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MFE 2 - Recitation

创建日期：2024-06-04 17:24:19

所属分类：数学mathematics

标签： MFE 2

Recitation

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MFE 2 - Recitation 1

Do the exercises on a separate sheet of paper (or a tablet) and be clean, precise, and
explain everything, like you would for homework. Finish the exercises that you did not
have time to cover at home. Check the answer keys provided, and ask for more details on
Campuswire.
Exercise I
Let ~a = 〈1, 2〉, ~b = 〈1,−1〉, and ~〈 − 3,−2〉. Compute:
1. ~a · ~a
2. ~a ·~b
3. ~b · ~a
4. ~a · (~b+ ~c)
5. ~a ·~b+ ~a · ~c
6. (2~a) ·~b
Exercise II
Let ~u = 〈−1,−1, 1〉 and ~v = 〈2, 5, 2〉.
1. Find all vectors orthogonal to both ~u and ~v.
2. Find all unit vectors orthogonal to both ~u and ~v.
3. Find all vectors orthogonal to both ~u and ~v that have length 3.
4. Is there a vector orthogonal to ~u, ~v, and the x-axis?
Exercise III
1. For the function f(x, y, z) = x2ey+2z, find −→∇f , the directional derivative of f at
(−1, 2, 1) in the direction of 〈1,−2, 1〉, the largest rate of change of f at (−1, 2, 1),
and the direction where it occurs.
2. Let
g(x, y) = x
2 + y2
2x .
Draw the level curve of g at level 2, and −→∇g(4, 0). Comment. Using only geometry,
compute the directional derivative of g at (4, 0) in the direction of 〈0,−1〉, of 〈1, 0〉,
and of 〈−1, 1〉. You may need to know that cos pi/4 = √2/2.

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