For all of you guys who wrote the provincial English exams today, I hope you did well. We covered a lot in class today so I STRONGLY RECOMMEND YOU READ THIS.
As you probably remember, yesterday we looked at the graph of a function f(t). At the end of class we were given five integrals, and were asked to draw the corresponding graphs. We spent the class going over the answers to these.
First of all, here is the graph f(t) and the given information, which the whole class was based around.
The graph of y=f(t): Is defined on the interval [0,4]
Has odd symmetry around the point (2,0)
On the interval [0,2], the graph is symmetric with respect to the line t=1
In yesterday’s class, we examined the graphs created by examining integrals on intervals, all using the function f(t). We discovered that f(t) was the derivative of all the different integrals we looked at. In other words, the graphs we created were all parent functions of f(t). In today’s class, we did the same thing, except that we had to do different transformations on the graph of f(t) and THEN find the graph of the parent function using the new graph.
I’m going to show the transformed graph for each question, followed by the graph of the parent function obtained from that graph.
1) Has somehow disappeared.. I'm very confused. Oh well. Anyway, the first question's basically what we did yesterday, no transformation involved. It just has the interval from x to zero, so in other words it the widths dx are negative, therefore the sign of the areas changes. The handy diagram below shows why. SEE YELLOW BOX. That's the thing to remember.
OH I FOUND IT! So.. never mind, here's the first question.
2)
3)
If you were to look for the derivative of g(h(x)), you would use chain rule. So you would get
g'(h(x))*h'(x)
= f(2x)(2)
4)
5)
And that's about all we did. Sorry this is up so late but the closer I get to a computer it seems the more technology-impaired I become. Oh right, and the next scribe is... Anh.
1 comment:
Hi Suzanne,
Your graphs are absolutely amazing. When I draw mine, one would think I am 93 years old with a severe case of shaky hands (I'm not :)). Anyway, I think I am taking a role of a person who asks questions all the time (is there a special word for such a person? English is not my first language, so I don't know all the words). Anyway, I have a few questions because I felt that I needed few more explanations in words. I'll list them in the order of problems (and most of them are really short):
1) You said there was no transformation involved, right? It seems to me that there might be. Can you try rewriting the expression for F_1? Here is a hint: you should use the yellow box that you said was very important to remember (except you have a typo there, so you should fix that first). Can you see the transformation?
3) In the second image for this problem you drew y=f(2t). Why did you do that?
4)From your graph I can see that f(|t|) is symmetric about the y-axis. I was then thinking that F_4 would be as well. But it's not. Can you explain why?
5)Here is a final question. Your F_5 looks beautiful. How did you know how to draw it? In particular, how did you know what to do around x=2?
Nice work. I am looking forward to these answers.
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