ivybeibeidn

2022-09-06

Having trouble finding the solution to ${e}^{x}-3{e}^{-x}-4x=0$. The answer is roughly $2.2$ but am not sure how to get there?

Quinn Alvarez

Beginner2022-09-07Added 13 answers

Let $f(x)={e}^{x}+3{e}^{-x}-4x$

Find $a,b$ such that $a<b$ and either $f(a)<0<f(b)$ or $f(a)>0>f(b)$ (i.e. find two points such that the function changes sign between them, meaning there is a solution somewhere in that interval - this only works because $f$ is a continuous function).

Take $c=\frac{1}{2}(a+b)$. If $f(c)=0$ then 𝑐 is the desired zero, and you can stop. Or, if $f(c)$ is smaller than the error you're willing to accept, then it's your approximation and you can stop.

Find the sign of $f(c)$. If $f(a)$ and $f(c)$ have the same sign, set $a:=c$, otherwise set $b:=c$ (i.e. pick a new $a$ and $b$ such that you still know that the zero is between them).

Go back to step $2$ with your new $a$ and $b$.

Find $a,b$ such that $a<b$ and either $f(a)<0<f(b)$ or $f(a)>0>f(b)$ (i.e. find two points such that the function changes sign between them, meaning there is a solution somewhere in that interval - this only works because $f$ is a continuous function).

Take $c=\frac{1}{2}(a+b)$. If $f(c)=0$ then 𝑐 is the desired zero, and you can stop. Or, if $f(c)$ is smaller than the error you're willing to accept, then it's your approximation and you can stop.

Find the sign of $f(c)$. If $f(a)$ and $f(c)$ have the same sign, set $a:=c$, otherwise set $b:=c$ (i.e. pick a new $a$ and $b$ such that you still know that the zero is between them).

Go back to step $2$ with your new $a$ and $b$.

Read carefully and choose only one option

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