yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Justification with the intermediate value theorem: equation | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Let g of x equal one over x. Can we use the intermediate value theorem to say that there is a value c such that g of c is equal to zero and negative one is less than or equal to c is less than or equal to one?

If so, write a justification.

So in order to even use the intermediate value theorem, you have to be continuous over the interval that you care about, and this interval that we care about is from x equals negative one to one.

And one over x is not continuous over that interval. It is not defined when x is equal to zero.

And so we could say no because g of x is not defined. Not defined, or I could say let me just say not continuous.

It's also not defined at every point of the interval, but let's say not continuous over the closed interval from negative one to one.

And we could even put parentheses: not defined at x is equal to zero.

All right, now let's ask the second question. Can we use the intermediate value theorem to say that the equation g of x is equal to three-fourths has a solution where one is less than or equal to x is less than or equal to two?

If so, write a justification.

All right, so first let's look at the interval. If we're thinking about the interval from one to two, well yeah, our function is going to be continuous over that interval.

So we could say g of x is continuous on the closed interval from one to two.

And if you wanted to put more justification there, you could say g is defined for all real numbers such that x does not equal zero.

I can write g of x is defined for all real numbers such that x does not equal to zero.

And you could say rational functions like one over x are continuous at all points in their domains.

At all points in their domain, that's really establishing that g of x is continuous on that interval.

And then we want to see what values does g take on at the endpoint, or actually these are the endpoints that we're looking at right over here.

g of one is going to be equal to one over one, which is one, and g of two is going to be one over two, which is equal to one over two.

So three-fourths is between g of one and g of two.

So by the intermediate value theorem, there must be an x that is in the interval from one to two such that g of x is equal to three-fourths.

And so yes, we can use the intermediate value theorem to say that the equation g of x is equal to three-fourths has a solution, and we are done.

More Articles

View All
Behind the Scenes With Director Everardo Gout | MARS
Presented by Acura. Precision crafted performance. Retro Rockets are about to fire in 1, 2, 3! Hello, my name is Ardo Good, and I’m the director of the miniseries. I was drawn to this project mainly because of two things. One is that I always try to look…
BREAKING: The Federal Reserve Rate Hike (Major Changes Explained)
What’s up, Graham? It’s guys here. So, we’ve just had a major announcement from the Federal Reserve that changes everything. With their 10th rate hike now going into effect, you’re going to want to hear this. After all, higher interest rates have already …
Symmetry of second partial derivatives
So in the last couple videos I talked about partial derivatives of multivariable functions, and here I want to talk about second partial derivatives. So, I’m going to write some kind of multivariable function. Let’s say it’s, um, I don’t know, sin of x * …
How Geographic Realities Keep Russia's Economy Behind
Two Russian-dominated multinational empires succeeded one another on the same territory, the first being called candidly the Russian Empire and the second the Soviet Union. Geographically, Russia is in some ways like the rest of Eastern Europe, but its na…
You Have to Protect Your House! | Life Below Zero
You don’t know when something unique is going to happen out here, but you better be on your toes. There he is, he’s on top of my Ridge now. After a brief stay in Dead Horse, SE has returned home to find a Wolverine lurking around CIC’s perimeter. To prot…
Why I Founded an Ocean Exploration Organization
When I was growing up, Jac Kisto had a big effect on my life. Fast forward, I learned how to dive. Um, and then about 11 years ago, I bought an ocean exploration vehicle. It’s a two-man submersible that goes down 1,000 meters, and I knew that I could give…