Calculus
posted by Isabella .
1. Use the Mean Value Theorem to find an xvalue where the instantaneous slope=average slope over the given interval:
f(x)=(9x^2)^(1/2) on [3,2]
I have done three chapters of Calculus already and I never have had to check for extraneous roots. So, my question is, when exactly do I check for extraneous roots? (ex/ when I was looking for critical points and I root my answer, I didn't have to check for extraneous roots)
2. Can someone show me how e^(ln((e^51)/5))=(e^51)/5?

Calculus 
Steve
by definition, e^(ln x) = x
for the MVT,
f(3) = 0
f(2) = √5
average rate is thus √5/5 or 1/√5
So, you want to find where the slope of f is 1/√5
f' = x/√(9x^2)
so, find x where f' = 1/√5
x/√(9x^2) = 1/√5
square both sides. (This is where an extraneous root might appear, because both (√a)^2 = a and (√a)^2 = a)
x^2/(9x^2) = 1/5
5x^2 = 9x^2
6x^2 = 9
x = ±3/√6
Now you have to check which solution you want.
f'(3/√6) = (3/√6)/√(9  9/6)
= 3/√6 / √(45/6)
= 3/√45
= 1/√5
And of course, f'(3/√6) = +1/√5
See the graphs at
http://www.wolframalpha.com/input/?i=plot+y%3D%E2%88%9A%289x^2%29%2C+y%3D1%2F%E2%88%9A5%28x%2B3%29%2C+y+%3D+1%2F%E2%88%9A5+%28x+%2B+3%2F%E2%88%9A6%29+%2B+%E2%88%9A%2845%2F6%29+where+3+%3C%3Dx+%3C%3D+2
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