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Free Response Skills Practice
Now that you have completed the course, it?s time to fit all these pieces together. You know a ton of Calculus. You know how to integrate, you know how to do derivatives. What you might not know yet, is how to break down a really complex problem, that comes at you from that field.
When you?re working on the free response section of the AP test, you?re going to find that some of the problems are pretty straight forward. It?ll say integrate this or do the rotation of this, and you?d immediately know what to do. But there?s always going to be some where you just wonder where in the world did they get this? How do I tackle this? This is just impossible. Then you panic and you freeze and you put your head down on the desk and you fail the test.
Now you don?t want that to happen, and it won?t happen. What you need to do is figure out how to analyze and breakdown a problem to guide what you?re going to do when you actually work on the Math. Here are the steps for breaking down a problem.
The first thing you want to do is scan it quickly. Don?t get bugged down on the details. Don?t see something like a word you recognize and immediately start doing some Math. No, read the whole thing first, it?s worth the time.
Once you?ve done that, then go back and identify and it seems grade school, but circle the keywords. It?ll help you find them more quickly when you need them later. Then and only then do you decide your approach to the problem, and actually solve it.
Let?s go ahead. I?m going to walk you through an example of this. So here is a little problem. We?ll go ahead and read it first. It says the function is defined for all real numbers and satisfies the following condition. I?m not circling anything yet, because seeing it twice will help you find things more quickly.
The function g is given by g(x) for all real numbers, where a is a constant. Find g'(0) and g''(0). Part b; the function is given by h equals the sine of kx times the function of x for all real numbers. Find h'. We are ready to go back and circle the key things.
Define for all real numbers. That?s technicality that we need in order to do the problem. Some initial conditions we?re going to need to use this at some point, we?re going to need this. We?re going to need that. The function well, we?re given a function almost certainly we?re going to need to use that. Find g'. There?s a cue as to what we have to actually do. Find g'. So it looks like we need to do a derivative. And g'' so we need to do a double derivative. But, g involves function f, and we don?t know function f. So that?s going to throw it for our loop. Let's break this down now.
Now we?re in our step three. Step three is trying to decide your method. Looking at function g, you can see this thing has two parts to it. We?re supposed to find the derivative of a specific point. This is one of the problems where what you?re expected to be able to do is to use the Chain Rule, the Product Rules etcetera and substitute it in initial condition. You?ll find something very similar to this in one of our episodes.
Next part, we are supposed to find h'. We?re supposed to also write an equation for a tangent line. So it looks like we have two tasks here. One of them is to do a derivative, and again this is the method where you?re doing a derivative from a chart. We?re also supposed to find the tangent line. We have an episode on tangent lines that you might want to review if you actually go and do this problem.
Now that we?ve analyzed our problem, I would like you to try it. It?s your turn. Go to the bonus material. You?ll find a practice problem in there that?s very similar to the AP free response questions that you are going to see. I?m going to give you five minutes to analyze this and I?ll wait here for you.
Well, here it is. I?m not going to read it out word for word for you. You?ve done that step already. I?m going to go right to circling the critical things. This graph is probably critical. I won?t circle the whole thing. Often times on the AP free response, even a problem like this where you?ll be allowed to use your calculator, they?ll often give you the graph. Good deal.
Let?s see oil in a storage tank, time interval. We are going to need this. They rarely give you any data that you don?t need. Time is measured in hours. It seems like a technicality, but yeah you?re probably going to need that.
When they grade these test, they look for you showing the work. They look for the proper form and often times they give you an extra point for putting in the correct units. well, oil enters the tank, that?s important. It means it?s going in. It?s increasing.
Rate is a very important word, extremely important word. If you miss that word, you?re going to get mixed up about this. F(t) isn?t the amount of oil on the tank, it?s how fast it?s going in. Keep that in mind, we?re going to be using that when we?re doing the problem. Oil drains and it?s also a rate. There?s a tricky thing here. It drains, but this is positive. It?s draining at +45 gallons per hour. When we put our problem together, we?re probably going to have to deal with that, because this is given as positive, yet it?s draining. If you look at the curve, here is the curve where it?s filling, the cosine function. Here is where it?s draining, a two-part linear function. Then both graphs is positives.
Let?s keep on going Of course we?re going to need this. The graphs of f and d they intersect at these two times we?re going to need that. And you might want to write those on the graph. You could have calculated these with your calculator by putting the functions into your calculator. Using the intercept function, but they?ll often give you this. It?s to save you a bit of time, because they?re not interested really in whether or not you can use your calculator. They?re interested in whether or not you know how to approach the problem.
Tank hold 2,000 gallons at t equals 0. You?ll have to probably use that at some point as well. here are the three parts. We?re going to go through this step by step, because we?re ready for part 3.
Part 3, we?re going to have to actually decide how we?re going to tackle this part of the problem. We should circle our key words on this as well. How many? Important little phrase, in other words quantity. We need to find the quantity, but wait a minute f(t) that wasn?t a quantity function. It doesn?t tell you how much is in the tank. It tells you how fast it?s filling the tank. So now we?ve got to decide our approach. Let?s see we have a function that tells you how fast the tank fills, but we?re supposed to find how many. How do you reverse our rate to find how many? You got it, it?s an integral. So our approach for this one is going to be to integrate a rate.
Let?s go into the next part. Find the time intervals. That?s important, time intervals tell us that we?re supposed to have, they?re things like these. Now the time is between some number and other number when the amount of oil in the tank is decreasing. Decreasing; important word.
Explain your answers. here is where that tricky graph comes in. This horizontal is a rate of draining. This titled line is also our rate of draining. Even though the graph does positive things on the graph, this is when the quantity is going down.
In this section here, the quantity of oil going into and leaving the tank is more than the quantity that?s going into the tank. You don?t need to love Calculus for this. You are actually given these intersections. You can do the problem without any work. You just have to explain why you get the answer you did. There?s one more section.
In this part here, it?s entering the tank faster than it?s leaving the tank, except for this little section right here. That?s end of this pot where the oil is leaving the tank more quickly. So our technique for here is really just to read the graph. That?s all it is.
Section c, at what time and I'm going to say our answer is supposed to be some kind of number involved in the time is the amount. There?s that word 'amount' again. That?s quantity. At what point is the amount of oil in the tank at it maximum. There is another key word; maximum. Well, amount, but once again like in part A, we weren?t given functions for the amount. We were given functions for the rate.
What we?re going to do with this one again is, to integrate a rate. Now this time, unlike the first part where you were just dealing with the filling rate, now you?re dealing with the filling and draining rates. This one will have to be broken up into sections. You?re going to have to take care of all these sections individually in order to solve this problem.
You?ll have to integrate separately in these four time intervals. Now I?m not going to go through this problem, but if you look through the bonus materials, I did work out the entire solution including my recommendations for how you would write this out on the AP test. On the section of the test where you?re allowed to use a calculator, you don?t have to show a lot of Math. What they grade you on is getting the correct set up, as well as getting the correct answer.
Well, there you have it. Don?t get overwhelmed by all this. The goal here is to build yourself a structure using the techniques that I?ve shown you on how to tackle a problem. What I recommend you do now is, follow the link to the College Board website. They have tons of free response questions on there. Just from the taking, all you have to do is go to the website, and you can download as many of them as you want.
In fact I recommend you download a lot of them. Not that I want you to solve all of the problems in there, what I want you to do is to repeatedly analyze problem after problem so that you can do it just like that. That will really set you up so that when you get to the AP test, you can analyze a problem quickly. The more of them you do, the more you?ll start to see that even though these problems seem like they?re from that field, they do tend to repeat a lot of the same concepts over and over again. Well, thanks for tuning in and don?t forget Calculus is fun.