1.6 Problem Solving: Interest, Mixture, Uniform Motion
Steps for Setting Up Applied Problems
STEP 1: Read the problem carefully, perhaps two or three times. Pay particular attention to
the question being asked in order to identify what you are looking for
2.2 Lines
I. Calculate and Interpret the Slope of a Line
Slope Formula
m: rise. = 33,15.
11"?
Find the slope of a line through the given points.
1. (4,2) andcfw_3,-2)
2. (5.3) and(7,3)
X\ Y: XI. y: X| \[u XI- Y.
2.2 -LI 3_3 0
3-1.! m 1-.5- -
Graphing
2.1 - Intercepts; Symmetry; Graphing Key Equations
Finding intercepts
- xintercepts: set y = O and solve for x ( xintercepts are also referred
to as roots or zeros of the equation.
yintercepts: set x = 0 and solve for y
Tests for Symmeth
To test the graph
1.5 - Radical Equations; Equations Quadratic in Form, Absolute Value Equations;
Factorable Equations
1. Equations containing radicals (extraneous solutions may occur when index of radical is
even)
even power. )
ex. 5t+4=2
(Jean )1:- 9.2
54s+ll = 1-|
El: =
12.1 Systems of Linear Equations
System of eguations: a collection of two or more equations, each containing one or more variables.
I. Solving Graphically
Exlcfw_ 2x+y=5
4x+6y=12
au: 5 le+CnL=l1
3:-a*s (93: 4416443.
-2.
B: Ex-l-Z
limo. Mae. inter: e
1.1 - Rectangular Coordinates; Graphing Utilities; Introduction to Graphing Equations
[. The Coordinate Plane
Ordered Pairs are represented in a plane using the rectangular or Cartesian Coordinate System:
10
u Lmelrun-lzs
Sm Cod i u lanai-III-
= ' Ill-ll
1.3 8: 1.4 - Quadratic Equations; Complex Solutions
Standard Form of a Quadratic Eguation
axz + bx + c = 0 where a, b, and c are integers and asto
Methods of Solving Quadratic Equations
I. Factoring
Set Quadratic = 0. Factor Set each factor =0. Solve
1.2 - Solving Equations Using a Graphing Utility; Solving Linear and Rational Equations
l. Solving Equations Using a Calculator
To solve an equation means to find all solutions of the equation. A solution is also called a zero or a
root.
To find solutions
Sections 1.1-1.7,2.1-2.2,12.1
Test 1 Study Guide
MAT 111 College Algebra
Character of the Solution
2
b -4ac > 0: two real solutions
b2-4ac = 0: repeated real solution
b2-4ac < 0: two complex solutions,
no real solutions
Symmetries
x-axis: replace y with y
92.305 Homework 4
Solutions October 9, 2009
Exercise 2.4.1. Complete the proof of Theorem 2.4.6 by showing that if the series 2n b2n diverges, then so does bn . Example 2.4.5 may be a useful reference. n=1 n=0 Proof: We will show that if 2n b2n diverges t
92.305 Homework 2
Solutions September 25, 2009
Exercise 1.3.2. (a) Write a formal denition in the style of Denition 1.3.2 for the inmum or greatest lower bound of a set. Answer: A number x R is the inmum or greatest lower bound of a set A R of real number
1.4.7. Finish the following proof for Theorem 1.4.12. Assume B is a countable set. Thus, there exists f : N ! B , which is 1 and onto. Let A B be an innite subset of B . We must show that A 1 is countable. Let n1 = min fn 2 N : f (n) 2 Ag. As a start to a
1.3.7. Prove that if a is an upper bound for A, and if a is also an element of A, then it must be that a = sup A. Since we are given that a is an upper bound for A, we need only check that part (ii) of Denition 1.3.2, namely that if b is any upper bound f
2.4.2. Dene the sequence (xn ) by x1 = 3 and xn+1 = 1 4 xn
(a) Prove that the sequence (xn ) converges. The rst three terms of the sequence are 3, 1, 1=3. We want to show that this sequence is decreasing (it will be bounded below by 0 because xn 3). So we
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 26, 2010
Assignment 29 1. Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. That is, cfw_fn is a uniformly convergent sequence of functions on a set
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 23, 2010
Assignment 28 1. Use the following argument to prove the Substitution Theorem (Theorem 7.3.8). Dene F (u) = u () f (x) dx for u I , and H (t) = F (t) for t J . Show that H (t) = f
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 21, 2010
Assignment 27 1. If f is continuous on [a, b], a < b, show that there exists c [a, b] such that we have f (c)(b a). This result is sometimes called the Mean Value Theorem for Integ
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 19, 2010
Assignment 26 1. Suppose f 0, f is continuous on [a, b], and that x [a, b].
b a f (x) dx
= 0. Prove that f (x) = 0 for all
Suppose not. That is, suppose that f (x) = c > 0 for some
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 12, 2010
Assignment 24 1. Suppose f is dened in a neighborhood of x, and suppose f (x) exists. Show that f (x + h) + f (x h) 2f (x) = f (x). h0 h2 lim Show by an example that the limit may
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 5, 2010
Assignment 23 1. A dierentiable function f : I R is said to be uniformly dierentiable on I := [a, b] if for every > 0 there exists > 0 such that if 0 < |x y | < and x, y I then f (x
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 31, 2010
Assignment 22 1. Suppose that f : R R is dierentiable at c and that f (c) = 0. Show that g (x) := |f (x)| is dierentiable at c if and only if f (c) = 0. Since f (c) = 0, we have g
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 29, 2010
Assignment 21 1. Let I R be an interval and let f : I R be increasing on I . If c is not an endpoint of I , show that the jump jf (c) of f at c is given by inf cfw_f (y ) f (x) : x