Finding maximum and minimum values of vel. and acc. of a particle on an ellipse

[JoeSabs]

Homework Statement

A particle moves around the ellipse ((y/3)^2)+((z/2)^2)=1 in the yz-plane in such a way that its position at time t is r(t)=(3cost)j+(2sint)k. Find the maximum and minimum values of |v| and |a|. (Hint: Find the extreme values of |v|^2 and |a|^2 first and take square roots later.)

Homework Equations

v(t)= (-3sint)j+(2cost)k
a(t)= (-3cost)j+(-2sint)k
|v|= sqrt(((-3sint)^2)+((2cost)^2))
|a|= sqrt(((-3cost)^2)+((-2sint)^2))

The Attempt at a Solution

I got those equations, but our teacher never showed us how to find the extrema of these equations. Coming out of a horrible calc II class, I'm not exactly sure how to evaluate these. Our current teacher has a bad habit of teaching the class after the homework's due...

 

[gabbagabbahey]

What do you know about local minimums and maximums of a function? What is the derivative of a function at a local max or min? v(t) and a(t) are both periodic functions. do periodic functions have any absolute maximums or minimums?

 

[JoeSabs]

Right, so the derivative of a local max/min is 0. does that mean i have to find the third derivative to find the max/mins for a(t)? As for periodic functions, they repeat, so they have a max and min that repeats. I'm not asked for the absolute max/min, so I'm assuming they want the relative.

so after plugging zero in for a(t) (to find relatives for v(t)) and a'(t) (to find relatives for a(t)) I got 3 and 2, respectively. I checked the back, and it states that each has a max and min of 3 and 2, respectively. What am I missing that I'm only getting half of the answer?

 

[gabbagabbahey]

Periodic functions won't have absolute max/mins but will have local max/mins. At a local max or min, the 1st derivative vanishes. At a local max, the second derivative is negative, and at a local min, the second derivative is positive.

So, just find out where the 1st derivatives of v(t) and a(t) are zero, then find out whether those values are maximums or minimums either by taking the second derivatives and determining whether they're pos or neg there or plugging the values of t into v(t) and a(t) and seeing when you get a larger v(t) or a(t) and when you get a smaller v(t) and a(t).

 

[JoeSabs]

I'm working backwards at the moment, plugging the answers given into the problem. They don't give derivatives which equal zero.EDIT:

Figured it out. Thanks!

My Prof Said I was wrong :( Can someone show me the work for this problem? I'm not sure but I think I may have made a sign error.

 

[HallsofIvy]

Show what you did and what answer you got. I'm a bit confused about why you "coming out of a horrible Calculus II course" would affect this problem. Most people learn to find max and min in Calculus I. It certainly shouldn't be the responsibility of your Calculus III teacher to show you how to find max and min of a function of one variable.