Using Newton's Method, determine the first approximation to the solution of:
e^(4x)= 10-2x
starting with x=0
what is x1= ??? P.S. ( the 1 is a subscript )
Let f(x)=e^(4x)-10+2x
We look for the solution(s) of f(x)=0.
Newton's method is essentially
xn+1=xn-f(xn)/f'(xn)
for
f'(x)=4e^(4x)+2
x0=0
x1=x0-f(x0)/f'(x0)
=0 - (1-10+0)/(4*1+2)
=1.5
x2=x1-f(x1)/f'(x1)
=1.25
x3=x2-f(x2)/f'(x2)
=1.018
x4=0.8036
...
x10=0.54669824240377
x11=0.54669824240377
To find the first approximation, x1, using Newton's Method, you will need to iterate the following steps:
1. Start with an initial value of x. In this case, x0 = 0 as given.
2. Calculate the value of the function f(x) = e^(4x) - (10 - 2x) at x = x0.
3. Find the derivative of the function f(x) with respect to x, denoted as f'(x).
4. Use the formula x1 = x0 - f(x0)/f'(x0) to calculate the first approximation value, x1.
5. Repeat steps 2-4 until you reach the desired level of precision or accuracy.
Let's go step by step:
Step 1: Start with x0 = 0.
Step 2: Calculate f(x0):
f(x0) = e^(4*0) - (10 - 2*0)
= e^0 - (10 - 0)
= 1 - 10
= -9
Step 3: Find f'(x):
f'(x) = d/dx(e^(4x)) - d/dx(10 - 2x)
= 4e^(4x) - (-2)
= 4e^(4x) + 2
Step 4: Use the Newton's Method formula to find x1:
x1 = x0 - f(x0)/f'(x0)
= 0 - (-9)/(4e^(4*0) + 2)
= 9/(4e^0 + 2)
= 9/(4 + 2)
= 9/6
= 1.5
Therefore, the first approximation to the solution is x1 = 1.5.