Question #2096b

1 Answer
Mar 26, 2017

Three solutions are (to 10dp):

# x_1 = -0.3714177525 #
# x_2 = 0.6052671213 #
# x_3 = 4.7079379181 #

Explanation:

Let:

# f(x) = e^x-5x^2 #

Our aim is to solve #f(x)=0#. First let us look at the graph:
graph{e^x-5x^2 [-5, 10, -30, 10]}

We can see that there are three solutions; one solution in the interval #-1 lt x lt 0#, one in #0 lt x lt 1#, and one in #4 lt x lt 5#. The root we find will depend upon our initial approximation #x_0#, and this value will require a little trial and error.

To find the solution numerically, using Newton-Rhapson method we use the following iterative sequence

# { (x_1,=x_0), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #

Therefore we need the derivative:

# \ \ \ \ \ \ \f(x) = e^x-5x^2 #
# :. f'(x) = e^x-10x #

Then using excel working to 10dp we can tabulate the iterations as follows:

Initial Value #x_0=-1#
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Initial Value #x_0=1#
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Initial Value #x_0=5#
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We could equally use a modern scientific graphing calculator as most new calculators have an " Ans " button that allows the last calculated result to be used as the input of an iterated expression.

And we conclude that the three solutions are (to 10dp):

# x_1 = -0.3714177525 #
# x_2 = 0.6052671213 #
# x_3 = 4.7079379181 #