About this video
A particle's position is p(t) = (t - a)(a - t)(t - a)(t - b) / sqrt(a^2 + b^2) with c > b > a > 0. The video compares p(c) with the number of times p(t) = 0. Sign analysis shows p(c) is negative, and the numerator is zero only at t = a or t = b, so there are 2 zeros, making the count the larger quantity.
You'll be able to
- find the sign of a product or quotient by checking the sign of each factor.
- find the zeros of a rational expression from its numerator's factors.
- count distinct zeros even when a factor repeats.
Key words
- factor
- One of the expressions multiplied together, like (t - a) or (t - b).
- zero of a function
- An input that makes the function's output equal 0.
- numerator
- The top part of a fraction; a fraction is 0 when its numerator is 0 and denominator isn't.
- sign analysis
- Working out whether an expression is positive or negative from the signs of its parts.
Transcript
Let's say we we're told that the position of a particle can be figured out from, or the position of a particle as a function of time is given by this crazy expression. The position could be positive or negative. And this expression is t minus a, times a minus t, times t minus a, times t minus b, all of that over the square root of a squared plus b squared.
And they also tell us that c is greater than b, is greater than a, is greater than 0. So given that information, we have two statements right over here. We have the position at times c on the left hand side. And over here, we have the number of times that our function is equal to 0, the number of times that p of t is equal to 0.
What I want you to do is pause the video right now and think about which of these statements provides a larger value. Is p of c greater than or less than the number of times p of t equals 0? And I guess a third and fourth option would be that you maybe don't have enough information to figure this out or maybe that these are equal.
So I encourage you to figure that out right now. Which of these are larger? Or do you not have enough information? Or are they equal? So I'm assuming you've given a go at it. So let's think about each of them. So let's think about what p of c is. So p of c is going to be equal to, right here p-- I don't want to arbitrarily switch colors, which I sometimes do.
p of c is going to be equal to, let's see. It's going to be c minus a. I'll do this all in this one color. It's going to be c minus a, times a minus c, times c minus a, times c minus b, all of that over the square root of a squared plus b squared. So what do we know about this quantity?
What do we know about this quantity right over here? Let me highlight all the Cs here. So c minus a, a minus c, c minus a, c minus b. Well, they tell us that c is larger than a and b and that they're all positive. So maybe we can come up with some statement about whether this thing is positive or negative, whether this expression is.
So what's c minus a going to be? Well, c is greater than a. So this is going to be positive. What about a minus c? Well, a is less than c. So this is going to result in a negative number. c minus a, well, this is going to be a positive again. And then c minus b is also going to be a positive.
c is greater than both b and a. And what do we have here in the denominator? Well, the square root of a squared plus b squared, well, this is just going to be a positive value. So what do we have going on here? Here in the numerator, I have a positive times a negative, times a positive, times a positive.
So what's that going to be? Well, that's going to be a negative. A positive, times a positive, times a positive is positive. And then you throw that negative in there. So you're going to get a negative over a positive. And what's a negative divided by a positive?
Well, that's going to be a negative. So we don't know what the actual value is. But all we do know is that this provides us with a negative value. So this ends up being a positive value. Then we could make a statement that this, we just say, hey, this is a negative value as well.
But we might not have enough information. So let's think about the number of times p of t is equal to 0. Well, p of t is equal to 0 whenever the numerator right over here is equal to 0. And when would the numerator equal 0? Well, I have the product of this 1, 2, 3, 4 expression.
So if any one of these expressions is 0, then the entire numerator is going to be 0. So let's think about how you can make these expressions 0. So the expression could be 0. This will be 0 if t is equal to a. This would also be 0 if t is equal to a. And this would also be 0 if t is equal to a.
And this would be 0 if t is equal to b. So there's two values for t that will make this numerator equal to 0, t equals a or t equals b. So there's two times. So the number of times that p of t is equal to 0 is two. So now let's answer our question. What is larger, the number 2, is the number positive 2, versus some negative number?
Well, 2 is larger than any negative number. So this is the larger quantity.
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