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Calculus 3 Final Exam at Columbia University, Exams of Calculus

Columbia CollegeCalculus

The final exam for Calculus 3 at Columbia University, taken on December 17, 2019. The exam consists of 8 questions divided into 3 parts, covering material from midterms 1 and 2, as well as material covered after midterm 2. The exam is worth 100 points and students have 150 minutes to complete it. Allowed materials include writing utensils, scratch paper provided, water and snacks, a non-graphing or programmable calculator, and a double-sided sheet of notes of A4 size. The exam covers topics such as lines, planes, parametric equations, limits, and ellipsoids.

Typology: Exams

2018/2019

Uploaded on 05/11/2023

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Columbia University
Department of Mathematics
UN 1201, Section 7– Final exam
Calculus 3
Instructor: Prof. Inbar Klang
December 17, 2019
Name:
UNI:
Section:
This exam contains 18 pages (including this cover page) and 8 questions. Part 1 consists of ques-
tions 1 and 2, which correspond to midterm 1 material. Part 2 consists of questions 3, 4, and 5,
which correspond to midterm 2 material. Part 3 consists of questions 6, 7, and 8, which correspond
to material covered after midterm 2. The total number of points is 100. You have 150 minutes to
complete the exam. Make sure to only have allowed materials with you: writing utensils; scratch
paper I provided; water and snacks; a calculator, but not graphing or programmable; a double-sided
sheet of notes of A4 size. All other items must be placed at the front of the classroom. Please show
your work. Good luck!
Distribution of points
Question Points Score
1 15
2 15
3 10
4 10
5 10
6 15
7 15
8 10
Total: 100
1
pf3
pf4
pf5
pf8
pf9
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Partial preview of the text

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Columbia University

Department of Mathematics

UN 1201, Section 7– Final exam

Calculus 3

Instructor: Prof. Inbar Klang

December 17, 2019

Name UNI::

Section:

This exam contains 18 pages (including this cover page) and 8 questions. Part 1 consists of ques-tions 1 and 2, which correspond to midterm 1 material. Part 2 consists of questions 3, 4, and 5, which correspond to midterm 2 material. Part 3 consists of questions 6, 7, and 8, which correspondto material covered after midterm 2. The total number of points is 100. You have 150 minutes to complete the exam. Make sure to only have allowed materials with you: writing utensils; scratchpaper I provided; water and snacks; a calculator, but not graphing or programmable; a double-sided sheet of notes of A4 size. All other items must be placed at the front of the classroom. Please showyour work. Good luck!

Distribution of points Question Points Score 1 15 2 15 3 10 4 10 5 10 6 15 7 15 8 10 Total: 100 1

Part 1

  1. Consider the two lines r 1 (t) =< 1 + 2t, t, 4 − t > and r 2 (t) =< 1 + t, − 3 t, 4 + 2t >. (a) (10 points) Find an equation for the plane that contains the two lines r 1 (t) and r 2 (t). (b) (5 points) Find cosine of the angle between the lines. (No need to simplify– for example, “cos(θ) = √ 12 √√^855 ” is an acceptable format for an answer.)
  1. (a) (10 points) Find an equation for the sphere consisting of points whose distance to (3is twice their distance to (0, 0 , 0). (For example, (1, 1 , 1) is such a point.) What are its, 3 , 3) center and radius? (b) (5 points) Find an equation in polar coordinates for the circle with center (03. , 0) and radius

(cot’d)

(cot’d)

  1. (10 points) Find a point on the surfaceto the plane 10x + 8y − 2 z = 0. z = 5sin(x) − 2 y^2 where the tangent plane is parallel
  1. (a) (5 points) Find parametric equations for the tangent line to the following parametric curveat the point (1, 3 , −1).

r(t) =< et, 3 − t^3 , 2 t − 1 > (b) (5 points) Compute the following limit, or prove that it doesn’t exist. lim(x,y)→(0,0)^3 xxsin (^2) + (yy 2 )

(cot’d)

(cot’d)

  1. Let f (x, y) = x^2 y + 2x^2 − y. (a) (10 points) Find the local minimum points, local maximum points, and saddle points of f. (b) (5 points) At the point (1maximal? Find the value of this directional derivative., 1), in which direction is the directional derivative of f (x, y)
  1. (a) (5 points) Suppose z = x^2 + y, x(s, t) = 2st, and y(s, t) = s + t. Find ∂z ∂s (1,^ 0) (That is, when s = 1 and t = 0.) (b) (5 points) Find all complex solutions to the equation w^2 − 6 w + 13 = 0.

(cot’d)