Are the following statements true or false?
1. If u⃗
is a unit vector, then fu⃗ (a,b)
is a vector.
2. ∇f(a,b)
is a vector in 3-dimensional space.
3. Suppose fx(a,b)
and fy(a,b)
both exist. Then there is always a direction in which the rate of change of f
at (a,b)
is zero.
4. If u⃗
is perpendicular to ∇f(a,b)
, then fu⃗ (a,b)=⟨0,0⟩
.
5. The gradient vector ∇f(a,b)
is tangent to the contour of f
at (a,b)
.
6. If f(x,y)
has fx(a,b)=0
and fy(a,b)=0
at the point (a,b)
, then f
is constant everywhere.
7. fu⃗ (a,b)
is parallel to u⃗
.
8. fu⃗ (a,b)=||∇f(a,b)||
1. False
2. True
3. False
4. True
5. False
6. True
7. True
8. False
1. True. If u⃗ is a unit vector, then fu⃗ (a,b) will also be a vector since it is a scalar multiple of u⃗.
2. False. ∇f(a,b) is a vector in 2-dimensional space, not 3-dimensional space. It represents the gradient of f at the point (a,b).
3. False. While it is true that in some directions the rate of change of f at (a,b) may be zero (called critical points), it is not always guaranteed. There may be directions where the rate of change is non-zero.
4. False. If u⃗ is perpendicular to ∇f(a,b), then fu⃗ (a,b) will not necessarily be equal to ⟨0,0⟩. It depends on the specific values and direction of u⃗.
5. True. The gradient vector ∇f(a,b) is indeed tangent to the contour (level curve) of f at (a,b). It points in the direction of the steepest increase of f at that point.
6. True. If fx(a,b) = 0 and fy(a,b) = 0 at the point (a,b), it means that the partial derivatives of f with respect to x and y are both zero at that point. This implies that f is constant in every direction at that point.
7. True. The vector fu⃗ (a,b) is parallel to u⃗ since it is a scalar multiple of u⃗.
8. False. fu⃗ (a,b) is not necessarily equal to the magnitude of ∇f(a,b) (||∇f(a,b)||). It depends on the specific values and direction of u⃗.