critical number

[samantha0417]

E(v) = (Cv^k)/(v-w)

Background: a turtle swims up stream at a speed (v) against a constant current w, the energy he uses is the equation above. C>0 and K>2

The question: find the critical pt (theres only one); is it a relative max or min.

My work

E'(v) = [kCv^(k-1)]/(v-w)

I said the critical point occurs with the turtle's speed is equal to zero. However, the second part of the question (part b) says that the critical number depends on k. I have no idea how this works. Don't you want E'(v)=0?

 

[soroban]

Hello, samantha0417!

You didn't use the Quotient Furmula for the derivative . . .

\(\displaystyle \L E(v) \:= \:\frac{Cv^k}{v\,-\,w}\)

Background: a turtle swims up stream at a speed \(\displaystyle v\) against a constant current \(\displaystyle w\).
The energy he uses is the equation above, for \(\displaystyle C\,>\,0\) and \(\displaystyle k\,>\,2\)

Find the critical point (there's only one).? .Is it a relative max or min?

\(\displaystyle \L E'(v)\;=\;\frac{(v\,-\,w)\cdot C\cdot k\cdot v^{k-1}\,-\,Cv^k\cdot1_}{(v\,-\,w)^2} \;= \;\frac{Cv^{k-1}\left[k(v\,-\,w) \,-\,v\right]}{(v\,-\,w)^2}\)


The derivative equals 0 if its numerator equals 0.
. . \(\displaystyle Cv^{k-1}\left[kv\,-\,kw\,-\,v\right] \:=\:0\)

Hence, either: \(\displaystyle \,Cv^{k-1}\:=\:0\;\;\Rightarrow\;\;\L v\:=\:0\)

. . . . . . . .or: \(\displaystyle \,kv - kw - v\:=\:0\;\;\Rightarrow\;\;(k\,-\,1)v \:=\:kw\;\;\Rightarrow\;\;\L v\:=\:\frac{kw}{k\,-\,1}\)


If \(\displaystyle v\,=\,0\), the turtle is not swimming at all.
. . He uses 0 energy . . . minimum.

Hence, \(\displaystyle v \:=\:\frac{kw}{k\,-\,1}\) is the velocity which uses maximum energy
. . and obviously its value depends on both \(\displaystyle k\) and \(\displaystyle w.\)

 

[daon]

v = w is also a cirtical number.