{Use of Tech} Finding all roots Use Newton’s method to find al...

Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = e⁻ˣ - ((x + 4)/5)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Use Newton's method to find all real roots of F of X is equal to the power of X minus 2 x 2 by first performing a preliminary analysis. or graphing if desired to choose good starting approximations. Stop iterating when two successive approximations agree to 5 decimal places. Fantastic. So it appears for this particular prompt we're asked to use Newton's method to help us find all real roots of this function of F of X, and we're asked to do so by performing a preliminary analysis or use a graphing graphing method. In order to help us choose good starting approximations, and we're also then asked to stop iterating when we have found two successive approximations that are equal to each other or agree with each other when rounded to 5 decimal places. Fantastic. So essentially we're trying to figure out all the real roots of this function of F of X. So with that said, let us take a moment here to recall a note that our function that we are examining is going to be F of X is equal to E of X minus 2 x 2. So, first off, Using our function, let's take it and plug it into some graphing software, either into our graphing calculator or some online graphing software that's easy to use, that we understand how to use. And when we do that, we will produce the following graph, which appears to be A it's like parabolic looking, but then it curves upwards. So almost like it's almost like there's two parabolic curves connected. So it's almost like a sideway, or actually more like a flipped on its side, like if you pushed an S over, it's like S shape, so an elongated S shape. That's been pushed sideways. Fantastic. So the curve is represented in blue and it's on our XY coordinate plane. So looking, like when we look at our graph, we can see that based on the graph alone, we can expect that there will be 3 real roots. And we know that these three real roots are going to be in parentheses negative 10 and 12 and parentheses 2.3. So we're gonna have 3 roots. So 3 routes. Total. Fantastic. So with that said, we can now start to solve for this problem officially, so we need to recall and use the iterative formula for Newton's method, which, as we should recall, states that X to the power of or rather I should say X to the subscript, not the power, so X to the subscript of N. Plus 1 and once again this is the iteration formula for Newton's methods. So X N + 1 is equal to X N minus F of X N divided by F F X N. So, as we should recall and note this F of X indicates the first derivative of F of X, because that's what that prime symbol means, the first derivative of FFX. So, with that said, We now need to find the derivative of F of X. So F of X, the first derivative of F of X is going to be equal to E to the power of X minus 4 X. Fantastic. So now we need to take this derivative and we need to plug it back. Into since since we've now found the derivative of FFX, we need to plug this back into our inner in now we need to put it plug it back into our Newton's method equation. So for the iteration formula, and when we do that, we will find that our final formula written altogether would be X subscript, N + 1 is going to be equal to X N minus E to the power of X subscript N. Minus 2 multiplied by X N to the power of 2, divided by E X N minus 4 multiplied by X N. Fantastic. So now we need to solve for our roots now using our iteration formula. So we're going to use this formula to help us perform our iterations. So with that said, we now need to focus our attention on finding the root that's in parentheses -10. So with that said, we need to note that using our graph, let's say that our initial guess X0, or X subscript 0 is going to be equal to -0.50. So with that said, we could say that X1, our first iteration is going to be equal to -0.50, because we're going to be taking a value of -0.50 and we're going to be plugging it in for all values of X subscript N. So when we do that, we will find -0.50 minus. E to the power of -0.50 minus 2 multiplied by 0.50 to the power of 2. Divided by E to the power of -0.50. And then we need to subtract 4 multiplied by -0.50. Fantastic. So when we plug in this big old long expression into our calculator, we should find that our final answer for the first iteration when rounded to five decimal places, is going to be equal to negative 0.54087. Fantastic. So I don't have to keep writing out our iteration formula every single time. I'm just gonna give you the final answers for each iteration that we do. So, for the second iteration, instead of plugging in a value of -0.50 and for X subscript N, we need to plug in a value of -0.54087. So we're In our iteration value for the first iteration that we did in for our value for X subscript and for our second iteration. So when we plug in a value of negative 0.54087 in for X subscript N, we will find that X2 is going to be equal to. 0 0.53984. So once again it's gonna give us -0.53984. And now we need to keep going because we need to have two approximations be equal to each other, so we need to perform a third iteration. So when we plug in a value of negative 0.53984 in for X subscript N, we will find that X3X subscript 3, our third iteration, is going to be equal to negative 0.53984, which matches our second iteration, which is perfect. So that means we've found two approximations when rounded to 5 decimal places that are equal to each other. So therefore, the two successive approximations with being rounded up to 5 decimal places that are equal to each other are from the 2nd and 3rd iterations. So that means that our route 1 final answer is going to be equal to negative 0.53984, and that is our first answer. Hooray, we did it. Now we need to solve for our second root. So once again, this is going to be our root in parentheses 1.2, and for 1.2, let us say that our initial guess using our graph, let's say that X0 is equal to 1.50. Fantastic. So similarly, for X equals 1, plugging in a value of 1.50 in for our X subscript N. We will get a value of 1.48794. Let me round to four decial places, noting that it's now a positive value. Similarly, for our second iteration X2, when we plug in a value of 1.48794 in in for our value of X subscript N, we will receive an answer of 1.48796. And then for our third iteration, once again, we're rounding everything to five decimal places. So for our third iteration, when we plug in a value of 1.48. 796 in for X subscript N, we will get a value of 1.48796 when we round to 5 decimal places. So that means, without a doubt, the two successive approximations that are equal to each other after being round rounded to 5 decimal places are going to be the 2nd and 3rd iterations. So that means our route to. Is going to be equal to 1.48796, and that's our second answer. Hooray, we did it. We're on our roll. So now we could solve for our third route. So now we need to note for our third route, we're focusing on the boundaries, parentheses or boundary rather, the range of 2.3, so 2.3, fantastic. And let us say for this particular case, let's say that our guess, our initial guess X0 is going to be equal to 2.50. Fantastic. So for X1, when we plug in a value of 2.50 N for X subscript N, we're going to receive a value of 2.64548 when we round to 5 decimal places and for our Second iteration, when we plug in a value of 2.64548 in for X subscript N, we're going to get a value of 2.61895 when we round to five decimal places, and when we do our third iteration. When we plug in a value of 2.61895 in for X subscript N, you'll get a value of 2.61787. And we have to keep iterating because we don't have two successive approximations that are equal to each other. So doing our 4th iteration. When we plug in a value of 2.61787 in for X subscript N, we'll get a value of 2.61787, which that means our 3rd and 4th iterations are equal to each other. So, therefore, the two successive approximations when rounded to five decimal places that are equal to each other, are going to be the 3rd and 4th iterations. So that means our route 3. It's going to be equal to 2.61787, and that is our third and final answer. Hooray, we did it. So looking up at the top, to quickly recap everything that we just went over together, that means that our final three answers, without a doubt, have to be negative 0. 53984 for our first route, and 1.48796 for our second route, and for our third route, it's gonna be 2. So we need to say for our third route, it's gonna be 2.61787. And that's it. Those are our 3 final answers. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = e⁻ˣ - ((x + 4)/5)

Key Concepts

Newton's Method

Preliminary Analysis

Graphing Functions

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