Calculus - 2WBB0

2WBB0 Calculus

• Area: Mathematics
• Topics: Complex Numbers · Differential Equations
• Related courses: Math I - 2DE20 · Math II - 5EMA0 · Systems - 5ESB0

Calculus_2.pdf

Preliminaries

Real Numbers
Intervals
Quadratic Equations
Cubic Equations
Higher Degree Equations
Equations involving rational functions
Roots
Inequalities
Powers
Quadrics
Functions and their graphs
Trigonometric Functions
Trigonometric Equations

Some Notes + P1-P7

How to Convert Repeating Decimals to Fractions: 9 Steps

Decimal to Fraction

composition:

To determine the domain of a rational function, we need to find the values of that make the denominator zero, as these values are excluded from the domain.

domain of domain
range of range


numerator to zero = how many solutions in equation
injective = one to one

• The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes.

• You can keep dividing function until denom>num ⇒ if remainder ⇒

use quad formula

• completion of squares ⇒ square to vertex form
• https://www.youtube.com/watch?v=EBbtoFMJvFc
• and

where
where

equation of a circle

domain: x possible input vals, range: y possible output vals

vertical translation:
horizontal translation: opposite
vertical stretching:
horizontal stretching: opposite
even function f
odd function f
if f and g are even →
if f and g are odd →

Roots

odd square root like: even: only positive

Inequalities

and

(cos,sin)
degree to radian:

Trigonometric Rules

just remember the identities on formula sheet given

&
cos even ⇒ sine odd ⇒
&
&

just remember for proving:

sine law:
cosine law: , ,

Evaluating Trig Functions

arcsin
arccos

since Q3 →

, , ,

slope=gradient, derivative of a function is its slope.

ALWAYS CHECK INEQUALITY VALUES!!!

Logarithms

and

• derivative https://www.youtube.com/watch?v=2A1xgHtTOCk
  
  
  
  
  
  
  

MOST IMPORTANT

Complex Numbers

Topic: Complex Numbers
https://medium.com/all-math-before-college/complex-numbers-4a1aeda2abb6

imaginary component: y
real component: x

• conjugate: what you see in first bullet point reverse of sign in middle
• where , eg:

Summery of Complex Numbers - Rectangular, Polar, Exponential Form

Geometric Operations

Think in terms of a circle, radius stretches, angle shifts when you stuff like multiplication etc…

Deriving Polar Formulas

: Euler Formula

for sine:
for cosine:

• always plot on IM/RE axis
  

conjugate root theorem polynomial: if it has a complex root it also has a conjugate of that complex as root

imaginary part of imaginary number = real number

so its set of all complex numbers are a circle around not origin but around with radius

imaginary part of imaginary number is a real value:

Having root means having 2 roots

Limits & Continuity

Limits
One sides Limits
Rules for Limits
Limits of Fractions
Squeeze Theorem
Limits with substitution
Limit with substitution
Limits to Infinity
Asymptotes
Standard Limits
Continuity

if limit from (-) and (+) then the general limit does not exist

• L’Hopital only when or
• when limit of and has trig manipulate fraction to be div by x
• you can see that its basically
• https://en.wikipedia.org/wiki/Squeeze_theorem
  
  
  
  
  asymptotes are limits to infinity.
  horizontally it must have both () asymptote
  vertically it can have either of the asymptote doesn’t have to have have both
  https://www.youtube.com/watch?v=WT7oxiiFYt8
  https://www.youtube.com/watch?v=NVQBZWTKygU
  
  
  right and left continuity means from negative and positive approach…
  
  2.6: Continuity - Mathematics LibreTexts
  
  composite functions → continuous
  

Intermediate Value Theorem. If f is continuous on the interval [a,b] and N is between f(a) and f(b), where f(a)≠f(b), f ( a ) ≠ f ( b ) , then there is a number c in (a,b) such that f(c)=N.

taylor: https://www.youtube.com/watch?v=EYjBnnUJTP8, https://www.youtube.com/watch?v=f_NvAc4Ffbg

Differentiation

Tangents
Derivative
Taking Derivatives
Derivatives of Trigonometric Functions
Mean Value Theorem
Higher order Derivatives
Implicit Differentiation
Linear Approximation
Taylor Polynomial
Taylor Series
Remainder of the Taylor series
Finding Taylor Series
l’Hopitals Rule
Limits with Taylor

• derivative of function you find slope by plugging value (demo), then in original function you can find x and y. and then write slope line
• with the slope you can find the tangent line.
• slope of normal to tangent:
• 
• If f is differentiable at x, then f is continuous at x.
  
  
  
  

mean value theorem: https://www.youtube.com/watch?v=Brat34fUe9o

minmax

Inflection Point

where it switches signs

idfk

Implicit Differentiation

More on Differentiation

to find an error of linear approximation: for certain between

for all the taylors we already have if they are of same form just plug in values
with limits the denom must also go to the same or higher power compared to num.

Big O

the max factor the function grows at.
at most
at most
at most cubically so also works

• Taylor Series you can find by looking at formula sheet and adjusting the function to be same.

find taylor series of tan(x)

Transcendental Functions

Inverse functions
Inverse functions of sin, cos and tan
Exponential and Logarithmic Functions
Logarithmic Equations
Differentiation of Exponential Functions

one to one: unique x→y pair

• for a function to be invertible it must be one to one. (think graphically).
• A function is a one to one if its derivative is always positive for all x.
• if not one to one you can restrict to domain which makes it one to one and find inverse.
• Think of implicit differentiation to find derivative of inverse function
• to prove a function is one to one you can use a derivative

& since & if -val then answer still always positive

see here how to solve

Tangent

,
⇒
⇒
⇒

• image/range of a function is a set of solutions it can produce.
• is a strictly increasing function

Growth Properties of Exp & In

but power beats logarithm, loses to exponential

Integration

piecewise means not continuous functions.

Fundamental Theorem of Calculus Example

since

• Another substitution example so

Steps for Integration by Substitution - Inverse of Chain Rule

Replace by
Replace by
Take care of remaining by rewriting them in terms of
Boundaries for boundaries for

you can keep the boundaries as in terms of but must indicate that

Integration by Parts - Inverse of Product Rule

⇒

• you can take partial integration more than once.
  

Integration of Rational Functions

1. First long division such that degree of numerator < degree denominator
2. Factor denominator in linear / quadratic factors (only possibilities!)
3. Decompose fractions into easier ones or complete the squares
4. Find anti-derivatives for separate terms

Improper Integral Example

doesn’t exist so:

if converges then so does
if diverges then so does

Differential Equations

Topic: Differential Equations

put constant only on one side

If f(x) in differential equation is equal to zero, it is said to be a homogenous. Otherwise, it is non-homogenous. An example of a homogenous differential equation is the second-order linear ODE mentioned above, while an example of non-homogenous differential equation is

first order linear differential equation: inhomo homo

First-Order Linear Equations

solved by: where and

Integration Factor:
Variation of the Parameter:

Integrating Factor: Multiple the equation by

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