2011 Calculus AB free response #1a

Problem 1. For 0 is less than or equal to t is less than or equal to 6, a particle is moving along the x-axis. The particle's position x of t is not explicitly given. The velocity of the particle is given by v of t is equal to all of this business right over here. The acceleration of the particle is given by a of t is equal to all of this business over here. They actually didn't have to give us that because the acceleration is just the derivative of the velocity. And they also gave us-- they don't tell us the position function, but they tell us where we start off, x of 0 is equal to 2. Fair enough. Now let's do part A. Is the speed of the particle increasing or decreasing at time t equals 5.5? Give a reason for your answer. It looks like they did something a little sneaky here because they gave us a velocity function and then they ask about a speed. And you might say, wait, aren't those the same thing? And I would say, no, they aren't quite the same thing. Velocity is a magnitude and a direction. It is a vector quantity. Speed is just a magnitude. It is a scalar quantity. And to see the difference, you could have a velocity-- and this isn't maybe particular to this problem because they don't give us the units-- but you could have a velocity of negative 5 meters per second. And maybe if we're talking about on the x-axis, this would mean we're moving leftward at 5 meters per second on the x-axis. So the magnitude is 5 meters per second. This is the magnitude. And the direction is specified by the negative number. And that is the direction. Your velocity could be negative 5 meters per second, but your speed would just be 5 meters per second. So your speed is 5 meters per second whether you're going to the left or to the right. Your velocity, you actually care whether you're going to the left or the right. So let's just keep that in mind while we try to solve this problem. So the best way to figure out whether our rate of change is increasing or decreasing is to look at the acceleration. Because acceleration is really just the rate of change of velocity. And then, we can think a little bit about this velocity versus speed question. So what is the acceleration at time 5.5? Get the calculator out. We can use calculators for this part of the AP exam. And I assume they intend us to because this isn't something that's easy to calculate by hand. So the acceleration at time 5.5, we just have to say t is 5.5 and evaluate this function. So 1/2-- I'll just write 0.5-- times e to the t over 4. Well t is 5.5. 5.5 divided by 4. And then, times cosine of 5.5 divided by 4 gives us 0.38. Did I do that right? We have 0.5 times e to the 5.5 divided by 4, times cosine-- oh, sorry. I made a mistake. That looked a little strange. It's not cosine of 5.5 divided by 4, it's cosine of e to the 5.5 divided by 4. So let's look at that. So that's one parentheses I close. And now that is the second parentheses that I've closed. And I get negative 1.3-- well, I just say roughly negative 1.36. So this is equal to, or approximately equal to, negative 1.36 if I round it. And we don't care about so much as the actual value. What we really care about is its sign. So the acceleration at time 5.5 is negative, which tells us that the velocity is decreasing. Now, you might be tempted to say, we're done. But remember, they're not asking us, is the velocity of the particle increasing or decreasing? They're asking us, is the speed of the particle increasing or decreasing? And if you're saying, hey, how did I know that, just remember acceleration is just the rate of change of velocity. If acceleration is negative, that means the rate of change of velocity is negative It is going down. But anyway, how do we address this speed issue? How do we think about it? Well, there's two scenarios. If our velocity is positive at time 5.5, so if we have a positive velocity, so let's say our velocity is 5 meters per second-- although they don't give us units here, so I won't use units. So let's say our velocity is 5. And then it's negative. So at some point our velocity is going to be something smaller. Then that means that the speed would also be decreasing. So if we have a positive velocity, then the fact that acceleration is negative means that both velocity and speed would be decreasing. On the other hand, if we had a negative velocity at time t equals 5.5, then the fact that it's decreasing means that we're getting even more negative. And if we're getting even more negative, then that means the speed is increasing. The magnitude is increasing in the leftward direction. So what we really need to do, beyond just evaluating the acceleration at time 5.5, we also have to evaluate the velocity to see if it's going in the left or the rightward direction. So let's evaluate the velocity. The velocity at 5.5-- and we'll just get our calculator out again. Velocity at 5.5-- this is our velocity function-- is going to be equal to 2 times the sine of-- let me write it this way. Just because I want to make sure I get my parentheses right. 2 times the sine-- let me write it this way-- 2 sine of e to the 5.5, that's our time, divided by 4. So I did that part right over here. And I'm going to close the 2 as well. And then plus 1. So this is our velocity. So our velocity at time 5.5 is negative. So negative 0.45. Negative 0.45 roughly. So this is negative. The velocity is negative. So we have this scenario where the velocity is negative, which means we're going in the left direction. And the fact that the velocity is also decreasing means that over time-- at least at this point in time-- as we go forward in time, it'll become even more negative. And it'll become even more negative if we wait a little bit longer. So that means that the magnitude of the velocity is increasing. It's just going in the leftward direction. So if the magnitude of the velocity is increasing although it's going in the leftward direction, that means that the speed is increasing. So the velocity-- so this is one of those interesting scenarios-- the velocity is decreasing. But the speed, which is what they're asking us in the question, speed is increasing. And if you wanted to do it really quick with all of this explanation I gave you, you would say, hey, look. What's acceleration? Is it positive or negative? You would evaluate it. You'd say, hey, it's negative. So you know velocity is decreasing. And then you would say, hey, what is velocity? Is it positive or negative? You evaluate it. You say it's negative. So you have a negative value that is decreasing. So it's becoming more negative. So that means its magnitude is increasing, or speed is increasing.