Sunday, November 4, 2012
November 5th - 9th
This week we are going to start chapter 4, which is all about applications of derivatives. It is also the last chapter we will learn this semester! In the first few lessons this week, we are going to learn how to find where functions are increasing, decreasing, and have maxima and minima without looking at the graph. The first derivative test is going to help us do this. We are also going to learn about curvature, or concavity, and points of inflection. We will learn how to find where a function is concave up or concave down, and we will be able to sketch a difficult curve without using a calculator. Watch the 2 videos at the right to learn a little more about the first derivative test and concavity and points of inflection. For your post, I'd like you to tell me what concavity is in your own words and then give me an example of an object or a function in real life that demonstrates concave up or concave down behavior.
Subscribe to:
Post Comments (Atom)
To me, concavity is where you have a function that doesn't have an x intercept. From the videos it is show that whether it is going up or down, it starts to level out across the x-axis, but never crossing it. A real world example is if a plan is diving and then levels out and never touches the ground. That would be an example of Concave up that is decreasing.
ReplyDeleteConcavity does not have to do with x-intercepts or the x-axis. It strictly deals with the curvature of the function. The function can cross the x-axis and still be concave up or down. You might be thinking of asymptotes Isaac.
ReplyDeleteConcavity is when a function is curving up or down. It may also be where the function has a relative maximum or minimum. When I think of an object that demonstrates concavity, an eye contact comes to mind as a perfect example. The function of a pendulum's path, such as a grandfather clock, while in motion also forms a concave up pattern.
ReplyDeleteConcavity is the up or down curve of a function where the function will always be above its tangent lines. An example would be a skateboarder's half-pipe. This exemplifies a decreasing and an increasing concave up function.
ReplyDeleteConcavity is concept of concave upward or concave downward of function (whether parts of graph's curving part is facing downward or upward). it does not exist domian values. For example, when someone throw a ball, it makes parabola which has concave down(both increasing and decreasing) pattern.
ReplyDeleteConcave is curving function which is upward or downward. For example, when a golf player hit the golf ball, the way of the golf ball is one of concave function. It has an increasing and a decreasing.
ReplyDeleteA concave function will curve upward or downward. An example of concavity is a snowboarder on a halfpipe. Another example would be the flight of a snowboarder coming off a jump.
ReplyDeleteConcavity describes the behavior of the slope of a function, it directly affects the slope but not the function itself.I kid going down a slide could represent a concave function. It would be concave down as the kid slides down.
ReplyDeleteNot quite Allie. A slide would be a decreasing concave up function. It probably makes more sense to you after our lesson today.
ReplyDeleteConcavity is when the function has an up or down curve, and will always be above its tangent lines. For an example, a rainbow would be a concave down function.
ReplyDeleteConcavity is the behavior of the slope. This directly affects the slope. Thus concavity is the second derivative of the function. As the slope increases, concavity increases. As slope decreases, concavity decreases. An example of this in real life would be surfing a HUGE wave!
ReplyDeleteConcavity is when a curve is either curved downward or upward. A parabola is a good example of this. A real world example of this is when a footbal is thrown. There is a downward concave motion created.
ReplyDeleteConcavity on a function is the curvature of the graph of the function, and whether or not it curves up or down, most of the time at a relative maximum or minimum. And example of a concave down function is if someone had a rock in a slingshot and shot the rock from the slingshot.
ReplyDeleteConcavity is simply the curvature of a line. It can either be up or down. Throwing a baseball in the air, then watching it come down would be a simple example of concave down. Another example would be hiking down into a valley than hiking up a mountain on the other side. This would be concave up. But if you walked to the top of the mountain, than down the other side, that would be concave down. The point of inflection would be where you were halfway up the mountain (where concave up changes to concave down).
ReplyDeleteDaniel I like your example, but you're not quite correct about the point of inflection. Climbing up and down a mountain would remain concave down the whole way. If you climbed up and down a mountain and then proceeded to hike down into a valley and back up again, the point where you got down to the bottom of the mountain and started hiking down into the valley would be the point of inflection!
ReplyDelete