MATH1241/1231 Formulas and definitions you might need PDF

Preview

Summary

You might need these definitions when studying 1231/1241, and pls use the formulas or else you will regret it...

Description

M0ATH1241 Cheat Sheet

Calculus Chapter 1 Sketching

*C gives the height above xy-plane Partial Differentiation

Geometric interpretation

Mixed Derivative Theorem

Continuous Rules

Tangent Planes

Total Differential Approximation

Chain Rule

Chapter 2

If m odd then use u=sin(x), if n odd use u=cos(x), if both odd either sub works and if both even use:

If we have

Use:

When given integrals with tan, sec use the fact that:

(pg 34 if need to prove) Fractional powers

Rational Functions f=p/q If degree on top is higher then f is proper, if degree on bottom is higher then f is improper Polynomial is irreducible if no real linear factors The strategy

For integrals like:

Let 𝑥 = 𝑢12 (12 being the lowest common multiple)

Chapter 3 Ordinary Differential Equation

Initial Value Problem

Separable ODEs Just separate the x’s and y’s and then integrate First Order Linear ODEs

To solve just use this method:

Exact ODEs

When you integrate with respect to x you will get constant that is a function of y and vice versa then compare both of your H(x,y). Or you could integrate with respect to x and you only need to the constant (that is a function of y). So you then partial differentiate with respect to y and then you could have C’(y)=something in the 𝜕𝐺 𝜕𝑦

= 𝐺 equation. Integrate to find C(y).

Note: Sometimes you will need to multiply through by something to make it exact (but if it can be zero then it might change the solution set so you need to check) Change of Variable Nothing to say really, why is it [X] idk

Second order linear ODEs with constant coefficients

(3.35) Homogenous Case

Every second order ODE has AT MOST two linearly independent solutions (not scalar multiples of each other) – so if 𝑦1 and 𝑦2 from above are not scalar multiples then that is every solution

The Non-Homogenous Case Algorithm to solve it

Guessing solution for step 2 (NOTICE: Not even a term in 𝑦𝑝 must be a solution to the homogenous)

Chapter 4 Taylor Polynomial

Taylor’s Theorem

Lagrange form for the remainder

Classifying Stationary Points

Sequences Real valued functions defined on a subset of the natural numbers {𝑎𝑛 }𝑛∈ℕ {𝑏𝑛 }𝑛∈ℕ = {𝑎𝑛 𝑏𝑛 }𝑛∈ℕ Classifying divergence

Rules of sequences

Above only works for FINITE limits Pinching for sequences

Chapter 5: Suprema and infima

(pg 121 for method to proving least/greatest bound)

Infinite Series

Tests for SERIES Convergence

Kth term test for DIVERGENCE

Integral test

Comparison test

Limit form of the comparison test

Ratio Test

Alternating series test

Error for alternating series

Absolute and Conditional Convergence

Rearrangement

Taylor Series

Power Series

Radius of Convergence

To see if convergent at end points, first find the radius of convergence then sub in end points and use normal tests Adding, Multiplying, Differentiating, Integrating power series

Chapter 5 Average value of a formula

Length of a line segment

Arc length when given in parametric form

Arc length in interval [a,b]

Arc length of a polar curve

Speed of particle at time t

Surface area of a curved surface

Surface area of revolutions using parametric, function or polar coordinates

Volume of a function rotated along x axis

Important:

Algebra Axioms

subspace

Span

Linear independence

The values of the scalars in the linear combination are non-unique if and only if S is a linearly dependent set

A set of vectors S is a linearly dependent set if and only if at least one vector in S is in the span of the other vectors in S.

Basis

Dimension

Coordinate vectors

Others:

Polynomials

Additional proof reminders

Notations

Linear maps

The domain and codomain of a linear map can be any vector spaces. In Section 7.5, we shall be concentrate specifically on linear maps from R n to R m. Unless otherwise stated, the following propositions and theorems are true for all linear maps

Property of linear maps

Linear map proof #

Linear map distributes

Linear map and basis value

Matrix that’s a Linear map

Matrix representation theorem

Linear map is a line to a line or a point

Kernel definition

Kernel of a matrix definition

Linear map and subspace

Nullity, kernel of linear transformation

Linearly independent and nullity

Image definition

Image and linear map

Rank definition and image with linear transformation

Rank-Nullity Theorem

Rank in terms of matrix

Addition of linear maps

Rank nullity theorem

Ax=b solutions in terms of ranks

Matrix representation theorem

Algorithm for matrix representation for a linear map

Linear maps

Multiplication of matrix

Combined Linear transformation and matrices

Linear map is one to one and onto, Inverse of linear maps

Linear maps and domain and codomain

Inverse of linear maps

Finite-dimensional vector and inverse and dimension

Inverse of a function

Definition of an inverse function

Additional stuff:

Laplace transformation

Rotational matrix

CHAPTER 3 Eigenvalue and eigenvector definition

Eigenvectors and eigenvalue for square matrices

Eigenvector and solution for (A-lambdaI)v

Determinant of matrices with complex lambda

Characteristic polynomial is sol of determinant

Square matrices and the amount of eigenvalues they have

Amount of eigenvalues = amount of linearly independent eigenvectors

Diagonal matrix formation

Diagonalisable definition

Powers of A in diagonal matrix

Powers of A formula

Solution for vexp(lambda t)

Solutions to the equation and their linear combination

Eigenvalue of a transpose matrix

Sum of columns for a square matrix

Chapter 4 Set stuff

Probability

Outcomes are called sample points

Rules of Probability

Conditional Probability

Statistical Independence

Random Variables

Essentially ascribing numbers to events

Cumulative distribution function

Discrete Random Variables

Mean, Variance, Standard Deviation of Discrete RVs

Special Distributions Binomial

Geometric

Sign Test...