## A little calculus help

Mechanics, discrete, statistics, ...
neonash7777
Posts: 75
Joined: Fri Aug 11, 2006 2:53 am
Location: ND
Contact:

### A little calculus help

Okay so I have two problems in calc that I need a little help with.
If you put solutions, please put in the hide tags, clues and tips can be left open. I prefer solutions for checking my work rather than using them to get answers, otherwise I would be screwed on the quiz haha ~Thanks

1. Differntiate f(x) = sinx^(cos x) + cosx^(sin x)

I tried implicit differntiation.
I tried to do the ln of both sides but I can't distribute the ln amongst the + sign. ln(a+b) is not = to ln a + ln b.
So how should I handle this?

2. In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation
ÃƒÂ¢Ã‹â€ Ã¢â‚¬Â P/ÃƒÂ¢Ã‹â€ Ã¢â‚¬Â t = r0 * (1 - (P(t)/Pc)) * P(t) - ÃƒÆ’Ã…Â¸P(t)

Where r0 is the birth rate of the fish, Pc is the maximum population the pond can sustain, and ÃƒÆ’Ã…Â¸ is the percentage of the population harvested.

A) What value of dP/dT corresponds to a stable population?

B) If the pond can sustain 10,000 fish, the birth rate is 5%, the harvesting rate is 4%, then find the stable population level.

C)What happens if ÃƒÆ’Ã…Â¸ is raised to 5%

~~~~~
I'm thinking for C that the population will die out, but there's nothing more than a hunch to that claim.
Phi, it's a whole "h" of a lot cooler than pi!

hk
Posts: 10445
Joined: Sun Mar 26, 2006 9:34 am
Location: Haren, Netherlands

### Re: A little calculus help

1) You can write: sin(x) as e^ln(sin(x), then sin(x)^cos(x)=(e^ln(sin(x)))^cos(x)=e^(cos(x)*ln(sin(x))
Use product and chain rule to find the answer.
Do the same with the other one after the + sign
Then use (f+g)'=f'+g'

2)
The key question is question A.
The population is constant if &Delta;P/&Delta;t=0.

For B:
fill in all given constants and require RHS to be 0. This gives a quadratic to be solved.

creativemind
Posts: 1
Joined: Sun Apr 12, 2009 2:39 pm

### Re: A little calculus help

no need to go for HK's method...

its very simple to solve f(x) = sinx^(cos x) + cosx^(sin x) , ya its implicit fucntion all u need to do is to find derivative of each summand separately and then add them up..

like put
sinx^(cos x)=u and
cosx^(sin x)=v
and we know that (u+v)'=u'+v'
where u' and v' can be calculated by taking log on both sides as ur mentioned....

cheers..!!!

hk