Find the points of intersection of the curves y=2sin(x-3) and y=-4x^2 + 2
tough question!
I used Newton's Method
and I let
y = 2sin(x-3) + 4x^2 - 2
then dy/dx = 2cos(x-3) + 8x
xnew = x - (2sin(x-3) + 4x^2 - 2)/(2cos(x-3) + 8x)
I started with x = 1
and my first xnew was.974691
make that your current x and find the next xnew
I quickly converged to
x = 0.974237
set your calculator to radians and test my answer, it works making both equations equal.
(error was .000000418 between LS and RS
Reiny found a positive root but there is a negative root x=-0.55 (approximately)
We can started with x=-1
To find the points of intersection between two curves, we need to set their equations equal to each other and solve for the values of x that make the equations equal.
Thus, we can set the two equations equal to each other:
2sin(x-3) = -4x^2 + 2
Let's solve for x:
1. Start by moving all the terms to one side of the equation to create a polynomial equation:
-4x^2 + 2 - 2sin(x-3) = 0
2. Since it's a polynomial equation, we can't solve it algebraically. We'll need to use numerical methods, such as graphing or approximation techniques.
One way to find the points of intersection graphically:
3. Graph both equations on the same coordinate system.
4. Observe the points where the two curves intersect. These are the x-coordinates of the points of intersection.
To find the points of intersection using an approximation method like Newton-Raphson:
3. Rewrite the equation in the form: f(x) = -4x^2 + 2 - 2sin(x-3) = 0.
4. Choose an initial estimate for one of the intersection points, let's say x1.
5. Apply the Newton-Raphson method to find an approximation for the root, x2.
6. Repeat steps 4 and 5 until the approximations converge on the actual roots.
7. Use the x-values obtained from the previous steps and substitute them into one of the original equations to find the corresponding y-values.
Therefore, finding the exact points of intersection requires either graphing the two equations and visually identifying the intersection points or using numerical methods to approximate the values of x and corresponding y-values at the points of intersection.