M303 Linear Algebra: Assignment 2
(1) Consider the following matrix.
1
A= 2
3
1
5
0
7
5 3
Let
7
u
= 3 .
2
Is u
in the null space of A? Is u
in the column space of A?
Solution: A vector u
nul(A) if and only if A u
= 0.
1
5
0
1
Au
= 7 2 + 3 0
The Definite Integral
M211
M211
The Definite Integral
1/6
5.2 Definite Integrals
By dividing the interval [a, b] into n subintervals of equal width
ba
, and taking sample points x1 , xn in the subintervals, where
x =
n
xi is in the i th subinterval, we ge
Indefinite Integrals
M211
M211
Indefinite Integrals
1/5
5.4 Indefinite integrals
More
general antiderivative of f (x) is denoted by
R
f (x) dx, and it is called an indefinite integral.
M211
Indefinite Integrals
2/5
5.4 Indefinite integrals
More
general an
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Your T.A.s Name:
M211
Test 2
20 October 2014
You must show your work to get full credit.
0
1. (5 points) Find the function f (x) and the point a for which f (a) = lim
h0
25 + h 5
.
h
f (x + h) f
Areas between curves
M211
M211
Areas between curves
1/3
6.1 Areas between curves
Let y = f (x) and y = g (x) be two continuous
functions such that f (x) g (x) for all a x b.
If A is the area between the curves y = f (x) and
y = g (x) and the lines x = a,
Average Value of a Function
M211
M211
Average Value of a Function
1/4
6.5 Average Value
The Average Value fave of a continuous function f from
a to b is given by
fave
M211
1
=
ba
Z
b
f (x) dx
a
Average Value of a Function
2/4
6.5 Average Value
The Mean Va
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Your T.A.s Name:
M211
Test 1
22 Sep 2014
You must show your work to get full credit.
1. (5 points) If f (x) = x 3 + 2x 5, evaluate
f (3 + h) f (3)
and simplify your answer.
h
2. (6 points) Find t
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Your T.A.s Name:
M211
Test 3
14 April 2014
You must show your work to get full credit.
1. (8 points) Tom and Jerry are standing at an intersection. Tom starts walking east at 3 mph and at the sam
Areas and Distances
M211
M211
Areas and Distances
1 / 10
5.1 Areas
By dividing the interval [0, 1] into n = 5 subintervals, estimate the area
under the curve y = x 2 from x = 0 to x = 1.
M211
Areas and Distances
2 / 10
5.1 Areas
By dividing the interval [
The Fundamental Theorem of Calculus
M211
M211
The Fundamental Theorem of Calculus
1/5
5.3 FTC1
The Fundamental Theorem of Calculus, Part 1,
FTC1
If f is continuous on [a, b] and g is defined by
Z x
g (x) =
f (t) dt
a
for a x b, then g is continous on [a,
MATH M211, Spring 2016
Calculus Syllabus
Class time and place
Section 12495
Time: MWF 10:10 11:00 am
Location: Swain East 140
Tuesday: Recitation
Instructor information
Instructor: Dr. P. Manoharan (Mano)
Office: Swain East 338
Office Hours: MWF 9:00 10:0
M211 Course Outline - Spring 2016
Text: CALCULUS: Single Variable Calculus; Early Transcendentals, 8th ed., by Stewart
Week
Dates
Sections
1
Jan. 11 15
1.1, 1.2, 1.4, 1.5
No class on Monday, January 18 Martin Luther King Jr. Day
2
Jan. 19 22
2.2, 2.3
3
Ja
MATH M119
Exam 1
Spring 2012
Part 1
Please choose the best answer for each of the following multiple choice problems. Please record your
answers on the cover sheet.
1. Find the average rate of change of f(x) = 2x 3 + 5x 7 on the interval x= 1 to x = 3.
a.
MATH M119
Spring 2011
Exam 1
Please do each of the following in the space provided. Where there is more than one step to a
problem, you MUST show your work to receive full credit.
1. A power function is a function that can be written in the form
y k x
p
.
Announcements
Exam 1 will be held Tuesday, February 5, from
2:30-3:45 pm in RH 100.
Whats on it?
All the topics covered in lecture, Webwork,
and quizzes up to that point.
1.9 Power functions,
polynomials and
proportionality
Retro Algebra
If n is a whole n
Announcements
The 1st Exam is one week from today.
February 5, 2012, from 2:30-3:45 pm in RH 100.
Seats will be assigned.
Announcements
The quiz will be at the beginning of the hour
today.
There will be a lecture after the quiz.
Limits
End behavior of pol
2.3 Interpretations of the
1st and 2nd derivatives
Approximations
Graphs and the
1st and 2nd derivatives
Recall: 1 Derivative
st
The derivative f(a) can be thought of as:
the slope of the function at x = a
the slope of the tangent line to f(x) at x = a
th
7.3 Definite integrals
Bringing numbers back
Yeah!
Them other digits dont
know how to act!
Recall: Basic rules of integration
n 1
x
n
x dx n 1 C
4
13
x
3
13
x
dx 4 C
Recall: Basic rules of integration
n 1
x
n
x dx n 1 C
x
1
dx ln x C
4
13
x
3
13
x
dx 4
Announcements
Exam One will be Tuesday, during class, in RH
100.
Please bring your university ID and a pencil.
Seats will be assigned.
Exam 1 info
Exam 1 has approximately 18 questions:
Some problems will require correct work
shown to receive credit
The r
3.1-3.2 Derivative formulas
Part I, Day 2
Polynomials,
Power functions,
Exponentials and Logs
Notation
The first derivative of f can be denoted in two
ways
df
dx
or
f
Notation
The second derivative of f can be denoted in two
ways
2
d f
dx 2
or
f
Adding fu
2.5 Profit, Cost, Revenue
Breakeven and Approximations
(a bit more than last time)
Breakeven
The breakeven point is the quantity produced
and sold which makes profit equal to zero.
Breakeven
(alt definition)
The breakeven point is the quantity produced
an
Midterm Exam Saturday
Starts at 9:00 am in BH 013.
Assigned seats.
Expect to be there 15 minutes early.
Bring pencil, ID and calculator.
Brief Review
Starting with Chapter 3
Derivative rules
Rule
Examples
Power rule
x
n
n x
k x
n
kn x
x
n 1
5
n 1
5 x
Announcements
Midterm exam is Saturday, March 2, 2013,
from 9:00-10:30 am.
Our section will be in BH 013.
You will be allowed to use a calculator such as a
TI-84 on the midterm exam.
Midterm will cover Chapters 1, 2, and 3.
Midterm Exam next Saturday
No n
Chapter 4
Using the derivative
4.1 Local maxima and minima
4.2 Points of Inflection
4.3 Global max and min
Recall: First Derivative
The first derivative can be thought of as
the slope of the function at x = a
And we can write the first derivative as
f or
7.2 Integration with substitution
This is like a puzzle.
Recall definitions
An antiderivative of f(x) is any function F(x) with
the property that F(x) = f(x).
The indefinite integral of f(x) is the generalized
antiderivative.
Example
Find the indefinite i
Midterm Exam Saturday
Starts at 9:00 am in BH 013.
Assigned seats.
Expect to be there 15 minutes early.
Bring pencil, ID and calculator.
Brief Review
Starting with Chapter 2
Fall 07
3) A supply curve is given by S(p) = 5p 200,
while the associated demand
Announcement
We will discuss Chapter 7 before Chapters 5 and
6.
Announcement
Next week, the quiz will be on THURSDAY (3/28)
instead of the Tuesday (3/26).
Section 7.2 is difficult, so we will need the whole
75 minutes to discuss it.
7.1 Antiderivatives
An
4.7 Logistic Growth
Another application of concavity
New Day, New Function
Suppose we are given a function defined for t 0
L
f (t )
kt
1 Ce
where L, C, and k are all positive constants to
be named later.
New Day, New Function
Suppose we are given a
func
Announcements
Exam Three will be on Thursday, April 11, during
class.
There will be a quiz today, Tuesday, April 9, at
the end of class, but no quiz on April 16.
Please make a note of this!
Announcements
Exam 3 is Thursday from 2:30-3:45 pm in RH
100.
The
5.1-5.3: Definite integrals and
approximations
Definite overlap with
previous material
5.1 Accumulated change
5.2 Definite integral
Recall
Velocity = rate of change of distance
In math-speak,
v(t) = D(t)
Reverse?
So, if we do the anti-derivative of veloci