Question

Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero.

1. At the point (1,0) in the direction of −j⃗ ,

2. At the point (−1,1) in the direction of (−i⃗ +j⃗ )/2√,

3. At the point (−1,1) in the direction of (−i⃗ −j⃗ )/2√,

4. At the point (0,2) in the direction of j⃗ ,

5. At the point (−2,2) in the direction of i⃗ ,

6. At the point (0,−2) in the direction of (i⃗ −2j⃗ )/5√,

Answer 1

1. At the point (1,0) in the direction of −j=

We are moving parallel to the contour at that point, so our z value would be unchangingat that instant, so the directional derivative is zero

2. At the point (−1,1) in the direction of (−i +j )/√2=

Moving towards lower z values, so the directional derivative is negative.

3. At the point (−1,1) in the direction of (−i -j )/√2=

Moving towards lower z values, so the directional derivative is negative.

4. At the point (0,2) in the direction of j=

Moving from contour z = 4 towards contour z = 2 means z is decreasing in that direction,so the directional derivative is negative.

5.At the point (−2,2) in the direction of i=

Moving from contour z = 8 towards contour z = 6 means z is decreasing in that direction,so the directional derivative is negative

6. At the point (0,−2) in the direction of (i −2j )/√5=

Moving from contour z = 4 towards contour z = 2 means z is decreasing in that direction,so the directional derivative is negative.

Answer 2

At the point (0,-2) the change in f with respect to x is zero since an infinitesimal increase in x does not change f. The change in f with respect to y isnegative because an infinitesimal increase in y decreases f. A directional derivative is the dot product of the gradient evaluated at a point, which we didabove, with the unit direction desired. The direction given is not a unit direction, but it has a positive x component and a negative y component. When thedot product is executed you have zero times that positive x component which is zero, added to a negative times a negative, which means that the directionalderivative is a positive number.