Use Newton's Method to solve the equation? #lnx+e^x=0#
We have:
# f(x) = lnx+e^x #
Our aim is to solve
graph{ln(x)+e^x [-5, 5, -10, 10]}
We can see that there is one solution in the interval
To find the solution numerically, using Newton-Rhapson method we use the following iterative sequence
# { (x_1,=1), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #
Therefore we need the derivative:
# \ \ \ \ \ \ \f(x) = lnx+e^x #
# :. f'(x) = 1/x+e^x #
Then using excel working to 10dp we can tabulate the iterations as follows:
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 we have very rapid convergence, and the solution is
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See below.
This gives rise to an iterative approximation procedure
In this case we have
or
#((x_k, f(x_k)),(0.5, 0.955574),(0.238107, -0.166189),(0.268497, -0.00692001),(0.269872, -0.0000118326),(0.269874, -3.4569*10^-11),(0.269874, 0.))#
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To solve the equation ( \ln(x) + e^x = 0 ) using Newton's Method, you would need to start with an initial guess ( x_0 ), then use the iterative formula:
[ x_{n+1} = x_n - \frac{\ln(x_n) + e^{x_n}}{\frac{1}{x_n} + e^{x_n}} ]
Continue iterating until the solution converges to the desired level of accuracy.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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