{Use of Tech} Finding intersection points Use Newton’s method ...

Question

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.

y = sin x and y = x/2

y = sin x and y = x/2

Accepted Answer

Find the approximate intersection points of Y equals cosine X and Y equals 2 X + 3, using Newton's method. Now, on the left here, I provide a graph that shows the two graphs intersecting each other. To solve this Let's first find a possible range where this intersection could be. We look at our graph. We see the intersection occurs. Between X equals -2, and X equals -1. So, we can make an approximation for our first point to be -1.5. Next, Let's form our function for F. So in Newton's method, The formula states that XN plus 1. S equals to XN plus F of XN divided by F of XN. So, we need our function F of X for this first. Since we're intersecting, let's set cosine X equals 2 x + 3. Now, I'll move everything to one side to get F of X equals cosine X minus 2X minus 3. Now, since we're saying X knot is -1.5, We can find X1. Which is X not or X0. Plus F of X0 divided by F of X0. Now F with the derivative of F, which is just negatives of X minus 2. Let's go ahead and plug in our values now. X1 is equals to -1.5. Plus cosine of -1.5. -2, multiplied by negative 1.5, -3. Divided by negative sign of -1.5 minus 2. If we simplify this, we get the value of -1.42944. That will be our value for X1. Let's do the same for our next iteration X2. X2 will be X1 plus F of X1, divided by F of X1. So we'll do the same thing as last time and plug in our most recent value. -1.42. 944 Plus cosine of -1.42944, minus 2 multiplied by -1.42944 minus 3. Divid By native son of native. 1.42944. -2 This is approximately 1.42967. Now, we can do X3 by repeating the process, X2 plus F of X2, divided by F X2. And if we replug this in again, we end up getting approximately 1.42967. Now, we noticed X2 and X3 are pretty much the same value. We can then confirm that our X value perroximation will be -1.42. 967 The y then we will plug in our x value into our equation. And in this case, We can just plug it into our linear equation. 2 multiplied by negative 1.42967. Plus 3. Which gives us the value. 0.14066. We can then confirm our intersection point. Will be 1.42967. 0.14066. And this should be our answer. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.