EXERCISESProblem 1:Use the definition of directional derivative to compute the directional derivative of the function, |
> with(linalg):f := (x,y) -> x/y: 'f(x,y)' = f(x,y);
f(x, y) = x/y
| at the point, |
> r := vector([6,-2]):
| in the direction of the vector, |
> v := vector([-1,3])/sqrt(10):
Solution 1Let's call Dvf the directional derivative. From the definition we have, |
> Dvf := Limit('(f(r+h*v)-f(r))/h',h=0);
f(r + h v) - f(r)
Dvf := Limit -----------------
h -> 0 h
| where in our case, |
> p := evalm(r+h*v): Dvf := Limit((f(p[1],p[2])-f(r[1],r[2]))/h,h=0);
1/2
6 - 1/10 h 10
----------------- + 3
1/2
-2 + 3/10 h 10
Dvf := lim ---------------------
h -> 0 h
| which simplifies to, |
> Dvf := Limit(simplify((f(p[1],p[2])-f(r[1],r[2]))/h),h=0);
1/2
10
Dvf := lim 8 ---------------
h -> 0 1/2
-20 + 3 h 10
| and the directional derivative is then, |
> Dvf := limit((f(p[1],p[2])-f(r[1],r[2]))/h,h=0);
1/2
Dvf := - 2/5 10
| Ofcourse the formula with the grad gives the same answer, |
> formula := innerprod(grad(f(x,y),[x,y]), v);
1/2
10 (y + 3 x)
formula := - 1/10 ---------------
2
y
> answer := subs({x=6,y=-2},formula);
1/2
answer := - 2/5 10