Use the contour diagram of in the figure below to decide if the specified direct
ID: 3346618 • Letter: U
Question
Use the contour diagram of in the figure below to decide if the specified directional derivatives below are positive, negative, or approximately zero.
(a) At point , in direction : is ? positive negative approximately zero
(b) At point , in direction : is ? positive negative approximately zero
(c) At point , in direction : is ? positive negative approximately zero
(d) At point , in direction : is ? positive negative approximately zero
(e) At point , in direction : is ? positive negative approximately zero
(f) At point , in direction : is ? positive negative approximately zero
Explanation / Answer
1. At point (?2,2), in direction ~i.
Moving from contour z = 6 towards contour z = 4 means z is decreasing in that direction,
so the directional derivative is negative.
2. At point (0, ?2), in direction ~j.
Moving from z = 4 towards z = 2, so directional derivative is negative.
3. At point (?1,1), in direction~i +~j
We are moving parallel to the contour at that point, so our z value would be unchanging
at that instant, so the directional derivative is zero.
4. At point (?1,1), in direction ?~i +~j.
Moving towards higher z values, so the directional derivative is positive.
5. At point (0, ?2), in direction ~i + 2~j.
Moving more in the y direction than x, or towards lower z values, so the derivative is
negative.
6. At point (0, ?2), in direction ~i ? 2~j.
Moving towards higher z values, so the derivative is positive.
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