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

2015 AP Calculus 2c | AP Calculus AB solved exams | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Let H be the vertical distance between the graphs of f and g in region s. Find the rate at which H changes with respect to x when x is equal to 1.8.

So, we have region s right over here. You can't see it that well since I drew over it. What you see in region s, the function f is greater than the function g; it's above the function g. So we can write H(x) as being equal to f(x) minus g(x).

What we want to do is find the rate at which H changes with respect to x. We could write that as H prime of x, but we want the rate when x is equal to 1.8. So H prime of 1.8 is what we want to figure out now.

We could evaluate f prime of 1.8 and g prime of 1.8. To do that, we would take the derivatives of each of these functions. We know how to do that; it's within our capabilities.

But it's important to realize when you're taking the AP test that you have a calculator at your disposal. A calculator can numerically evaluate derivatives and can numerically integrate. So whenever they want us to find the area or evaluate an integral where they give the endpoints or evaluate a derivative at a point, well, that's a pretty good sign that you could probably use your calculator here.

What's extra good about this is we have already essentially input H(x) in the previous steps. In part A, I had defined this function here, and this function is essentially H(x). I took the absolute value of it, so it's always positive over either region, but I could delete the absolute value if we want.

So, let me delete that absolute value and have to get rid of that parentheses at the end. Notice this is H(x). We have our f(x), which is 1 + x + e^(x^2) - 2x, and then from that, we subtract g(x).

So we have g(x), which was a positive x^4, but we're subtracting x^4. Let me show you g(x) right over here. Notice we are subtracting it, so y1, as I've defined in my calculator, is now H(x).

Now, I can go back to the other screen and evaluate its derivative when x is equal to 1.8. I go to math, I scroll down, and we have nDerivative right here. So, I click enter.

Then, what I'm going to take the derivative of well, the function y sub one that I've defined in my calculator. I can go to variables, y variables; it's already selected function, so I'll just press enter and select the function y sub one that I've already defined.

So, I'm taking the derivative of y sub one with respect to x, and I'm going to evaluate that derivative when x is equal to 1.8. That’s simple!

Then, I click enter, and there you have it. It's approximately -3.812.

And we're done! You know, one thing that you might appreciate from this entire question, and even question one, is they really want to make sure that you understand the underlying conceptual ideas behind derivatives and integrals. If you understand the conceptual ideas of how to use them to solve problems and you have your calculator at your disposal, these are not too hairy. These can be done fairly quickly!

More Articles

View All
AI in your life
So in this video we’re going to talk about where we all have artificial intelligence or AI in our lives. And so before I go into where we’re already seeing it and where we’re likely to start seeing it more and more, I want you to pause this video and thi…
Ireland’s Underwater World | National Geographic
[Music] [Applause] [Music] The first time I saw it, I just thought, “Oh, how my father would have loved this.” Growing up, I was mesmerized by Cousteau films from the underwater world, and I thought, “Well, that couldn’t be Ireland; that must be some exot…
Job Security in an Insecure Time | America Inside Out
When you found out you’d been hired by GE, what was your reaction? “I didn’t believe it at first. It really didn’t sink in until I got the first paycheck, and I thought, ‘I’m really in here.’ You’d walk across the parking lot, look all the way down the A…
Stop Caring What People Think | The Stoic Way
It never ceases to amaze me. We all love ourselves more than other people, but care more about their opinion than our own. People spend so much energy on worrying about what other people think about them. The thing is: it’s a huge waste of time; especiall…
Graphing two variable inequality
So what I would like to do in this video is graph the inequality negative 14x minus 7y is less than 4. And like always, I encourage you to pause this video and see if you can graph it on your own before we work through it together. So the way that I like…
Phil Libin at Startup School 2013
Wow, good morning! Uh, I think this is literally the largest number of people I’ve ever spoken to. Pretty sure it is. Uh, very cool, very impressive. Thank you for coming. Um and to everyone, uh I’m super excited to talk to you. You know, Paul asked me t…