Math 225: Calculus III	Quiz 4    2/18&2/20/96

<< "Calculus`Math225`"; i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};

1. Imagine yourself walking on top of the graph of
    f[x_,y_] = x^2 y + x^3 + x y^2
When you are over the point {1,1}, which direction is the steepest climb: the positive x-direction or the positive y-direction? (Justify your answer!).

Solution

We need only compare the partial derivatives of f with respect to x and y evaluated at the point {1,1}, since these numbers give the slopes of f in the poisitive x- and y-directions.

f[x_,y_] = x^2 y + x^3 + x y^2;

D[f[x, y], x]

% /. {x -> 1, y -> 1}

D[f[x,y],y]

% /. {x->1, y->1}

The derivative in the x-direction is greater, so that direction is the steeper climb.

2.  Let z = y^3 Sin[x^2 + y]. Suppose x == g[u,v],
y == h[u,v], g[0,0] == Sqrt[Pi], h[0,0] = Pi, and
D[x,u] == -2, D[x,v] == 3, D[y,u] == 5, D[y,v] == -1.
Compute D[z,u] at {u,v} = {0,0}$.

Solution

This is a direct application of the chain rule. We use the function Dt[] which assumes the variables depend on each other.

Clear[x,y,z]

z = y^3 Sin[x^2 + y];

Dt[z, u]

Now we plug in the given values:

% /. {x->Sqrt[Pi], y->Pi, Dt[x,u]->-2, Dt[y,u]->5}