1. Let’s go back to the first example we looked at with the intersection points between = and = 3.
Use Newton’s Method to find the second intersection point.
2. You’re going to find any absolute extrema for the function =5sin( −2) −( −2)2− on the interval [0,2]. In order to find critical points, you’re going to need to solve ′=0, and this is where you’ll use Newton’s Method.
(Notice that the derivatives ′ and ″ might be annoying to find: feel free to confirm your derivatives with others!)3
3. You’re going to find any absolute extrema for the function =5sin( −2) −( −2)2− on the interval [0,2]. In order to find critical points, you’re going to need to solve ′=0, and this is where you’ll use Newton’s Method.
(Notice that the derivatives ′ and ″ might be annoying to find: feel free to confirm your derivatives with others!)
4. We might recall that it could be nice to have a general idea of what this function looks like on this interval. Let’s graph the function on the interval [0,2].
5. We might have a better idea on how to find these absolute extrema. Make note, though, that we should show that the -values we find for the absolute extrema actually match with the absolute maximum/minimum -value.
6. Let’s investigate the following function: = 3−2 +2. We’re going to try to find the -intercept. Try using Newton’s Method a couple of times with the following initial guesses: 0, 1, a value between 0 and 1, and a value bigger than 1
7. Here’s a fun one. Let’s consider the function
=−288+2880 −11314 2+21785 3−20523 4+7560 5
This function has 5 real -intercepts. We could do some algebra and stuff and find all 5 of these, but let’s use Newton’s Method.
The hint I’ll give is that the -intercepts are all between =0 and =4. Use Newton’s method with different starting points to try to find all of the -intercepts.
8. Are you frustrated yet? You’ve likely found that this is annoying or hard to do. Let’s go back a step and try plotting the function on the interval [0,4] in order to get some visual context. This might help us find the remaining -intercepts using Newton’s Method.
Finish up by finding whatever -intercepts you hadn’t found yet.
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