Calculus III Midterm 1 Practice Exam: Columbia University MATHV1201, Study notes of Calculus

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COLUMBIA UNIVERSITY

MATHV

Calculus III

Fall 2016

Midterm 1 (practice)

Instructor: Guillaume Barraquand

Your name:

UNI:

Question: 1 2 3 4 5 6 Total

Points: 12 10 10 5 7 6 50

Score:

Instructions:

• Please write your UNI on every page.
• Unless stated otherwise, Show your work in every question.
• Please write neatly, and put your final answer in a box.
• Books, notes, calculators, smartphones or any other electronic devices are not allowed.

Exercise 2............................................................................... Consider points in R^3 with coordinates A(1, 0 , 1), B(− 1 , 2 , −4), P (− 1 , 1 , 0), Q(1, 0 , 3) and R(1, 0 , −2). (a) (3 points) Find a parameteric representation of the line going through A and B.

(b) (4 points) Find an equation of the plane passing through P, Q and R.

(c) (3 points) Find the distance from the point A to the plane passing through P, Q, and R.

Exercise 3....................................................................... 5 points

(a) (4 points) Find an equation for the set of points equidistant to the plane z = − 2 and the point A(2, 3 , 2). (b) (1 point) What type of surface is it?

Exercise 4....................................................................... 5 points Assume that a parallelogram has edges parallel to the axis and is such that the points (4, 2 , 0) and (5, − 2 , 1) are some of its vertices. Find the volume of that parallelogram.

Exercise 6............................................................................... Find the complex conjugate and the modulus of the complex numbers

(a) (2 points) (1 − i)^3

(b) (2 points) 1 −

3 i 1 +

3 i

(c) (2 points) (2 − 3 i)(2 + 3i)

In case you need more space