Consider the function f (x comma y )equals2 x squared minus 5 y squared minus 7 and the point (negative 2 comma negative 1 ). a. find the unit vectors that give the direction of steepest ascent and steepest descent at p. b. find a vector that points in a direction of no change in the function at p.
given the function
we calculate the gradient
for each term we consider all variables different to the one we are derivating as constants. For each term we have
gives the direction of maxium increase.
a) with x = -2, y= 1
which magnitude is
so the unitary vector in the direction of the steepest ascent is
and the unitary vector in the direction of steepest descent is
finally, the vector in no change direction is basically doing one of the following possibilities with :
if we have a vector <a,b> the perpendicular vector (direction of no change) will be either <-a,b> or <a,-b>
so i will select <-a,b>
The inverse is
ps, the ^ means "to the power of"
vertex form is
basically complete the squaer
move 7 over to oth er side
look at inner to coplete square
make sure n umber in front of x^2 is 1, it is
now divide the constant in front of the x term by 2 and square it (-6/3=-3, (-3)^2=9)
now add that to both sides
comlete the squaer