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Studying for a test? Prepare with these 3 lessons on AP Calculus AB solved exams.

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# 2015 AP Calculus 2c

Video transskription

- [Voiceover] Let h be
the vertical distance between the graphs of
f and g and region s. Find the rate at which h
changes with respect to x when x is equal to 1.8. We have region s right over here. You can't see it that
well since I drew over it. And what you see when we're in region s, the function f is greater
than the function g. It's above the function g. We can write h of x, we could write h of x, as being equal to f of x minus g of x and what we wanna do is we wanna find the
rate at which h changes with respect to x. We could write that as h prime of, we could say 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 wanna figure out. We could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and 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. And a calculator can numerically integrate and can numerically evaluate derivatives. And so when they actually 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. And what's extra good about this is we have already
essentially inputted h of x in the previous steps
and really in part A. I defined this function here, and this function is essentially h of 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. Delete that absolute value, then I have to get rid of
that parentheses at the end. Let me delete that. And so notice this is h of x. We have our f of x, which was 1 plus x plus e
to the x squared minus 2x, and then from that we subtract g of x. G of x was a positive x to the
fourth but we're subtracting so negative x to the fourth, let me show you g of x right over here. G of x is right over here, and notice we are subtracting it. So the y1, as I've defined
it in my calculator, and I just pressed this y
equals button right over here, that is my h of x. Now I can go back to the other screen, and I can evaluate its derivative
when x is equal to 1.8. I go to math. I scroll down. We have n derivative right here and so click Enter there. And then what are we gonna
take the derivative of? Well the function y sub 1
that I've already defined in my calculator. I can go to variables, y variables, it's already selected function
so I'll just press Enter. And I'll select the function y sub 1 that I've already defined. So I'm taking the derivative of y sub 1. I'm taking the derivative
with respect to x. And I'm going to evaluate that derivative, when x is equal to 1.8. That simple. And then I click Enter. And there you have it. It's approximately -3.812. So approximately -3.812. And we're done. One thing you might appreciate
from this entire question, and even the question 1, is that they really wanted to make sure that you understand the
underlying conceptual ideas behind derivatives and integrals. And if you understand
the conceptual ideas, of how do you use them to solve problems, and you have your calculator at disposal, then these are not too hairy, that these can be done fairly quickly.