# For what range of x is (e^x-1)/2x=0.5 correct to 15 decimal digits?

• ver_mathstats
In summary, to find the range of x for which (e^x-1)/2x=0.5 is correct to 15 decimal digits, we can rewrite the equation as 0.5 + x/4 + x^2/12 + ... and focus on making the x/4 term small enough to not affect the first 15 decimal digits. This means that the value of x must be specified with enough precision to make the x/4 term insignificant.
ver_mathstats
Homework Statement
For what range of x is (e^x-1)/2x=0.5 correct to 15 decimal digits?
Relevant Equations
(e^x-1)/2x=0.5
We have ex=1 + x + x2/2 + x3/3! + ...

ex - 1 = x + x2/2 + x3/3! + ...

(ex - 1)/(2x) = 0.5 + x/4 + x2/(2⋅3!) + ...

((ex - 1)/(2x)) - 0.5 = + x/4 + x2/(2⋅3!) + ...

After this, I am unsure of how to proceed to find my error any help would be appreciated thank you. Would we just be trying to isolate x, but that seems incorrect?

ver_mathstats said:
Homework Statement:: For what range of x is (e^x-1)/2x=0.5 correct to 15 decimal digits?
Relevant Equations:: (e^x-1)/2x=0.5

We have ex=1 + x + x2/2 + x3/3! + ...

ex - 1 = x + x2/2 + x3/3! + ...

(ex - 1)/(2x) = 0.5 + x/4 + x2/(2⋅3!) + ...

((ex - 1)/(2x)) - 0.5 = + x/4 + x2/(2⋅3!) + ...

After this, I am unsure of how to proceed to find my error any help would be appreciated thank you. Would we just be trying to isolate x, but that seems incorrect?
Yes, that's incorrect. You're not going to be able to solve for x in any of those equations.
It's better to write your last equation as ##\frac{e^x - 1}{2x} = 0.5 + \frac x 4 + \frac {x^2}{12} + \dots##. The dominant variable term on the right side is the ##\frac x 4## term. For small values of x, the ##x^2## and higher-degree terms will be relatively insignificant. If you can make ##\frac x 4## small enough, the contributions of the higher-degree terms shouldn't make any difference.

So how small should ##\frac x 4## be so that its contribution won't affect the first 15 decimal digits of your approximation?

Last edited:
How precisely do you have to specify ##x##?

SammyS

## 1. What is the equation in question?

The equation is (e^x-1)/2x=0.5.

## 2. What is the significance of 15 decimal digits?

Fifteen decimal digits is the standard for high precision calculations in scientific and mathematical fields.

## 3. How do you solve the equation?

The equation can be solved using algebraic manipulation and logarithmic functions.

## 4. Can the equation be graphed?

Yes, the equation can be graphed to show the range of x values that satisfy the equation.

## 5. How do you determine the range of x that satisfies the equation?

The range of x can be determined by finding the roots of the equation and using a calculator or computer program to find the values that satisfy the equation to 15 decimal digits.

• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
11
Views
1K
• Precalculus Mathematics Homework Help
Replies
8
Views
482
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
845
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Engineering and Comp Sci Homework Help
Replies
7
Views
1K
• Precalculus Mathematics Homework Help
Replies
4
Views
1K
• Precalculus Mathematics Homework Help
Replies
3
Views
749
• Calculus and Beyond Homework Help
Replies
32
Views
2K