The Intermediate Value Theorem is pretty straightforward. A continuous function can not skip values - meaning that if the function went from 10 to 200,000 it would hit every single value inbetween those numbers. This seems really obvious to me and I'm not sure why they would dedicate an entire section just to that. The Bisection Method, used to find zeros within specific intervals is also really straightforward.
Problems:
1. Show that the function f(x)= x^2 / x^7+8 takes on the value 0.10. To solve this problem I chose a pretty obvious interval: [0,1]. Plugging in 0 for x, gave me 0^2/ 0^7+8 which equals 0/8 or 0. So obviously there is a zero in this interval, but since the question isn't asking to find a zero, I plugged in 1 for x. This gave me 1^2/ 1^7 +8 = 1/9. 1/9 converted to a decimal = 0.1 repeating, which is really, really close to 0.10.
2. Carry out the Bisection Method for cos θ - 2*sinθ to determine whether or not there are zeros in the interval [1.25, 1.5].
There can't be a zero in this interval because if you plug 1.25 in for θ, you get -1.58. If you plug 1.5 in for θ, you get -1.92. From -1.58 to -1.92 there are no zeros. However, there are zeros in the interval [3.5,3.75]. When 3.5 is plugged in for θ the answer is -0.2349 and when 3.75 is plugged in for θ you get 0.3226. Since the answers go from negative to positive there has to be a zero in that interval.
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