Since I don't have graphing calculator.. how to find the answer of this?

Question

A graphing calculator is recommended.

Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by looking at a graph to find initial approximations. (Enter your answers as a comma-separated list.)

root(5-x^3) = e^x^2

Answer 1

I think you may find the site desmos.com useful here.

There you will find a graphing calculator tool, and you can plot both sides of the equation.

Type sqrt for inputting the square root. It will 'autocomplete' the sqrt to a root symbol template.

sqrt 5 - x^3 (graphed in red for me)

and

e^(x^2) (graphed in blue for me)

The intersections can be clicked on to see the solutions (-0.939, 2.414) and (0.859, 2.09), so the x-values are the approximate solutions: x = -0.939 and x = 0.859.

Let me know if I can explain this any further. Thank you!

Answer 2

go to desmos.com. Write the equation y=(5-x^3)^.5 - e^(x^2). You can actually just click on the zeroes of the function on the graph.

The assignment is to find a starting x₀ near enough to use Newton's Method to approximate the local zero:

x_n+1 = x_n - f(x_n)/f'(x_n) The y expression is f(x) f'(x) = .5(5-x³)^-.5 (-3x²) - 2xe^{x^2}

You can apply Newton's Method by hand to find x₀,x₁,... You need a reasonable x₀ for Newton's Method to converge to the root you want (You will have to do 3 NM calculations.) When it does converge, it converges quickly (x₂ will probably be accurate enough)