Okay, wow. First off, let me just clarify two things:
1) This class is not as terrifying as I thought.
2) I cannot explain things very well at all, but I’ll try.
In my opinion, Average Speed and Instantaneous Speed are both pretty easy. Since I probably can’t explain in 300 words how to do just one of those, I’ll explain both.
Let’s start with an example problem for Average Speed. (This is probably not accurate, by the way) A wolf runs at y=93+2.3(t)^2 miles in t hours. What is it’s average speed during the first 5 seconds?
Now, break this down into steps, it’s a lot simpler.
1) State the slope formula: (y2-y1)/(x2-x1)
2) Put the time you need to find into parenthesis or brackets: [0, 5] These are your two x’s.
3) Plug x1 into your formula: y=93+2.3(0)^2 = 93 93 is your y1
4) Plug x2 into your formula: y=93+2.3(5)^2=150.5 150.5 is your y2
5) Plug back into your slope formula: (150.5-93)/(5-0)
6) Simplify: 11.5 miles per hour
Now for Instantaneous Speed. Let’s use the same problem. A wolf runs at y=93+2.3(t)^2 miles per hour. What is the instantaneous speed for t=3?
1) State the Instantaneous Speed formula:
Lim f(t-h)-f(t)
H->0 h
2) In the original problem, replace y with f(t): f(t)=93+2.3(t)2
3) Plug in everything.
Lim 93+2.3(3-h)^2-93+2.3(3)^2
H->0 h
4) Foil the parenthesis.
Lim 93+2.3(9-6h+h^2)-93+2.3(9)
H->0 h
5) Start simplifying.
Lim (93+20.7 –13.8h+2.3h^2) – (93 + 20.7)
H->0 h
6) Simplify even more.
Lim 2.3h^2-13.8h+41.4
H->0 h
7) Cancel some h’s.
Lim 2.3h-13.8+41.4
H->0
8) Simplify as much as possible and then plug in 0 for h.
Lim 2.3(0)+27.6
H->0
9) Your answer is what is left: 27.6 miles per hour.
Sorry for all of the decimals! I didn't think that there would be that many when I created the problem. I think all of that is right. I HOPE all of that is right. If someone finds a mistake, other than typing error (I typed this in word and tried to transfer....fail), please feel free to make me look like an idiot! =]
Also, for what I don’t understand: Fractions in Derivatives. When there is a radical, do you make that into a fractional exponent? And when there is a fraction as the exponent, what do you do? In general, fractions as exponents confuse me…badly. Can someone break it down into 3 year old terms for me? Please and thank you!!
~Ashley
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Very good reflection Ashley.
ReplyDeleteFractions in derivates...very tricky issue.
ReplyDeleteWhen there is a radical, you can definitely change it to a fractional exponent, usually it makes it easier to find the derivative. When you have an exponent as a fraction, you treat it like any other number.
x^(1/2)+3 is a good example. To find the derivative you just multiply x by (1/2) then subtract 1 from the exponent leaving you with... x^(-1/2) as the derivate, which simplifies to 1/(x^(1/2))
I hope this helps you better understand fractional exponents.
Okay, so when you have the 1/(x^(1/2)) left, you DON'T put a radical at the bottom? Oh wait, typing that out made me realize how stupid that was. But, is there ever a time when you change the ^(1/2) into a radical??
ReplyDeleteI don't believe that you're supposed to turn it into a radical unless specified (ask B-Rob).
ReplyDeleteAnd a heads up, I don't know if you do this, but I'll forget the negative of an exponenent, not flip it, and the negative mysteriously vanishes. So you might want to double check your work (which you should all the time!) just to make sure you didn't kill it.
okay so with the derivitive things i get confused to but milky helped me figure it out a little...so if you have something like 1/x^2the derivative you do 1/2x^-1 = -2x^-1=1/-2x^1 which is the answer
ReplyDelete