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Wednesday, January 21, 2009

Finding anti derivatives!

First class of the "finding anti derivatives" unit was today, and it wasn't so bad. We also talked about the test we had yesterday, and by far, the free responses were a lot harder then expected, so we went over the answers in class. The test seemed to be quite long, but in fact, it was about 2/3 its original size! Now that's a fun fact. The answers to the free response questions are as follows, and I will try my best to explain them.
Okay so this here is the first free response questions, and is the easiest of the two we went over in class. The answer would be to sketch the graph y = (Sinx)/x. Where x cannot equal zero which means its a discontinuity at x = 0, but it also says that the graph exists at 1 where x = 0, so it pretty much fills in this discontinuity. Confusing? Yeah, I know. Part b asks for what values of x does A have a local minimum, and A is the parent function of the graph we drew, so the local minimum of the parent function would be the point where it the x values decrease then increase, and on the derivative, it would be where the values are less than zero, to when they begin to be greater than 0, this would be at 2pi, and it would repeat every cycle. Part c asks for the coordinates of the first inflection point of the parent function right of the origin, this would be at 3pi/2 where the derivative graph decreases then increases. Finally, part d asks to solve the following equation accurate to one decimal point.
This is quite simple, because we just use our calculators, and keep integrating the graph until the area we get is 1. It's pretty much trial and error, and the answer should be 1.1
I honestly don't know how to explain the answer to the 2nd free response question, so I'm not going to try, but Mr. K gave us an alternate solution, which we will learn about later on in this unit. Here it is:
Part b asks us to find where k = 2 and this is simple enough, just plug in the value 2 where k is supposed to be, the answer is 4.

Part c is here:
It's pretty much a rates problem, where we are given the rate of k in respect to time, and we have to find the rate of A(x) (parent function) in respect to time. We know when k = 2, A =4 from part b, and we have the area of A as a function of k from part a. We pretty much derived the equation of A and multiplied it by the rate given, and plugged in 2 for k, and the answer is 4.

That was it for the test questions. Mr. K also introduced us to a new site entitled Mathway which pretty much solves all our questions on calculus, of course you should use this for help, and not answers for upcoming exercises. Now onto our actual unit, "finding anti derivatives" we went over many derivative rules, and we anti derived them and don't forget to add +C, when anti deriving an equation, because the C stands for a constant, and this constant disappears when it is derived, so don't forget! We also anti derived some questions toward the end of class, and we also looked at a list of partial anti-derivatives rules. Here it is! Our next scribe will be Rence.

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