{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 = 4√x and y = x² + 1

y = 4√x and y = x² + 1

Accepted Answer

Using Newton's method, find the approximate intersection points of Y equals 2 square roots of 3 x + 1 and Y equals 3 X2. I have provided the graph on the left side here. And we noticed that there are 2 intersection points. One of them seems to be between. X equals 1 and X equals 2. And the other one seems to be between X equals -1, and X equals 0. So let's try and find these points. The first set are equations equal to each other. We have 2 square roots. 3 X plus 1, S equals +23 X squared. Now, let's simplify this. If you square both sides first. We get 4 multiplied by 3 X + 1. Equals 9 to 4th. Giving us 12 x plus 4 equals 9 X to the 4th. If we get everything on one side of the equation, we get F of X equals 9 of the 4th. -12 x minus 4. So now we need the formula for Newton's method. This tells us that if we have X N + 1, This equals to X of N. Minus F of Xin divided by F of Xin. We can then find F of our equation. Which is 36 X cubs. 12. We can now make our Newton's method formula. XN + 1 is equals to 9 x 4th. -12 X minus 4. Divided by 36 X cubes minus 12. We also have the X subn in front of that fraction to XN + 1 equals XN minus. 9 XN to the 4th, minus 12 XN minus 4, divided by 36 XN cubed minus 12. So, now we know there's a point between -1 and 0. Let's make an initial guess. Of X 0 equals -0.5, which is right in the middle of -1 and 0. This will give us X1 equals negative. 0.5 -9 multiplied by -0.5. Race to the 4th -12, multiplied by -0.5. -4 All divided By 36 multiplied by -0.5. Race the 3rd, minus 12. If we simplify that, we get approximately 0.34470. Now, for X2, we will take the value we got from X1. And plug it into the equation. So we'll take our negative 0.34470. And plug it in for all of our X sub in. So we have 9 multiplied by -0.344. 70 To the 4th, minus 12, multiplied by -0.34470. Minus 4 Divided by 36, multiplied by negative 0.34470. To the 3rd minus 12. This is approximately -0.32515. Now X3 will use the previous value. Which is approximately 0. 32497 And if we go even further to X4, we end up getting about the same value. Now because we have duplicate values here, we can then confirm. That the first X value will be negative 0.32497. We can now plug in why. Why? We notice is the equation 3X2. So we can say 3, multiplied by negative 0.32497 squared. What is approximately 0.31681. We can now repeat this with our other point. So disappoint Between 1 and 2. Let's guess a value. Let's say X0 is equals to 1.2. We will repeat the same process as before. Or X1 is equals. To 1.2 -9, multiplied by 1.2, rates to the 4th. -12, multiplied by 1.2, minus 4. Divided by 36, multiplied. By 1.2, raises the third minus 12. Which is approximately 1.19477. If we repeat, X2. Will give us 1.19473, and X3 will also give us 1.19473. We can then plug this value in for Y. Which is 3, multiplied by 1.19473 squared. Which is approximately 4.28215. We now have both points for our answer. Our first point Will be -0.32497. 0.31681. And our second point. Will be 1.19473. 4.28215 OK, these are the two answers to our problem. I hope to help you solve the problem. Thank you for watching. Goodbye.

{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 = 4√x and y = x² + 1

Key Concepts

Newton's Method

Intersection Points

Graphing and Analysis

Watch next

{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 = 4√x and y = x² + 1

Key Concepts

Newton's Method

Intersection Points

Graphing and Analysis

Watch next

{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 = 4√x and y = x² + 1