Back To CourseMath 104: Calculus
14 chapters | 116 lessons | 11 flashcard sets
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Let's review some of the things we know about derivatives. The derivative is the slope of the function. If the derivative is positive, the slope is increasing. If it is negative, the slope is decreasing. If the derivative is zero, the function isn't changing, so it might be a minimum or a maximum. The second derivative is the slope of the derivative. If the second derivative is positive, the function is concave up; we have a 'cup.' If the second derivative is negative, we have something that's concave down, a 'frown.' If the second derivative is zero, the function might be at an inflection point if we're changing from a cup to a frown. So what does this mean?
Remember that dream I had where my father put a GPS on my car? Let's say that instead of going all high tech and putting a GPS on there, he put an accelerometer on my car. So he can't tell where I am, but he can tell how fast I'm going. And let's say he can graph my speed by velocity, going away from home, as a function of time. The first time he does this, the graph looks something like this. My velocity starts out at some positive value and is constant for some set of time. Then, my velocity jumps and is zero for some period of time. And then my velocity jumps again and is negative for some period of time. Can he reconstruct, roughly, where I am at any given point in time? Can he graph my position from this velocity?
Let's say he tries to plot my position as a function of time knowing I start out at home, at the origin here, at time=0. For the first 20 minutes, I'm going a constant, positive velocity. So I know he could plot my position as a function of time just by drawing something with a constant, positive slope. Here, you can imagine that I'm leaving the house at a nice 35 miles per hour. Because my slope is constant, my velocity is going to be constant. Now 20 minutes in, my velocity goes from constant, 35 miles per hour, to zero, and for the next 20 minutes my velocity is zero. That means my position isn't changing. Obviously I'm not going anywhere; my velocity is zero. Let's say I'm rock climbing during this period of 20 minutes. After that's over, my velocity becomes negative. So about 40 minutes in, I decide I'm going home and I start driving with a constant slope back home. It's constant because my velocity is constant. Not too bad.
Let's say he does this again the next day. Now I've got my velocity going away from home as a function of time, and it looks like nothing's constant on this graph. Let's mark out the interesting points. I'm going to mark this where my velocity is zero, and everywhere else my velocity is positive. When the velocity is zero, I know that I've stopped; if my velocity isn't changing, I'm obviously not driving. When my velocity's positive, I'm going away from home, because I'm driving forward.
So what can we tell from this graph? At the very beginning, my velocity is not that large. This plot has my velocity as being somewhat close to zero - it's not zero, it's positive, but it's not a big number. Maybe I'm just puttering along. As I go along the graph, my velocity is getting larger and larger, so not only am I moving away from home, I'm moving away from home faster and faster. At this point, my velocity is constant, so I'm still driving away from home, but I'm driving away from home at a constant velocity. Let's say I'm on the freeway. Then my velocity starts to go toward zero, so I start slowing down some. I'm still moving away from home, but I'm just slowing down. Then I stop. Maybe I'm at the mall shopping, getting my father back for all those times he's tracked me.
Okay, once I'm done at the mall, I get back in the car and start driving. Again my velocity starts out small, close to zero. I'm still moving away from home but slowly. Then I get faster, and faster, and faster, really fast now. So my velocity is really high. Now, as I move along in time, my velocity isn't changing as a function of time; it has flattened out. Here, I'm going on the freeway; I'm just speeding this time. And I get off the freeway and my velocity goes back toward zero as I slow down. When my velocity hits zero, I've stopped. Now I'm using his credit card at the movies! So in all of this, even though my velocity was slowing down, because it was still positive, I was moving away from home. All this time, I haven't been getting any closer to going home. I could take this one step further and look at my acceleration, which is going to be the derivative of my velocity.
When my acceleration is positive, my velocity is increasing, right? Here, when my velocity is increasing, my acceleration is going to be greater than zero. When my velocity is staying the same, when I'm on the freeway not speeding, my acceleration is zero. I'm not changing velocities at all; the slope here is zero. Then I start slowing down. As my velocity starts to decrease, my acceleration is negative. Then I'm parked, and I have no velocity and no acceleration. I get on the freeway, and my acceleration is positive because I speed up to get with traffic, then I speed past traffic. When I'm speeding along on the freeway, I'm not accelerating or decelerating; my acceleration is constant. When I get off the freeway, rip into my parking spot and slam on the brakes at the movies, my acceleration is negative. When I'm at the movies, my acceleration is zero. If I tried to go back and graph my position as a function of time, anywhere that my acceleration is positive I have something that is concave up - remember, this is our cup - because my velocity is increasing. When the acceleration is negative (this is my second derivative of my position), I have something that is concave down, a frown.
Let's try to look at my position as a function of time, and let's just look at it for this first half. At that point, I leave city limits and I'm way out in the country. In this first part here, I'm moving away from home and my velocity is increasing, so I have something that is concave up. When my velocity is constant but positive, my distance as a function of time is going to be increasing at a constant rate (because my velocity is positive). So right here, my slope is constant and I have a straight line. When I get off the freeway, my velocity starts to decrease. I'm still moving; I'm just slowing down. In this part, I have a concave down region. Now I'm pulling into my parking spot at the mall and my velocity isn't changing as a function of time. So my distance is going to be constant as a function of time.
Let's review what we can learn from the derivative of a function. We know that if f `, the derivative of a function, is positive, our original function is increasing. If it's negative, our original function is decreasing. This is because f ` is the slope of the function. If f ` is zero, your function might be at a maximum or a minimum. But as we just saw, I might be parked at the mall or the movies. The derivative of f `, f ``, is the slope of the derivative. This is going to tell you about the concavity of your function. If f `` is positive, your function is concave up and looks like a cup. If f `` is negative, f is concave down; it's a frown. And if f `` is zero, f might be an inflection point. This is where we'd go between something that's concave up and concave down. With all of these, you can glean a lot of information about a function just by looking at its derivative.
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Back To CourseMath 104: Calculus
14 chapters | 116 lessons | 11 flashcard sets