Final answer:
To approximate the positive root of the equation 3sinx=x^2 using Newton's method, define a function f(x)=3sinx-x^2 and use the iteration formula to find the improved estimates of x. Repeat the process until the desired level of accuracy is achieved.
Step-by-step explanation:
To approximate the positive root of the equation 3sinx=x^2, we can use Newton's method. First, let's define a function f(x)=3sinx-x^2. Then, we can use the iteration formula x2 = x1 - f(x1)/f'(x1), where x2 is the improved estimate, x1 is the previous estimate, f(x1) is the value of the function at x1, and f'(x1) is the derivative of the function at x1.
Let's start with an initial estimate, x1=2. Substitute x1 into the function f(x)=3sinx-x^2 to get f(x1). Then, calculate the value of the derivative f'(x) at x1. Finally, use the iteration formula to find x2. Repeat this process until you achieve the desired level of accuracy, in this case, six decimal places.
For example, let's take x1=2. We have f(x1)=3sin(2)-(2)^2=-1.625 and f'(x1)=3cos(2)-2x1=-5.0101. Using the iteration formula, x2=2-(-1.625)/(-5.0101) = 2.32432. Repeat this process with x2 as the new x1 until you reach the desired accuracy.