MTH299 - Homework 8
Question 1. Define the function, f : R R via the assignment
f (x) = (x 5)2 + 9,
and define the set E as
E := cfw_x R : f (x) 5.
Prove that the set E is bounded.
Question 2. Let Ar = (r, r). What is
R : x > 0.
rI Ar and
MTH299 - Homework 1
Question 1. exercise 1.10 (compute the cardinality of a handful of finite sets)
Question 2. exercise 1.20 (compute the union of two sets)
Question 3. exercise 1.23 (compute the intersection of two sets)
Question 4. exerci
MTH299 - Homework 9
Question 1. Assume that E R. Let A be the statement E is not an open set. Let B be the statement
E is a closed set. Give an example of E which demonstrates that the implication
A = B
is FALSE. (This is a very important th
MTH299 - Homework 7
Question 1. Define the sets
A = cfw_q P2 : q(x) 0 for all x R,
B = cfw_p P2 : p(x) = a(x x0 )2 + b, where a, b, x0 R, and a 0, b 0.
Prove that A = B.
Question 2. Let f : R Z be the integer floor function
f (x) = bxc := ma
MTH299 - Homework 3
Question 1. There are eight different functions f : cfw_a, b, c cfw_0, 1. List them all. Diagrams will suffice.
Question 2. Show that the function f : R2 R2 defined by the formula f (x, y) = (x2 + 1)y, x3 ) is
MTH299 - Homework 10
Question 1. Prove that P4 is a vector space over R.
Question 2. Define the set V = cfw_x R2 : x2 = 21 x1 . Sketch a picture of the set V inside of the plane,
R2 . Is V a vector space over the scalar field R? Use the usua
MTH299 - Homework 10
Name: Mukangwa Masamba
Homework 10; Due Wednesday, 11/9/2016
Medium Justification Questions. Provide brief justifications for your responses.
Question 1. Define the function L : R R as L(x) = ex . Is L a linear function? Briefly justi
The following information is for X Com pa nys two products A and B:
Total contribution margin 36,540
Reversing differentiation. In many problems, especially physical ones, we are
interested in some function F (x), but we only know its derivative F 0 (x) = f (x).
We need to reverse the differentiation process to find t
Limits at Infinity
Vertical asymptotes. We say a curve has a line as an asymptote if, as the curve
runs outward to infinity, it gets closer and closer to the line. Closer and closer
reminds us of limits, and indeed we have seen that x
APPENDIX E SIGMA NOTATION
A convenient way of writing sums uses the Greek letter 2 (capital sigma, corresponding to .
our letter S) andis called sigma notation.
m Definition If am, am+1, . . . , an are real numbers and m
More Uses for Integrals
Review. The integral
f (x) dx has four levels of meaning.
Physical: Suppose y, z are physical variables determined as continuous functions
of an independent variable x, so that y = f (x) and z = F (x). If y
Rectangle example. Suppose we have 40 meters of fence to make a rectangular
corral. What length and width will fence off the largest area? The range of
possiblilities is illustrated below:
It appears that the square with
Derivatives and Graphs
Increasing and decreasing functions. We will see how to determine the important features of a graph y = f (x) from the derivatives f 0 (x) and f 00 (x), summarizing our Method the last page. First, we consider w
Reversing the Chain Rule. As we have seen from the
R b Second Fundamental
Theorem (4.3), the easiest way to evaluate an integral a f (x) dx is to find an
antiderivative, the indefinite integral f (x) dx = F (x
Man vs machine. In this section, we learn methods of drawing graphs by hand.
The computer can do this much better simply by plotting many points, so why
bother with our piddly sketches? One reason is that calculus tell
Area and Distance
Review. The derivative of y = f (x) has four levels of meaning:
Physical: If y is a quantity depending on x, the derivative
of change of y per tiny change in x away from a.
is the rate
Geometric: f 0 (
The Definite Integral
Precise definition. We have defined the integral a f (x) dx as a number approximated by Riemann sums. The integral is useful because, given a velocity function,
it computes distance traveled; given a graph, it
Mean Value Theorem
Vanishing derivatives. We will prove some basic theorems which relate the
derivative of a function with the values of the function, culminating in the
Uniqueness Theorem at the end. The first result is:
Roots of equations. We frequently need to solve equations for which there is
no neat algebraic solution, such as:
f (x) = x3 + x 1 = 0 .
In this case, the best we can ask is an approximate solution, accurate to a specif
Stewart 4.1, Part 2
Notation for sums. In Notes 4.1, we define the integral a f (x) dx as a limit of
approximations. That is, we split the interval x [a, b] into n increments of size
x = ba
n , we choose sample points x1 , x2 ,
Absolute maxima and minima. In many practical problems, we must find
the largest or smallest possible value of a function over a given interval.
Definition: For a function f (x) defined on an interval x [a, b],
(ai + bi )
for any constant c
for integers 1 K n 1.
n(n + 1)
1 2 1
n + n
n(n + 1)(2n + 1)
Definite Integration Formulas
For any real constants a, b, and c:
= c (b a)
b 2 a2
b 3 a3
We also have the following linearity property of integrals, which can be used to combine the
Tangent linear function. The geometric meaning of the derivative f 0 (a) is the slope of
the tangent to the curve y = f (x) at the point (a, f (a). The tangent line is itself the graph
of a linear function y = L(x
Practice Quiz 1 - Solutions
1. The population of otters in a river watershed is modeled by
where t is the
P t t 2 10t 200
number of years since 2005.
a. What time does the otter population reach a minimum?
P t 2t 10 0 2t 10
t 5 2010
b. At w