21256: Implicit partial dierentiation
Clive Newstead, Thursday 5th June 2014
Introduction
This note is a slightly dierent treatment of implicit partial dierentiation from what I did in
class and follows more closely what I wanted to say to you. Im doing
18.01 Exam 2
Name:
Problem 1:
/20
Problem 2:
/40
Problem 3:
/30
Problem 4:
/10
Total:
/100
Instructions: Please write your name at the top of every page of the exam. The exam is closed
book, calculators are not allowed, but you are allowed to use your pre
18.01 EXAM 1
SEPTEMBER 18, 2003
Name:
Problem 1:
/20
Problem 2:
/25
Problem 3:
/10
Problem 4:
/20
Problem 5:
/15
Problem 6:
/10
Total:
/100
Please write the hour of your recitation.
Hour:
Instructions: Please write your name at the top of every page of th
Academic Integrity
This is a very important topic, and one about which I feel quite strongly.
For the purposes of this course adhering to Carnegie Mellon's Statement on
Academic Integrity means primarily one thing: making sure that work you turn
in for cr
21111: Calculus I
Scheduling Information
Exam Time:
Lecture 1: 11:3012:20 (regular class time) in PH 226C (regular class
room).
Review Session:
The class period on Monday 26 September will be used for exam review.
Homework: You may pick up your graded H
Assignments: Week #1
Reading:
Monday: Review Techniques of Differentiation (Chapter 3).
Wednesday: Section 6.1.
Friday: Section 6.2.
Exercises: Due in Recitation on Thursday 1 September Tuesday 6
September.
To find the exercises for Week #1, first read th
Assignments: Week #5
Reading:
Monday: Exam Review.
Wednesday: Exam #1.
Friday: Section 6.8.
Additionally: Read the section "Confusion about Notation" from Eric
Schechter's The Most Common Errors in Undergraduate Mathematics.
Exercises: There will be no as
Assignments: Week #2
Reading:
Monday: Labor Day. No class.
Wednesday: Section 6.3.
Friday: Section 1.5, Examples 1 and 2. Section 1.8, Examples 5 and 6.
Section 2.6, Examples 1, 2 and 3.
Exercises: Due at the start of recitation on Thursday 8
September Tu
Assignments: Week #4
Reading:
Monday: Section 6.6.
Wednesday: Section 6.7.
Friday:
Additionally: Read the section "Algebra Errors" from Eric
Schechter's The Most Common Errors in Undergraduate Mathematics.
Exercises: Due in recitation on Thursday 22 Septe
18.01 Exam 3
Name:
Problem 1:
/20
Problem 2:
/30
Problem 3:
/25
Problem 4:
/25
Total:
/100
Instructions: Please write your name at the top of every page of the exam. The exam is closed
book, calculators are not allowed, but you are allowed to use your pre
18.01 Exam 1
Name:
Problem 1:
/15
Problem 2:
/10
Problem 3:
/15
Problem 4:
/20
Problem 5:
/25
Problem 6:
/15
Total:
/100
Instructions: Please write your name at the top of every page of the exam. The exam is closed
book, calculators are not allowed, but y
21256: Lagrange multipliers
Clive Newstead, Thursday 12th June 2014
Lagrange multipliers give us a means of optimizing multivariate functions subject to a number of
constraints on their variables. Problems of this nature come up all over the place in rea
21256: Unconstrained extrema
Clive Newstead, Wednesday 11th June 2014
Unconstrained extrema in two dimensions
Local extrema
Intuitively, a neighbourhood of a point (a, b) is a set in which there is some positive distance so
that you can move in any direc
21256: Partial dierentiation
Clive Newstead, Thursday 5th June 2014
This is a summary of the important results about partial derivatives and the chain rule that you
should know.
Partial derivatives
Given a function f of n variables, x1 , x2 , , xn , the
Homework 1
Part I. For the rst part, we will try to understand the idea of the area under the curve as a function and
the fundamental theorem of calculus.
1. Consider the function f (x) = 2x.
i. Graph this function on an xyplane.
1
ii. Determine
f (x) dx
Homework 4
Part I. Sketch the region enclosed by the curves, and then nd its area.
1. y = x
y = x2 2
2. x = y 4
y = 2x
y=0
3. Use calculus to nd the area of the triangle with the vertices (0, 0), (3, 1), (1, 2).
4. If the birth rate of a population is b
Homework 3
Part I. (From 7.1) Use integration by parts to solve the following integrals.
1.
x cos(5x) dx (use u = x and dv = cos(5x) dx)
2.
sin1 (x) dx (use u = sin1 (x) and dv = dx)
Part II. (From 5.5 and 7.1) For the problem followings, use usubstituti
Homework 2
Part I. Evaluate the integral by making the given substitution.
1.
e3x dx where u = 3x.
2.
x3 (2 + x4 )5 dx where u = 2 + x4 .
3.
cos3 () sin() d where u = cos().
4.
dt
where u = 1 6t.
(1 + 6t)4
Part II. Evaluate the integral.
5.
sin(2x) dx
7.
Scheduling Information
Exam Time:
11:3012:20 (regular class time) in PH 226C (regular class room) on
Monday 17 November.
Review Session:
Time: in class on Friday 18 November.
Place: PH 226C (regular class room).
Homework: Check with your TA to find out w
21112: Calculus II
Scheduling Information
Exam Time:
Tuesday 13 December, 8:3011:30
Exam Place:
DH A310
Review Session:
Review will be held in class on Wednesday and Friday, during the last
week of classes. The recitations during the last week of classe
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