Surely someone here has experience in these problems.
I would appreciate any help on the following problem:
Suppose I am descending a mountain, and for every 3 meters I travel NW, I climb 1/2 a meter, and for every 2 meters I travel NE, I descend 1/4 of a meter.
a)What direction should I start for fastest descent?
b)If I travel in the direction of fastest descent at 2 meters/sec., what will be my rate of descent?
c)derive an expression for this rate of descent as a function of the direction traveled and the speed in that direction.
d)in what directions should I start in order NOT to go up or down?
I know how to find directional derivatives and gradient vectors but I am used to questions in a format where I am given the functions. I know the gradient vector gives the direction of fastest increase of such a function (travelling perpendicularly down level curves). However, I am not very good at applying my knowledge!
I can start it, perhaps, but I wouldn't be surprised if i was heading in the wrong direction already (bad pun): well i know the units for my directional derivative are in meters in altitude per meters travelled, one directional derivative might look like a general gradient vector times 1/sqrt(2) times the unit vector <-1,1>=(1/2)/3, and perhaps the other directional derivative looks like another general gradient vector with 1/sqrt(2) times the unit vector <1,1>=(-1/4)/2, assuming i have laid out my N,E,S,W directions according to the + y axis, + x axis, - y axis, and -x axis, respectively. Did I do all this wrong and/or could I have help from here?
Call the function f so that its gradient is . Taking the positive y axis as north and the positive x-axis east, a unit vector NW would be . The derivative in that direction is given by . A unit vector NE would be so the derivative in the direction is .
Solve those two equations for and .
a)What direction should I start for fastest descent?
b)If I travel in the direction of fastest descent at 2 meters/sec., what will be my rate of descent?
c)derive an expression for this rate of descent as a function of the direction traveled and the speed in that direction.
d)in what directions should I start in order NOT to go up or down?
I know how to find directional derivatives and gradient vectors but I am used to questions in a format where I am given the functions. I know the gradient vector gives the direction of fastest increase of such a function (travelling perpendicularly down level curves). However, I am not very good at applying my knowledge!