So this week in Calculus we reviewed Monday and Tuesday for the multiple-choice test on Wednesday and the free response test on Thursday. On Friday, we reviewed limits because a majority of the class did not do well on that.
Limits:
Infinite limits:
1) if degree of top = degree of bottom:: the answer is the top coefficient over bottom coefficient.
2) top degree > bottom degree = +- infinity
3) top degree < bottom degree = 0
Finite Limits:
Plug in x
Examples:
limit x à infinity x / (x^3 + 9) = 0
limit x à 3 x / (x^3 + 9) = 3 / 36 = 1 / 12
Something I still do not completely understand is related rates. I get how to do them, just don’t understand some of the properties.
Example: Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is dA/dt contant? Explain.
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Well this problem isn't too too hard. Basically you work it like any other related rates problem. You take the derivative of the area of a circle.
ReplyDeleteThe area of a circle is pir^2 right? So the derivative is
dA/dt = 2rpi dr/dt
So, now it says if dr/dt is constant...is da/dt constant? Well lets say dr/dt is 1 for our purposes.
Now we plug in different values for the radius and see if we get the same thing for da/dt (which would make it constant)
Plugging in 1 for r gives you 2pi
Plugging in 2 for r gives you 4pi
2pi is not the same as 4pi, therefore dA/dt is not constant because when the radius changes, so will it.
When dealing with related rates, always identify what you are given.
ReplyDeleteSo lets look at an example problem:
The radius of a circle is 5. and the radius is increasing at a rate of 10. They want to know the rate at which A is increasing.
the Formula would be A=(pi)r^2
da/dt= (pi)2r dr/dt
da/dt= (pi) 2(5) (10)
so da/dt would equal to 100.
hope this helps!
Okay you'd look at this problem like every other related rates problem.
ReplyDeleteThe area of a circle is pir^2, correct?
so taking the derivative you get dA/dt = pi2rdr/dt.
Now it asks if dr/dt is constant would dA/dt be constant? Well you plug in different values for r and see if you get the same answer. If you do it is constant (obviously) and if not it is not constant.
dA/dt is not constant as the radius changes.