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

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!