Review SIDE NOTES:
Math 157 Calculus I for the Social Sciences
Lecture 1
based on “Calculus Early Transcendentals Differential &
Multi-Variable Calculus for Social Sciences”
Review
This text, including the art and illustrations, are available under
the Creative Commons license (CC BY-NC-SA), allowing anyone to
reuse, revise, remix and redistribute the text. For more info visit
http://creativecommons.org/licenses/bync-sa/4.0/
Calculus Early Transcendentals Differential & Multi-Variable Calculus for So-
cial Sciences was adapted by Petra Menz and Nicola Mulberry from Lyryx’ lec-
ture notes by Michael Cavers licensed under a Creative Commons Attribution
Non-Commercial Share-Alike 3.0 Unported License (CC BY-NC-SA 3.0). The
adaptation ensures congruency between the text Calculus Early Transcenden-
tals Differential & Multi-Variable Calculus for Social Sciences and the lectures
for MATH 157.
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Math 157 Calculus I for the Social Sciences
Lecture 1
Review
Learning Outcomes – Detailed:
The successful student will be able to
1. Interpret and work within the Cartesian coordinate system.
2. Describe the equation of a line given two points, or a point and a slope, or
other information that will lead to either of the two previous givens.
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
What is this Course about?
Calculus is a branch of mathematics that deals with rates of change
Used throughout many areas of science
Three basic ideas: limits, derivatives and applications
There are a variety of resources available to help you through this course:
I Visit the Applied Calculus Workshop.
I Attend office hours.
I Read the course notes, and read your lecture notes.
I Watch videos on khanacademy.org, and read other online lecture notes (e.g., Paul’s
Online Notes).
I Check out an SFU peer tutor.
I Read Recommendations for Success in Mathematics.
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
How to take Lecture Notes
Listen to the Instructor, who
I explains the concepts;
I draws connections;
I demonstrates examples;
I emphasizes material.
Copy the presented lecture material
I by arriving to the lecture prepared;
I using telegraphic writing, i.e. packing as much information into the smallest
possible number of words/ symbols (do you really need to copy all the
algebraic/manipulative steps?).
Mark up your notes immediately while listening and copying using a
system such as offered here:
! pay attention (possible exam material)
? confusing (read course notes or visit ACW )
→ practice (using course notes and online assignments)
underline/highlight key concepts
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Sets
A set is a collection of distinct objects considered as a whole.
{a, b, 1, 2} is a set of four objects, namely, a, b, 1 and 2.
For a set S, we use the notation x ∈ S to mean that x is an element contained in the set S.
The intersection between two sets S and T is denoted by S ∩ T and is the collection of all
elements that belong to both S and T .
The union between two sets S and T is denoted by S ∪ T and is the collection of all
elements that belong to either S or T (or both).
Example
Let S = {a, b, c} and T = {b, d}.
Then a ∈ S but d 6∈ S.
Then S ∩ T = {b} and S ∪ T = {a, b, c, d}.
Note: We don’t write the element b twice in S ∪ T even though b is in both S and T .
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Numbers
What is a number? Our answer to this question has expanded over time. . .
N - the natural numbers, {1, 2, 3, . . .}.
Z - the integers, {. . . , −3, −2, −1, 0, 1, 2, 3, . . . }.
n o
p
Q - the rational numbers - ratios of integers r = q
: p, q ∈ Z, q 6= 0 .
R - the real numbers - can be written using a finite or infinite decimal expansion.
C - the complex numbers - allow us to solve equations like x 2 + 1 = 0.
All rational numbers are real numbers - they have a repeating decimal expansion.
√
Real numbers not ending in a repeating pattern are called irrational ( 2,π,e).
N ⊂ Z ⊂ Q ⊂ R ⊂ C.
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Laws of Exponents
Definitions: If m, n are positive integers, then
1. x n = x · x · · · x (n times)
2. x 0 = 1, for x 6= 0
1
3. x −n = xn
, for x 6= 0
√
n √ m √
4. x m/n = x m or n x , for x ≥ 0, where n denotes the nth root.
Combining: If a, b are real numbers, then
1. x a x b = x a+b
xa
2. xb
= x a−b , for x 6= 0
3. (x a )b = x ab
Distributing: If a, b are real numbers, then
1. (xy )a = x a y a , for x ≥ 0, y ≥ 0
a a
2. yx = yx a , for x ≥ 0, y > 0
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
OPTIONAL Solving Quadratics
The Quadratic Formula
The solutions to ax 2 + bx + c = 0 (with a 6= 0) are:
√
−b ± b 2 − 4ac
x= .
2a
OPTIONAL Example—if not covered, try it on your own and check with the ACW
Solve x 2 − 9 = 0.
Solution:
Method 1: Factor as a difference of squares:
Method 2: Use the Quadratic Formula:
Method 3: Solving for x gives:
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Inequalities and Intervals
a < b means a is to the left of b (i.e., a is strictly less than b)
a ≤ b means a is to the left of or the same as b (i.e., a is less than or equal to b)
a > b means a is to the right of b (i.e., a is strictly greater than b)
a ≥ b means a is to the right of or the same as b (i.e., a is greater than or equal to b)
The notation (a,b)= {x ∈ R | a < x < b} is what we call the open interval from a to b
and consists of all the numbers between a and b, but does not include a or b.
On the real number line we depict this as:
The notation [a,b]= {x ∈ R | a ≤ x ≤ b} is what we call the closed interval from a to b
and consists of all the numbers between a and b and including a and b.
On the real number line we depict this as:
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Different Types of Intervals
Combining brackets together we get different possible types of intervals (here, we assume a < b):
= {x ∈ R | a < x < b} = {x ∈ R | a ≤ x ≤ b}
= {x ∈ R | a ≤ x < b} = {x ∈ R | a < x ≤ b}
= {x ∈ R | x > a} = {x ∈ R | x ≥ a}
= {x ∈ R | x < b} = {x ∈ R | x ≤ b}
= R = all real numbers
10
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Absolute Values
The absolute value of a number x is written as |x|.
|x| represents the distance x is from zero.
Mathematically, we define it as follows:
x if x ≥ 0,
|x| =
−x if x < 0.
√
x 2 = |x|.
The function f (x) = |x| can be visually represented by taking the straight line g (x) = x
and “flipping up” the negative part of it to make it positive:
y y
g (x) = x 4 f (x) = |x| 4
2 2
x x
−4 −2 2 4 −4 −2 2 4
−2 −2
−4 −4
11
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
Review SIDE NOTES:
Equations of Lines (Linear Functions)
A linear function is any function of the form y = mx + b, where the constant m represents
the slope and the constant b represents the y -intercept of the line.
The slope m of a (nonvertical) line through points (x1 , y1 ) and (x2 , y2 ) is:
change in y
Slope Formula: m = rise y2 −y1
run = change in x = x2 −x1 = ∆x .
∆y
I Point-Slope Form: y − y1 = m(x − x1 ).
I Slope-Intercept Form: y = mx + b.
I General Form: Ax + By + C = 0.
12
LEGEND: ! pay attention for exam ? confusing, read course notes
underline key concepts → need to practice
