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

Finding specific antiderivatives: exponential function | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

We're told that F of 7 is equal to 40 + 5 e 7th power, and f prime of X is equal to 5 e to the X. What is F of 0?

So, to evaluate F of 0, let's take the anti-derivative of f prime of X, and then we're going to have a constant of integration there. So we can use the information that they gave us up here that F of 7 is equal to this. This might look like an expression, but, well, it is an expression; but it's really just a number. There's no variables in this, and so we can use that to solve for our constant of integration. Then we will have fully known what f of X is, and we can use that to evaluate F of 0.

So let's just do it. If f prime of X is equal to 5 e to the X, then F of X is going to be equal to the anti-derivative of f prime of X. So the anti-derivative of 5 e to the X dx.

And this is the thing that I always find amazing about exponentials. Actually, let me just take a step. I'll take that 5 out of the integral so it becomes a little bit more obvious. And so the anti-derivative of e to the X, well, that's just e to the X because the derivative of e to the X is e to the X, which I find amazing every time I have to manipulate or take the derivative or anti-derivative of e to the X.

So this is going to be 5 e to the X + C, and you can verify: take the derivative of 5 e to the X + C. The derivative of 5 e to the X, well, that's 5 e to the X, so that works out, and the derivative—well, and the derivative of C is zero, so you wouldn't see it over here.

So now let's use this information to figure out what C is so that we know exactly what f of X is, and then we can evaluate F of 0. So we know that F of 7, so when X is equal to 7, we're going to—that this expression is going to evaluate to this thing: 40 + 5 e to the 7th power.

So, 5 * e to the 7th power plus C is equal to 40, is equal to 40 + 5 e to the 7th power. And all I did is said, okay, F of 7—well, if this is f of X, let me write this down—if this is F of seven, if this is f of X, I just replace the X with a seven to find F of seven.

We know that F of seven is also going to be equal to that; they gave us that information. But when you just look at this, it's pretty easy to figure out what C is going to be. You can subtract 5 e to the 7th from both sides, and you see that C is equal to 40.

And so we can rewrite F of X. We can say that F of X is equal to 5 e to the X plus C, which is 40. And so now from that, we can evaluate F of 0. F of 0 is going to be 5 * e to the 0 power + 40.

e to the 0 is 1, so it's going to be 5 * 1, which is just 5 + 40, which is equal to 45. And we're done.

More Articles

View All
5 Ways To Have 10x More Energy Throughout The Day
Hey, it’s Joey. Welcome to Better Ideas! Have you ever wanted to have just like uncomfortable amounts of energy? Do you lack the necessary energy to carry out basic daily tasks, like going to the gym, doing your homework, doing the laundry? A lot of peopl…
THE FED JUST RESET THE MARKET | Major Changes Explained
What’s up guys, it’s Graham here and, uh, welp, it just happened. The Federal Reserve completely just shocked the market right now with the 75 basis point rate hike, setting off yet another chain reaction that’s about to impact the entire market at the co…
Private jet expert destroys noobs
What celebrity owns the most expensive private jet in the world? Oh man, so there’s two people ahead of this guy, but you don’t know them and I don’t know them. One is Sultan of Brunei, $22 million. Okay, Sultan of Brunei was from like 20 years ago. This …
Chernobyl - What It's Like Today
That is Chernobyl nuclear reactor number four. It melted down on April 26, 1986. So, what happened was so much heat was generated inside that reactor that it basically blew the top off, spreading radioactive isotopes throughout this whole surrounding area…
Laura Overdeck on reducing math anxiety and connecting math with real life | Homeroom with Sal
We’re seeing questions come on YouTube, uh, ask Laura and I anything, and we have team members who are looking at them, and we’re going to surface, uh, them. And actually, I’ll start with a question from YouTube, and that did help. Thanks, Laura. So this…
Answering Presuppositionalism: Basic
Theists who subscribe to the presuppositionalist school of thought say that atheists can’t account for inductive reasoning. They claim that, in fact, whenever an atheist uses inductive reasoning, she is borrowing from the Christian worldview, because acco…