Explanation:
In order to prove that affirmation, we define the function g over the interval [0, 1/2] with the formula 
If we evaluate g at the endpoints we have
g(0) = f(1/2)-f(0) = f(1/2) - f(1) (because f(0) = f(1))
g(1/2) = f(1) - f(1/2) = -g(0)
Since g(1/2) = -g(0), we have one chance out of three
- g(0) > 0 and g(1/2) < 0
- g(0) < 0 and g(1/2) > 0
- g(0) = g(1/2) = 0
We will prove that g has a zero on [0,1/2]. If g(0) = 0, then it is trivial. If g(0) ≠ 0, then we are in one of the first two cases, and therefore g(0) * g(1/2) < 0. Since f is continuous, so is g. Bolzano's Theorem assures that there exists c in (0,1/2) such that g(c) = 0. This proves that g has at least one zero on [0,1/2].
Let c be a 0 of g, then we have

Hence, f(c+1/2) = f(c) as we wanted.
Answer:
r=13
Step-by-step explanation:
Answer:
You would plot the following points, remembering that they’re in the form (x,y) = (time, height):
(1,35) (2,60) (3,92) (4,120) (5,175)
The ordered pairs would be closer together, and there would be 4 times more of them if you were to measure every 15 seconds instead. You would have a line with more points that were closer together.
Answer:
9700
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