{4(0.49)^3}{3} = \frac{4(0.117649)}{3} = \frac{0.470596}{3} = 0.156865$ $+ \frac{2(0.49)^4}{3} = \frac{2(0.05764801)}{3} = \frac{0.11529602}{3} = 0.038432$ $+ \frac{4(0.49)^5}{15} = \frac{4(0.028236441)}{15} = \frac{0.112945764}{15} = 0.007530$

Summing these terms: $P_5(0.49) = 1 + 0.98 + 0.4802 + 0.156865 + 0.038432 + 0.007530$ $P_5(0.49) \approx 2.6630$

Approximation Using $\frac{1}{P(0.49)}$

Using four-digit rounding arithmetic: $\frac{1}{P(0.49)} = \frac{1}{2.6630} \approx 0.3755$

Exact Value and Errors

The exact value of $e^{-0.98}$ is: $e^{-0.98} \approx 0.3753$

Absolute Error

$\text{Absolute Error} = | \text{Exact Value} - \text{Approximate Value} | = | 0.3753 - 0.3755 | = 0.0002$

Relative Error

$\text{Relative Error} = \frac{\text{Absolute Error}}{\text{Exact Value}} = \frac{0.0002}{0.3753} \approx 0.0005$

Summary

• Approximate Value using $\frac{1}{P(0.49)}$: $0.3755$
• Exact Value of $e^{-0.98}$: $0.3753$
• Absolute Error: $0.0002$
• Relative Error: $0.0005$

Would you like more details or have any questions?

Related Questions

1. What is the general form of the Maclaurin series for $e^x$?
2. How does the Taylor series differ from the Maclaurin series?
3. What are some applications of Maclaurin series in real-world problems?
4. How can we improve the accuracy of our polynomial approximations?
5. What are some limitations of using Maclaurin series for approximations?
6. How do we determine the radius of convergence for a power series?
7. What is the significance of the fifth-degree polynomial in approximations?
8. How can we extend these concepts to multivariable functions?

Tip

Always use the exact value and keep intermediate results in higher precision to minimize rounding errors.