Solve the problem.
-Use Newton's method to find the value of $x$ for which $e ^ { - x } = x ^ { 2 } - 2 x + 5$. Find the answer correct to five decimal places.
A) -2.99294
B) -2.99107
C) -2.99263
D) -2.99288

Question

Quizplus · Accepted Answer

We begin by setting up Newton's method:

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$

where $f(x) = e^{-x} - x^2 + 2x - 5$, and $f'(x) = -e^{-x} - 2x + 2$.

Choosing an initial guess of $x_0 = -3$, we get

\begin{align*}
x_1 &= x_0 - \frac{f(x_0)}{f'(x_0)} \
&= -3 - \frac{e^3 + 6}{e^3 - 4} \
&\approx -2.9839
\end{align*}

Repeat this process until the difference between two successive approximations is less than $10^{-5}$:

\begin{align*}
x_2 & \approx -2.99287 \
x_3 & \approx -2.99288
\end{align*}

Therefore, the answer is D).

Solve the Problem xxx For Which e−x=x2−2x+5e ^ { - x } = x ^ { 2 } - 2 x + 5e−x=x2−2x+5

Solve the Problem $x$ For Which $e ^ { - x } = x ^ { 2 } - 2 x + 5$