THE SLOPE OF THE TANGENT LINE
Why? Because tangent lines tell us a lot about curves, even complicated ones. On a piece of paper, draw a curve that goes up and comes down a few times. Choose a bunch of points on the curve and draw the tangent lines at those points? What do you notice?
1. When the curve seems to be going uphill (looking from left to right), what do you notice about the tangent line? Positive or negative slope?
2. Likewise, as the curve goes downhill, the slopes will have the same sign. Do all downhill slopes have the same value?
3. At the points where the curve turns around, what are the slopes of the tangent lines?
You should notice that the tangent lines are useful in describing the curve. That's what mathematicians are all about... finding numerical ways to describe behavior, in this case, function behavior. Instead of saying that the curve seems to be "going uphill" sharply at a certain point, we can say that the slope of the tangent line is "positive 4" or "positive 20" or whatever. This tells us that, yes, the curve is going uphill and it also tells us exactly how steep it is. Efficient, huh?!
Watch the video link below, close to 10 minutes long. At about 4 minutes into the video, the teacher will bring up an example of an object in free fall. Answer these questions for this week's post:
a. Why does the speed of the object differ on time intervals (2, 3) and (3, 4)?
b. What rudimentary method does he use to find the slope of the tangent line? Why is this inadequate?
c. In BC times (before calculus), you needed two points to find a slope. First, the man uses 3 and 3.1 as the two x-values; then, he uses 3 and 3.01. What concept allows us to use smaller than 3.000000001 as an x-value?
So, here's the very important point: the DERIVATIVE of a function = SLOPE OF THE TANGENT LINE.
Well, here's the link. Get some popcorn: Instantaneous Rate of Change
a.Because it is picking up speed as it falls. like on the curve the space between the seconds grows the further it goes out.
ReplyDeleteb.he uses a secent, the reason why it is inadequate is because the secent is never close enough to the tangent to be equel. you can never have an exact answer.
c. The concept of getting closer and closer to zero difference between two numbers. 3.001 is further from 3 then 3.000000000001 is. So the closer that second point to 3 for example, the better our secent is.
a. It does not fall the same speed the entire time. It increases speed throughout the fall.
ReplyDeleteb. He uses a secant, but this can never produce a perfect answer. It will get close, but the method is too simple for a tangent graph.
c. The smaller the difference between the numbers, the closer the secant will be.
a. Because it increases speed as time goes on.
ReplyDeleteb. He uses a secant. It is inadequate because the secant is not perfectly equal to the tangent but it will be close.
c. As the difference between the numbers gets smaller, the closer the secant will be. 3.000000001 is closer to 3 than 3.01 so it would be better to use 3.000000001 for the x value to get a closer secant.
a. The speed of the object differs because its speed is increasing as the time value increases.
ReplyDeleteb. He uses a secant. The problem with using a secant is that he will never get an exact value answer, it will come close but not be exact.
c. As the difference between the two values decreases, the value of the secant being closer to the exact value of the tangent increases.
a. as an object falls for longer periods of time, its speed increases.
ReplyDeleteb. he uses the secant of the line which will give him a very close approximation, but never the exact value.
c. the concept that allows us to do this is that the closer the one number is to the other, the more accurate the secant line will be to the actual tangent line. (.000000001 is closer to 0 than .001 is, and is therefore more accurate)
A) As more time elapses, more distance is going to be covered because of the formula given.
ReplyDeleteB) He uses the secant and its an approximation so it is not adequate.
C)The concept is that because the closer and closer you get to 3 the more accurate the secant line will be to the tangent, and you will get a closer answer.
A. because gravity is always in acceleration, and therefore cannot be constant.
ReplyDeleteB. He uses a secant, which can get very close to the tangent but will never be equal.
C. If the numbers are as close as possible to eachother, almost the same value, the line will almost be a tangent.
So far, so good. I'm glad to see posts coming in. I'm still waiting to see a key word for part three. If you're reading this, think about vocabulary before you post a comment.
ReplyDeletea. because as the object falls, it is continually building speed, and the slope of the curve is changing.
ReplyDeleteb. he uses the graph and its intervals to check the slope of the tangent. this is inadequate because the graph will not always allow for such use.
c. the more zeroes behind the decimal, the smaller the number, and the more accurate. if .001 is close to zero, then .000000000001 is even closer, and therefore making the slope of the tangent line we are after that much more accurate
a. As it falls it gets faster
ReplyDeleteb. He uses secant, which gets almost the same number.
c. The more zeroes you add, the more close it is to 3. Therefor it is closer to the real number
A) As time goes on, the speed will increase.
ReplyDeleteB) He uses the secant in order to find it, which gives a value that's pretty close, but isn't as precise.
C) It just shows the values as if they would infact be closer to 3. Using the value closer to the 3 would produce a better line closer to the tangent.
-Michael Ferrer, Period 6