Use Newton's method to approximate a root of the equation 3sin(x)=x as follows.
Let x1=1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
for x2 I got -1.454 which is right but I can't get x3
See:
http://www.jiskha.com/display.cgi?id=1283924839
To use Newton's method to approximate a root of the equation 3sin(x) = x, we follow these steps:
Step 1: Choose an initial approximation, denoted as x1.
Given that x1 = 1, we can move on to the next step.
Step 2: Determine the equation for the tangent line to the graph of the equation at the point (x1, f(x1)).
The equation for the tangent line can be found using the derivative of the equation, f'(x).
Let's differentiate both sides of the equation 3sin(x) = x with respect to x:
3cos(x) = 1
Step 3: Find the equation of the tangent line.
Using the point-slope form of the line, we have:
y - f(x1) = f'(x1)(x - x1)
Substituting in the values, we get:
y - (3sin(x1)) = (3cos(x1))(x - x1)
Step 4: Solve the equation for the value where the tangent line intersects the x-axis.
To find this intersection point, we set y = 0 and solve for x. This will give us the next approximation, denoted as x2.
0 - 3sin(x1) = (3cos(x1))(x2 - x1)
-3sin(1) = (3cos(1))(x2 - 1)
x2 - 1 = -3sin(1) / (3cos(1))
x2 = -3sin(1) / (3cos(1)) + 1 β -1.454
Now, to find the third approximation x3, we follow the same steps:
Step 5: Find the equation of the tangent line at the point (x2, f(x2)).
Differentiating the equation 3sin(x) = x and substituting x with x2, we get:
3cos(x2) = 1
Step 6: Determine the equation of the tangent line.
Using the formula for the tangent line, we have:
y - f(x2) = f'(x2)(x - x2)
Substituting the values, we get:
y - (3sin(x2)) = (3cos(x2))(x - x2)
Step 7: Solve the equation to find the value where the tangent line intersects the x-axis.
Setting y = 0, we can solve for x to obtain x3:
0 - 3sin(x2) = (3cos(x2))(x3 - x2)
Here lies the challenge - solving this equation algebraically is non-trivial, and it may not have a simple analytical solution. Therefore, we need to use numerical methods or a calculator to find the value of x3.
One way to solve this equation numerically is by using iteration:
- Take an initial guess for x3, denoted as x3_0.
- Substitute this guess into the equation: 0 - 3sin(x2) = (3cos(x2))(x3_0 - x2).
- Solve for x3_1.
- Repeat these steps by taking x3_1 as the new guess, until you reach a desired level of accuracy or convergence.
Alternatively, you can use a graphing calculator or software to graph the equation and visually find the intersection point of the tangent line and the x-axis to approximate x3.