Summary
Summary Mathematics 114
- Course
- Institution
- Book
A summary of all the concepts covered in the semester. It includes all the formulas, rules and proofs necessary to understand the concepts covered in the module.
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
Solution for James Stewart Calculus (ch1 - ch11)
-
Complete Solution Manual Calculus 8th Edition Questions & Answers with rationales (Chapter 1-17)
-
Solution Manual for Calculus 8th Edition / All Chapters 1 - 17 / Full Complete 2023
Seller
Follow
hollymadison
Reviews received
Content preview
WEEK ONENUMBERS
Natural numbers:
- Positive whole numbers 1, 2, 3, …
Integers:
- Whole numbers -1, 0, 1, …
Rational numbers:
- Ratios of integers with non-zero denominator.
!
- Numbers of the form " where both n and m are integers, and m is non-zero integer.
Real numbers:
- All numbers on number line, including numbers like 𝜋 and √2.
COORDINATE GEOMETRY AND LINES
Distance formula:
$(𝑥# − 𝑥$ )# + (𝑦# − 𝑦$ )#
Gradient / slope:
%! & %"
(! & ("
Point-slope form of a line:
y - 𝑦$ = m(x - 𝑥$ )
y = mx + c
slop-intercept form of straight line:
Ax + Bx + C = 0, where A ≠ 0 and B ≠ 0
Two lines with gradients 𝑚$ and 𝑚# respectively, are parallel if 𝑚$ = 𝑚# , and are
&$
perpendicular if 𝑚$ 𝑚# = -1, or equivalently, 𝑚$ = " , provided 𝑚$ ≠ 0 and 𝑚# ≠ 0.
!
INEQUALITIES
Rules for inequalities:
1. Exactly one of the following is true: a < b, a = b, b < a
2. If a < b and b < c, then a < c
3. If a < b, then a + c < b + c
4. If a < b and c > 0, then ac < bc
5. If a < b and c < d, then a + c < b + d
6. If a < b and c < 0, then ac > bc
$ $
7. If 0 < a < b, then ) > *
For real numbers a, b, c and d:
1. If a ≤ b and b ≤ c, then a ≤ c
2. If a ≤ b, then a + c ≤ b + c
3. If a ≤ b and c ≤ d, then a + c ≤ b + d
4. If a ≤ b and c ≥ 0, then ac ≤ bc
5. If a ≤ b and c ≤ 0, then ac ≥ bc
$ $
6. If 0 < a ≤ b, then ) ≥ *
ABSOLUTE VALUE
𝑎 𝑖𝑓 𝑎 ≥ 0
|a| = 0
−𝑎 𝑖𝑓 𝑎 < 0
Which means that |a| is defined to be a when a ≥ 0 and is defined to be -a when a < 0.
,Properties of absolute values:
For all a, b 𝜖 R and n 𝜖 Z:
1. √𝑎# = |a|
2. |ab| = |a||b|
) |)|
3. | | = when b ≠ 0
* |*|
4. |an| = |a|n
5. If a > 0 then |x| = a if x = a or x = -a
6. |x| < a if -a < x < a
7. |x| > a if x > a or x < -a
8. |x| ≤ a if -a ≤ x ≤ a
9. |x| ≥ a if x ≥ a or x ≤ -a
10. |a + b| ≤ |a| + |b| this is the triangle identity
Proofs for triangle identity:
i) We have:
-|a| ≤ a ≤ |a|
-|b| ≤ b ≤ |b|
Hence, adding these two identities we get
-|a| + -|b| ≤ a + b ≤ |a| + |b|
⟺ - (|a| + |b|) ≤ a + b ≤ |a| + |b|
⟺ |a + b| ≤ |a| + |b|
ii) Since:
|a + b|2 = (a + b)2 = a2 + 2ab + b2
And
(|a| + |b|)2 = |a|2 + 2|a||b| + |b|2 = a2 + 2|ab| + b2
It follows that
(|a| + |b|)2 - |a + b|2 = 2|ab| - 2ab = 2(|ab| - ab)
And hence since |ab| ≥ ab we have that
(|a| + |b| - |a + b|)(|a| + |b| + |a + b|) = (|a| + |b|)2 - |a + b|2 ≥ 0
Therefore since (|a| + |b| + |a + b|) > 0; unless a = b = 0 (in which case the
identity is trivially true), it follows that
|a| + |b| - |a + b| ≥ 0
And hence that
|a + b| ≤ |a| + |b|
ANGLES
Use radians [rad] as unit for angles.
Relationship between radians and degrees is given by equation 180° = 𝜋rad.
,
It follows that an angle 𝜃 in degrees corresponds to 𝜃 rad in radians while an angle 𝜙 in
$-.
$-.°
radians corresponds to 𝜙 , 0)1 in degrees.
Note: when write an angle in radians we usually leave out the unit.
Conversion of some common angles.
TRIG FUNCTIONS
23345678 ;1<)=8!7
Sin 𝜃 = 9%3478!:58 Cot 𝜃 = 23345678
;1<)=8!7
Cos 𝜃 = 9%3478!:58
23345678
Tan 𝜃 = ;1<)=8!7
9%3478!:58
Sec 𝜃 =
;1<)=8!7
9%3478!:58
Csc 𝜃 =
23345678
,TRIG IDENTITIES
$
Csc 𝜃 = >?@ A
$
Sec 𝜃 = BC> A
$
Cot 𝜃 = DE@ A
>?@ A
Tan 𝜃 =
BC> A
BC> A
Cot 𝜃 = >?@ A
Sin2 𝜃 + cos2 𝜃 = 1
Tan2 𝜃 + 1 = sec2 𝜃
1 + cot2 𝜃 = csc2 𝜃
EVEN AND ODD IDENTITES
Sin(-𝜃) = -sin(𝜃)
Cos(-𝜃) = cos(𝜃)
PERIODIC IDENTITES
Since 𝜋 represents one full rotation around a circle we have:
Sin (𝜃 + 2𝜋) = sin(𝜃)
Cos(𝜃 + 2𝜋) = cos(𝜃)
ADDITION AND SUBTRACTION FORMULAS
DOUBLE-ANGLE FORMULAS
HALF-ANGLE FORMULAS
PRODUCT FORMULAS
, GRAPHS OF TRIG FUNCTIONS
sec(x)
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller hollymadison. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.78. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
49160 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now