Help finding the midpoint of a closed interval & more

[Queens]

Hey everyone,

This is a review of some stuff I learned in high school, but I haven't actually done anything calculus related in about 2 years, and to be honest it looks foreign to me, if someone could help jog the old noodle it would help tremendously.

The first question is as follows;

1. There is a useful way of describing the points of the closed interval [a,b], as usual we assume that a<b.

a)Consider the interval [0,b], for b>0. Prove that if x lies in [0,b], then we have x=tb for some t with 0≤t≤1. What is the significance of the number t? What is the midpoint of the interval [0,b]?

b) Prove that if x ∈ [a,b], then we have x = (1-t)a+tb for some t with 0≤t≤1. What
is the midpoint of the interval [a,b]? What is the point 1/3 of the way from a to b?

c) Prove conversely that if 0≤t≤1 then (1-t)a+tb is in [a,b].



I have given this problem much though, and even asked some engineer buddies but the answer seems to elude all of us, perhaps t=inverse of b and the significance of that is... something.

I'm really grabbing at all I can get here, if anyone could help explain this to me I would appreciate it.

Thanks so much

 

[Mark44]

Hey everyone,

This is a review of some stuff I learned in high school, but I haven't actually done anything calculus related in about 2 years, and to be honest it looks foreign to me, if someone could help jog the old noodle it would help tremendously.

The first question is as follows;

1. There is a useful way of describing the points of the closed interval [a,b], as usual we assume that a<b.

a)Consider the interval [0,b], for b>0. Prove that if x lies in [0,b], then we have x=tb for some t with 0≤t≤1. What is the significance of the number t? What is the midpoint of the interval [0,b]?

t is just a number between 0 and 1 (inclusive). Try thinking about this with some concrete numbers. Suppose you're working with the interval [0, 1]. What's the midpoint of the interval?

Now suppose you're working with the interval [0, 5]. What's the midpoint now?

Can you make the leap to finding the midpoint of the interval [0, b]?

b) Prove that if x ∈ [a,b], then we have x = (1-t)a+tb for some t with 0≤t≤1. What
is the midpoint of the interval [a,b]? What is the point 1/3 of the way from a to b?

Try this out with some specific numbers, say [3, 7]. Can you see why x = (1 - t)*3 + t*7 hits every point in [3, 7] for some value of t? If t = 0, which point in the interval do you get? If t = 1, what point do you get now?

c) Prove conversely that if 0≤t≤1 then (1-t)a+tb is in [a,b].



I have given this problem much though, and even asked some engineer buddies but the answer seems to elude all of us, perhaps t=inverse of b and the significance of that is... something.

I'm really grabbing at all I can get here, if anyone could help explain this to me I would appreciate it.

Thanks so much