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Math 116 — First Midterm Exam (October 5, 2012), Exams of Mathematics

Baddi University of Emerging Sciences and TechnologiesMathematics

The instructions and problems for the first midterm exam of math 116. The exam covers various topics in mathematics, including calculus, vectors, and coordinate systems. Students are required to describe and sketch surfaces, find intersections of curves, calculate limits, and convert between different coordinate systems.

Typology: Exams

2012/2013

Uploaded on 03/07/2013

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Math 116 — First Midterm
October 5, 2012
Name:
Instructor: Section:
1. Do not open this exam until you are told to do so.
2. This exam has 8 pages including this cover AND IS DOUBLE SIDED. There are 6 problems.
Note that the problems are not of equal difficulty, so you may want to skip over and return
to a problem on which you are stuck.
3. Do not separate the pages of this exam. If they do become separated, write your name on
every page and point this out when you hand in the exam.
4. Please read the instructions for each individual problem carefully. One of the skills being
tested on this exam is your ability to interpret mathematical questions.
5. Show an appropriate amount of work (including appropriate explanation). Include units in
your answer where that is appropriate. Time is of course a consideration, but do not provide
no work except when specified.
6. You may use any previously permitted calculator. However, you must state when you use
it.
7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch
of the graph that you use.
8. Turn off all cell phones and pagers, and remove all headphones and hats.
9. Remember that this is a chance to show what you’ve learned, and that the questions are
just prompts.
Problem Points Score
1 18
2 20
3 22
4 14
5 14
6 12
Total 100
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Download Math 116 — First Midterm Exam (October 5, 2012) and more Exams Mathematics in PDF only on Docsity!

Math 116 — First Midterm

October 5, 2012

Name:

Instructor: Section:

  1. Do not open this exam until you are told to do so.
  2. This exam has 8 pages including this cover AND IS DOUBLE SIDED. There are 6 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.
  3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out when you hand in the exam.
  4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions.
  5. Show an appropriate amount of work (including appropriate explanation). Include units in your answer where that is appropriate. Time is of course a consideration, but do not provide no work except when specified.
  6. You may use any previously permitted calculator. However, you must state when you use it.
  7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph that you use.
  8. Turn off all cell phones and pagers, and remove all headphones and hats.
  9. Remember that this is a chance to show what you’ve learned, and that the questions are just prompts.

Problem Points Score

Total 100

  1. [18 points]

a. [6 points] Describe and sketch the surface defined by z + 88 = − 9 x^2 + 54x − 4 y^2 − 16 y.

b. [6 points] Write down the parametrization r(t) for the intersection of this surface with the surface z = x^2.

c. [6 points] Calculate the equation for the tangent plane to the parabolic bowl at (4, − 2 , 0).

  1. [22 points] Given a point in rectilinear coordinates (x, y, z) there is a function f (x, y, z) = (r, θ, w) which gives us the cylindrical coordinates, and another function g(x, y, z) = (ρ, θ, φ) which gives us the spherical coordinates. The Martians have a slightly different way of describ- ing vectors (for a full description, read “The Martian Chronicles”). They use (a, b, c) which satisfy (x, y, z) = M (a, b, c) = (3a cos(b) sin(c), 4 a sin(b) sin(c), 7 a cos(c)). a. [8 points] Calculate the Jacobians of f and M.

b. [6 points] Plot the rectilinear coordinate (1, 1 ,

  1. and convert to cylindrical and spherical coordinates.

c. [8 points] Sketch the Martian equation a = 2.

  1. [14 points] Calculate the following limits, or demonstrate that they do not exist:

a. [7 points]

f (x, y) =

{ (^) y 4 x^4 +y^2 if (x, y)^6 = (0,^ 0) 0 , if (x, y) = (0, 0). What is lim(x,y)→(0,0) f (x, y) or does it not exist?

b. [7 points] Does lim (x,y)→(0,0)

ln(1 − x^2 − y^2 ) + x^2 + y^2 x^2 + y^2 exist? If so, what is it?

b. [7 points] Circle the plot of the level curves of f (x, y). Briefly explain your choice.

  1. [12 points]

a. [6 points] Sketch the image of the unit square under f (x, y) = (x − y^2 , y).

b. [6 points] Sketch the level curves for f (x, y) = x

2 4 +^

y^2 9 with^ c^ = 0,^ 1.