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MFE 2 – Homework 1
To upload on Gradescope by Thursday, February 11th, 5:00 pm EDT.
The below homework specifications will be enforced. If the specifications are not respected, points
might be deducted, or the homework assignment may not be accepted for grading.
Guidelines for your work
• Write your name (as on the roster) and NetID on the first page.
• If you write on paper, use clean and new sheets of paper and take as much space as necessary.
• Number your pages in the top-right corner, such as 1/3, 2/3, 3/3.
• Use a draft and hand in your final version. Make sure that it
– is clean and legible;
– has each problem clearly indicated;
– does not have anything crossed out or contain notes in the margins;
– has solutions in which all steps are clearly shown and explained, including all steps of the
computations;
– has grammatically correct complete sentences, including punctuation and spelling;
– is written using correct mathematical terminology and notation;
– has final answers in exact forms (do not approximate unless otherwise stated).
• You may consult your classmates or other resources (including Campuswire and office hours)
for ideas on the problems; however, the solutions you turn in must be in your own words and
must reflect your own understanding. Your solutions and write-ups will be checked for textual
similarities. You may not copy from, reword, or paraphrase another student’s work or any other
resource material; such conduct will be treated as a violation of academic integrity. Remember
that you will not learn anything by simply copying, rewording or paraphrasing another person’s
work.
Guidelines for Gradescope
• You can either write on blank or lined paper, use a tablet, or type your assignment in LaTeX.
• Your work should be uploaded as a single PDF file (not as separate photos).
• If you write down on paper, scan your work using a scanner or an app such as Camscanner. Make
sure that the scans are not blurry and are in portrait mode.
• When you upload this file, match each exercise with the corresponding pages.
1
Exercise I: Vectors
1. Consider the vectors ~u = h−2, 3, 1i and ~v = h1, −1, 4i. Compute ~u + ~v , ~u − ~v and
(~u + ~v ) · (~u − ~v ).
2. Consider the vectors ~u = h3, 2i and ~v = h3, −2i. Compute ~u + ~v , ~u − ~v and (~u + ~v ) ·
(~u − ~v ). What do you notice?.
3. In general, for any two vectors ~u, ~v , compute and simplify (~u + ~v ) · (~u − ~v ) (your
final result should not contain any dot product).
4. Under what condition are ~u + ~v and ~u − ~v orthogonal (perpendicular)? Does this
agree with what you observe in Questions 1 and 2?
Exercise II: More vectors
−→ −→
1. Consider an equilateral triangle ABC, with side length 2. Compute AB · AC in
exact form (no approximation). You can draw a picture, and use classical facts
about equilateral triangles. You may also need to look up the value of the cosine of
specific angles.
2. Find all vectors that are orthogonal to both ~u = h2, 2, −1i and v = h0, 4, −1i. Then
find all such unit vectors.
3. Assume that you have a function f (x, y), and that you know that a level curve is
→
−
the straight line y = 2x + 3. You also know that | ∇f (2, 7)| = 5. What are the
→
−
possible values of ∇f (2, 7)? You can illustrate with a picture.
Exercise III: Gradient
Let
f (x, y, z) =
ln y + 1
.
x + z2
1. Compute the gradient of f .
2. Compute the directional derivative of f at (0, 1, 2) in the direction of the vector
h1, 2, 2i.
3. Find the maximum value of D~u f (2, 1, 1) and for which ~u it is attained.
2
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