Count iblis helped me solve this in another thread. But I am looking for an easier method. Can anyone help?
What I have is
e^x - 2x = e - 2
Solve for x. (The answer is 0.351)
Thanks.
That is a transcendental equation that has to be solved by iteration. A better approximation to the answer is 0.35139.
I got that by using a hand calculator, trying various numbers between 0.351 and 0.352, and interpolating until e^2 - 2x was equal to e - 2 = .71828.
0.35136 is a better answer, to 5 significant figures
5^(2x+1)=X^x
To solve the given equation, e^x - 2x = e - 2, we need to use numerical methods since it is a transcendental equation. While there are various methods available, let's discuss a commonly used technique called the Newton-Raphson method.
The first step is to rewrite the equation in the form f(x) = 0. In this case, we have f(x) = e^x - 2x - e + 2. Now, we can use the Newton-Raphson method to find the root of this function.
The method involves an iterative process that starts with an initial guess and refines it until a satisfactory solution is obtained. Here are the steps involved:
1. Choose an initial guess for the root value, let's say x_0 = 0.5.
2. Calculate the function value and its derivative at this point: f(x_0) and f'(x_0).
In this case, f(x) = e^x - 2x - e + 2, so f'(x) = e^x - 2.
3. Use the formula: x_1 = x_0 - f(x_0) / f'(x_0).
Calculate the new value of x, x_1, using the initial guess and the function values.
4. Repeat steps 2 and 3 until the desired level of accuracy is achieved.
Update x_i using the formula: x_i+1 = x_i - f(x_i) / f'(x_i).
By iteratively applying the Newton-Raphson method, we can approach the solution to the equation. In this case, the desired solution is x = 0.351. The method will converge towards this value as we perform more iterations.
Although it is possible to perform these calculations manually, it is much more convenient to utilize numerical tools such as calculators, spreadsheet software, or programming languages to efficiently solve these equations.