MATER039N Engineering Mathematics 2 Exam Review

These flashcards cover key vocabulary and concepts from the Engineering Mathematics 2 lecture notes, focusing on theoretical aspects relevant to the exam. Topics include events, differential equations, partial derivatives, distributions, matrices, complex numbers and vector vector of lines.

Event A ∪ B

The event A ∪ B means that at least one of the events occur.

Fundamental Solutions

The set of fundamental solutions of a second order linear differential equation is defined as two linearly independent solutions.

Second Order Partial Derivatives

If the second order partial derivative functions of function f ⊂ R2× R exist in a neighbourhood of a ∈ Df and are continuous at a then f′′ xy(a) = f′′ yx(a)

Poisson Distribution

The number of trucks going through in a crossing during 1 hour obeys a Poisson distribution.

DE Solutions

If {φ1, φ2} is a set of fundamental solutions of the DE x′′ + a1(t)x′ + a0(t)x = 0, then for any solution φ there exist c1, c2 ∈ R such that φ = c1φ1 + c2φ2.

Jacobi Matrix

In the Jacobi matrix, the partial derivatives of the first coordinate function of f are in the first row.

Eigenvalue of Matrix A

If As = 0, where A is a 2 × 2 matrix and s ∈ R2 is a non-zero vector, then A = 0 is a FALSE statement.

Probability Mass Function

No, P(X = 0) = 1/3, P(X = 1) = 1/3, P(X = 2) = 1/2 cannot be the probability mass function of X because probabilities must sum to 1.

Event Space Cardinality

When flipping a fair coin three times, the cardinality of the event space is 8.

Wronski Determinant

The Wronski determinant of functions φ1 and φ2 is defined as W(t):= | φ1(t) φ2(t) ; φ′1(t) φ′2(t) |.

Differentiable Functions

If function f ⊂ Rn× R is differentiable at an inner point a of its domain, then f′ x(a) = f′ y(a) = 0 is FALSE.

Poisson Distribution

Number of events occurring during a unit time can be modeled with Poisson distribution.

DE Fundamental Solutions

For any solution φ there exist c1, c2 ∈ R such that φ = c1φ1 + c2φ2, given {φ1, φ2} is a set of fundamental solutions of the DE x′′ + a1(t)x′ + a0(t)x = 0.

Level Curve

The level curve of f belonging to c is Sc := {(x, y) ∈ R2 | f(x, y) = c}.

Matrix Eigenvector

If As = 0, where A is a 2 × 2 matrix and s ∈ R2 is a non-zero vector, then A = 0 is a FALSE statement.

Cumulative Distribution Function

The cumulative distribution function F is monotonically increasing.

Fair Coin Flip

The correct answer is The cardinality of the event space is 8.

Unique Solution

We say that the initial value problem has a unique solution if there exists a solution φ , such that all other solutions are a restriction of φ.

Partial Derivatives

If function f ⊂ R2× R is partially differentiable with respect to both variables and directional derivates of f at a exist.

Null Hypothesis

The two means are equal is the null hypothesis of the two sample t-test?

Solution of DE

If φ0 is a non-trivial solution of the DE x′ = f(t)x and c ∈ R is arbitrary then cφ0 is a solution of the DE.

Jacobi Matrix

The Jacobi matrix of function f is x ∈ Df, J : x → [ ∂f1/∂x1(x) … ∂fm/∂x1(x) … ∂f1/∂xn(x) … ∂fm/∂xn(x) ] .

Complex Number FALSE Statement

If z = a + bi is an arbitrary non-zero complex number, zz = z2 is a FALSE statement.

Cumulative Distribution Function Limit

Given the cumulative distribution function F, the limit as x approaches −∞ of F(x) = 0.

Fair Coin Flipping

The cardinality of the event space is 8. is the correct answer when flipping a fair coin three times.

General Solution

The general solution of the first order homogeneous linear differential equation x′ = f(t)x (t ∈ I) can be given with the formula φ(t) = ce^(∫ f(t) dt) (c ∈ R).

Partial Derivatives

If function f ⊂ R2× R is partially differentiable with respect to both variables, the directional derivates of f at a exist.

Normal Distribution

The height of a population can be modelled with normal distribution.

Non Trivial Solution

If φ0 is a non-trivial solution of the DE x′ = f(t)x and c ∈ R is arbitrary then cφ0 is a solution of the DE.

Local Maximum

The function f ⊂ R2× R has a local maximum at a ∈ Df if there exists δ > 0 such that for all x ∈ Df , x ∈ Df ∩ kδ(a), f(x) ≤ f(a).

Vector of the line e

If v is paralel to n, then e is perpendicular to S.

Cumulative Distribution

Being a step function is NOT characteristic of the cumulative distribution function of a continuous random variable?

Event A ∩ B Meaning

The event A ∩ B means that both events occur.

Wronski Determinant

The Wronski determinant of functions φ1 and φ2 is defined as W(t):= | φ1(t) φ2(t); φ′1(t) φ′2(t) |.

Partial Differentiable Function

If function f ⊂ R2× R is partially differentiable with respect to both variables, then the directional derivates of f at a exist.

Poisson Distribution

Number of events occuring during a unit time can be modelled with Poisson distribution.

Function System

sin t, 3 sin t are linearly DEPENDENT.

Jacobi Matrix

False Statement of the Matrix

For all x ∈ R2 Ax = 5x holds is the FALSE statement.

Cumulative Distribution Function Limit

The limit as x approaches ∞ of F(x) = 1

Coin Flip Twice

Ω = {hh, ht, th, tt} is correct answer

Integrable differential equation

Integrable differential equation is NOT the type of the differential equation x′ + 4x = 0?

False statement

f has a local extremum at a. is the FALSE statement

Binomial Distribution

Sampling with replacement can be modelled with binomial distribution

Solution to DE

If φ0 is a non-trivial solution of the DE x′ = f(t)x and c ∈ R is arbitrary then cφ0 is a solution of the DE.

Inner Point

There exists δ > 0 such that kδ(p) ⊂ A. (inner point of A)

Arbitrary Non Zero Complex Number

1/z = a/(a^2 + b^2) - b/(a^2 + b^2) i

Expected Value

The random varible has only positive values is the FALSE statement.

Mutually Exclusive Events

P(A ∪ B) = P(A) * P(B) is the False statement

Wronski determent

W(t):= | φ1(t) φ2(t); φ′1(t) φ′2(t) | is the wronski determinant

false Statement

f′ x(a) = f′ y(a) = 0 is the FALSE statement

Normal Distirbution

Normal distribution is the law

Characterisitc equation

x′′ + a1x′ + a0x = 0 has the characteristic equation λ2 + a1λ + a0 = 0

Point within neighborhood.

kδ(a):= {x ∈ Rn | ∥x − a∥ < δ } is the equation

Directional vector

either e and S does not have a joint point or e lies on S is the law

Step Function

Being is a step function is a property

Mutually Exclusive

A ∩ B = ∅ expresses that A and B are mutually exclusive events?

Formula to solution

φ(t) = ce^(∫ f(t) dt) (c ∈ R) is the expression

Differentiable expression

f′ u(a) = is the expression ⟨grad f(a), u⟩.

Binomial Distribution

Sampling with replacement can be modelled with binomial distribution.

The non-trivial Solution to the DE

If φ0 is a non-trivial solution of the DE x′ = f(t)x and c ∈ R is arbitrary then cφ0 is a solution of the DE

Gradient of f

The gradient of f is NOT defined at a.

Directional vector

a coordinate of v is 0 does the equation doesn't exist

Probability Density Function

Its limit at infinity is 1 is NOT characteristic of a probability density function?

Mutually Exclusive Events

A ∩ B = ∅. expresses that A and B are mutually exclusive events

Wronski determinant

W(t):= | φ1(t) φ2(t); φ′1(t) φ′2(t)| is the wronski function of functions function.

Partial Derivative

the directional derivates of f at a exist exist.

Poisson Distribution

Number of events occuring during a unit time can be modeled with Poisson distribution

Linear differential Equations.

x′′ + a1x′ + a0x = 0 one of the following differential equations has the characteristic equation λ2 + a1λ + a0 = 0?

Jacobi Matrix

x ∈ Df, J : x

Line vector .

e

Random Varibale

monotonically increasing function

Probability theory notation .

Ω = {hh, ht, th, tt}.

Maths constant coeficients equation.

λ2 + a1λ + a0 = 0.

math constant equations rule.

f′′ xy(a) = f′′ yx(a).

Length of convos.

Exponential

maths constant for equations.

c1φ1 + c2φ2 is a solution of the DE.

Maths description point.

there exists δ > 0 such that kδ(p) ⊂ A.

eigen value.

A = 0.

Probability function.

f(x):= { −x, if 0 ≤ x ≤ 1, 0, otherwise.

The false statement.

P(A ∪ B) = 0.

the statement

for c1, c2 ∈ R, the equality c1φ1 + c2φ2 = 0 holds only if c1 = c2 = 0.

If the 2d derivative exists.

f′′ xy(a) = f′′ yx(a).

Obey of event.

Poisson .

The Characteristic equation.

x′′ + a1x′ + a0x = 0. equation.

A point.

there exists δ > 0 such that kδ(p) ⊂ A.

Arbirtray complex number.

zz = z2.

Functions.

F is monotonically increasing.

The false statement.

P(A ∪ B) = P(A) · P(B)

The solution

φ, such that all other solutions are a restriction of φ..

Second partials function

f′′ xy(a) = f′′ yx(a)

Variable x and numbers .

Hypergeometric.

Functions

sin t, 3 sin t

Radius equations.

x ∈ Rn | ∥x − a∥

5 is egen value.

2 × 2 matrix,

function X

and P(X = 2) = 1 / 2, are the probability mass function of X?.

Properties functions .

Fucntions solution

monotonically increasing.

Test .

F-test

2d derrivatives .

−2φ1 and is not a solution