Let’s go back to the first example we looked at with the intersection points between 𝑦=𝑒𝑥 and 𝑦=𝑥3.

1. Let’s go back to the first example we looked at with the intersection points between 𝑦=𝑒𝑥 and 𝑦=𝑥3.

Use Newton’s Method to find the second intersection point.

2. You’re going to find any absolute extrema for the function 𝑦=5sin(𝑥−2)𝑒−(𝑥−2)2−𝑥 on the interval [0,2]. In order to find critical points, you’re going to need to solve 𝑦′=0, and this is where you’ll use Newton’s Method.

(Notice that the derivatives 𝑦′ and 𝑦″ 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 𝑦=5sin(𝑥−2)𝑒−(𝑥−2)2−𝑥 on the interval [0,2]. In order to find critical points, you’re going to need to solve 𝑦′=0, and this is where you’ll use Newton’s Method.

(Notice that the derivatives 𝑦′ and 𝑦″ 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 𝑥-values we find for the absolute extrema actually match with the absolute maximum/minimum 𝑦-value.
6. Let’s investigate the following function: 𝑦=𝑥3−2𝑥+2. We’re going to try to find the 𝑥-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

𝑦=−288+2880𝑥−11314𝑥2+21785𝑥3−20523𝑥4+7560𝑥5

This function has 5 real 𝑥-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 𝑥-intercepts are all between 𝑥=0 and 𝑥=4. Use Newton’s method with different starting points to try to find all of the 𝑥-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 𝑥-intercepts using Newton’s Method.

Finish up by finding whatever 𝑥-intercepts you hadn’t found