Identifying intercepts of parametric planar motion functions
About this video
The video finds the x-intercept of a parametric function g(t) with y(t) = 2ln(5t) and x(t) = e^(2t + 5). It sets y(t) = 0 to get ln(5t) = 0, so 5t = e^0 = 1 and t = 1/5. Substituting t = 1/5 into x(t) gives x = e^(27/5), so the x-intercept is the point (e^(27/5), 0).
You'll be able to
- find an x-intercept of a parametric function by solving y(t) = 0 for t.
- solve a simple natural log equation like ln(5t) = 0.
- substitute the t value back into x(t) to get the x-intercept.
Key words
- x-intercept
- The point where a graph crosses the x-axis, so y = 0 there.
- parametric function
- A function where x and y are each given in terms of a parameter t.
- natural log
- ln is log base e; ln(a) = b means e^b = a.
Transcript
- [Instructor] We're asked consider the parametric function g of t is equal to all of this business. What is the x-intercept of g of t? So, pause this video. Have a go at that. Let's just remind ourselves what's going on. So, I don't know what this function is going to look like, but if we were to plot it in the xy-plane, so x, y.
As, and I'm not claiming though I'm about to draw, is actually accurate to this function. I have no idea what it looks like. But as we have various values of t, we're going to get different xy pairs. And then we're gonna have some point, they're saying that there's an x-intercept here of g of t.
So, this is some value here where y is going to be equal to zero and then x is going to be equal to some value. And actually, that's the value they want if they want the x-intercept. So, what I'm going to do is say, all right, at what point does y of t equal zero, at what t value?
And then I can substitute that back in to figure out what x value that's going to get us. So, let me set this to zero. So, if I have 2 times the natural log of 5t is equal to a zero, well, this is the same thing as saying, well, one, I could, if I want, divide both sides by 2 and then I'll just get the natural log of 5t is equal to a zero.
And then this is another way of saying that e, 'cause natural log is just log base e, that e to the zero power is equal to 5t or e to the zero power is just one. So one is equal to 5t. And if we just divide both sides by 5, we get that t is equal to 1/5. Now, we're not done.
We didn't say at what t value do we hit the x-axis. We wanna know where do we intersect the x-axis. So, what we wanna do is take this t equals 1/5, and then substitute it back in over, let me do another color just for fun, substitute it back in over here. So let's do that.
So, if we say x of 1/5 is going to be equal to e to the 2 times 1/5 power plus 5. This is going to be equal to e to the 2/5 power plus 5, which we could view as, well, I'll just write it this way, 25 over 5, which is equal to e to the 27th over 5 power, whatever that is going to be.
It's a little bit more than e to the fifth power. That is where we're, so it's actually a pretty large number (chuckles) where we are going to intersect the actual x-axis. Or another way of saying it is we are going to intersect the x-axis at the point. The x value is going to be e to the 27 over 5, and the y value is going to be equal to zero because we're actually at the x-axis.
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