Question #90cf3

2 Answers
Sep 26, 2016

To find the roots of equations like e^x = x^3, I recommend that you use a recursive numerical analysis method, called Newton's Method

Explanation:

Let's do an example.

To use Newton's method, you write the equation in the form f(x) = 0:

e^x - x^3 = 0

Compute f'(x):

e^x - 3x^2

Because the method requires that we do the same computation many times, until it converges, I recommend that you use an Excel spreadsheet; the rest of my answer will contain instructions on how to do this.

Enter a good guess for x into cell A1. For this equation, I will enter 2.

Enter the following into cell A2:

=A1-(EXP(A1) - A1^3)/(EXP(A1) - 3*A1^2)

Please notice that the above is Excel spreadsheet language for

x_2 = x_1 - (e^(x_1)-x_1^3)/(e^(x_1)-3x_1^2)

Copy the contents of cell A2 into A3 through A10. After only 3 or 4 recursions, you can see that the method has converged on

x = 1.857184

Sep 26, 2016

We can use the Intermediate Value Theorem to see that each pair has at least one point of intersection.

Explanation:

f(x) = e^x-x^2 is continuous on the entire real line.

At x=0, we have f(0)=1.

At x=-1, we have f(-1) = 1/e-1 which is negative.

f is continuous on [-1,0], so there is at least one c in (-1,0) with f(c)=0.

g(x)=e^x-x^3 is continuous on the entire real line.

At x=0, we have g(0)=1.

At x=2, we have g(2) = e^2-8 which is negative.
(Note that e^2 ~~ 2.7^2 < 7.3 < 8.)

g is continuous on [0,2], so there is at least one c in (0,2) with g(c)=0.