In calculus last week, we learned about derivatives, instantaneous speed, and average speed. We learned thirty-six formulas for derivatives, and also learned some for instantaneous speed and average speed. Although we learned many formulas, we only used about ten last week for derivatives. The two most important formulas that we learned, in my opinion, were quotient rule and product rule.
The thing that I am most comfortable with is average speed. The difference between instantaneous and average speed, is that instantaneous speed occurs only at one instant, while average speed occurs over a certain period of time. For average speed, you’re given a problem where an anvil falls off the roof onto Ryan G.’s head and then you are asked to find the average speed during the first 2 seconds of falling if y=16t^2 describes the fall. Your x’s in the problem would be (0, 2) because you are asked to find the average speed in the first two seconds. Then, to find your y’s, you would simply plug in your first x to the problem y=16t^2, which is 0, to find your first y. (y=16t^2, y=16(0)^2, y=0) After that, you plug in your second x to find your second y. (y=16t^2, y=16(2)^2, y=16(4), y=64) Therefore, your y’s in the problem would be (0, 64). After that, you use slope formula to find your answer. (y2-y1/x2-x1, 64-0/2-0, 64/2=32) So, your average speed would be 32 m/s.
The thing that we learned in calculus that I was most confused with was simplifying the derivatives. For example, after using the quotient rule or product rule to solve a derivative, after I plugged in everything I wouldn’t know what to do. I know that you take the derivative of the bottom times the top minus the derivative of the top times the bottom all over the bottom squared, but after I finish that first step I get really mixed up with all the cancellation rules and how to simplify it. Also, I do not know when the problem is completely simplified. I know that the easiest way to become comfortable with it is just by practicing, but if anyone can help that would also be nice. :)
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Sarah,
ReplyDeletegood explanation of average speed.
Sarah,
ReplyDeleteAbout simplifying: Start off with looking at any 1s or 0s you may have, because they are the easiest to simplify. After that then see if you can multiply or add anything at the top, if you can do so. After you do that, if it looks like the numbers at the top may be cancelled out by numbers at the bottom, then you can foil the bottom out and simplify.
The way I usually do it is to keep simplifying until it looks easy to solve when pluging in a number for x.
Sarah,
ReplyDeleteLike Ryan said look at all of that stuff but make sure when you are doing a problem with the quotient rule that you remember you can only cancel something off of the bottom if it can be taken from both sides of the problem from the top such as you can simplify something like
15x^2(sinx)-5x^3(cosx)
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5x
You can simplify the 15x^2, 5x^3, and 5x because it can be taken from both sides to get 3x(sinx)-x^2(cosx). Theres no more fraction because when you do this the 5x at the bottom no equals 1 after cancellation.
Now if any of the numbers such as the 15 or one of the 5's is a different number where all three are not divisible by each other then you can not do this.
hope this helps