GE 111 Final Tutorial

“The greatest joy for a man is to defeat his enemies, to drive them before him, to take from them all they possess, to see those they love in tears, to ride their horses, and to hold their wives and daughters in his arms.”

- Genghis Khan (1162-1227)

By the end of this tutorial, you will be Genghis Khan, your knowledge will be a horde of Mongol warriors, and your GE111 exam will be an unprotected farming community in Central Asia.

Overview- Exponentials & Logarithms- Linear Data Graphing- Non-Linear Data Graphing- Trigonometry & Rotation 30%- Velocity- Material Balance- Matrices 70%- Complex Numbers

Exponentials & Logarithms

Exponentials : bx = y

y is equal to the base multiplied by itself x times

35 = 3 * 3 * 3 * 3 *3 = 243

Logarithms : logbx = y

y is equal to the power the base would be raised by to equal x

by = xlog3243 = 5

The ONE most important thing about Exponentials and Logarithms is:

Knowing the RULES

Exponential Rules

•Product Rule: bx * by = bx+y

•Quotient Rule: bx / by = bx-y

•Addition/Subtraction: bx + bx = 2 bx

•Multiplication: (bx)y = bxy

•Zero-th Power: b0 = 1

Logarithm Rules

•Product Rule: logb(x*y) = logbx + logby

•Quotient Rule: logb(x / y) = logbx - logby

•Power Rule: logb(xc)= c*logbx•Zero-th Power Rule: logb1=0•Change of Base Rule: logby = logay / logab

Exponential & Logarithm Examples

•(16)-3/4

•(x4)3 * (y5)-5 / (x7)

•3log(2x) – 2log(4x)

•log(x) / log(e)

Graphing Linear Data

• The linear form is y = mx+by is the dependent variablex is the independent variablem is the slope, or the rise/run relationshipb is the constant, or y-intercept of the relationship

x (independent) axis (abscissa)

y (dependent) axis

(ordinate)b

The TWO most important things about Graphing Linear Data are:

Formatting a GraphFinding your Variables

Proper Graph Format

Axis Label, Symbol (units)

Axis

Lab

el, S

ymbo

l (un

its)

Descriptive Graph Title

equation

Appropriate Scale

Proper Graph Paper

Variable Search Strategies

Slope, m: found by dividing rise/run, b = Δy / Δxy-intercept, b: found by setting x = 0, or reading from chart, or substituting in a pair of x and y variables with a known slopey & x: found by substitution into y = mx + b formula

Linear Graphing ExampleThe following data represent the time/velocity relationship for a skier travelling down a hill.

a) Plot this data on a rectilinear graphb) Determine the equation of the relationshipc) Estimate the average acceleration

Time, t(s) Velocity, v (m/s)

5 50

10 95

15 140

20 185

25 230

Non-Linear Data Graphing

• Two non-linear forms are used in GE111

•Exponential: y = b*emx (semi-log)

•Power: y = b*xm (log-log)

The TWO most important things about Graphing Non-Linear Data are:

Choosing the correct formConverting between forms

Converting between Forms

Exponential Base e: ln(y) = ln(b) + mx

Exponential Base 10: log(y) = log(b) + mx

Power: log(y) = log(b) + m*log(x)

Non-Linear Graph Examples

(midterm 2013)A fiberboard factory uses charcoal screens to keep airborne formaldehyde levels below 300 μg/m³. The relationship between the age of the screen (t, in days) and concentration removed from airflow (Cr, µg/m³) is exponential: Cr = Ci*e^(kt), where Ci is the initial concentration.

The concentration in the outflow airstream is thus Co=Ci-Cr. For a newly installed screen, the outflow concentration was measured at 130 μg/m³ for a time of 2.00 days, given a Ci of 500 μg/m³.

Given a new screen, how many hours will it take for the outflow airstream (Co) to reach a formaldehyde concentration of 300 μg/m³?

Trigonometry & Rotation

Trigonometry: uses identities sin(), cos(), and tan()as well as Pythagoras, and the Laws of Sines and Cosines

Rotation: uses trigonometry identities, angles in degrees and radians, and angular and tangential velocities

Basic Trigonometry

αAdjacent

Opposite

Hypotenuse

SOH CAH TOAsin(α) = Opposite/Hypotenusecos(α) = Adjacent/Hypotenusetan(α) = Opposite/Adjacent

Area = base * height/2

Pythagoras’ Theoremx2 + y2 = z2

Basic Rotation

α

S = arc length

r = ra

dius

S = α*r

(Degrees) * (π / 180) = (Radians)

The ONE most important thing about Trigonometry and Rotation is:

Knowing and Understanding the Equations

Trigonometry Equations

Law of Sines: a2 = b2 + c2 -2bc cos(A)

Law of Cosines: sin(A) = sin(B) = sin(C) a b c

B

CA

c a

b

Rotation Equations

θ = s/rω = θ/tf = ω/(2π)v = ω*r

θ = angular displacement (radians)s = arc length (m)r = radius (m)ω = angular velocity (radians/s)v = tangential velocity (m/s)f = frequency (rotations/s)

Trigonometry & Rotation Example

(midterm 2013)If the Earth is a perfect sphere with a radius of 6357 km and the distance between the Earth and sun is 150 million km, find:a) The angular rotation velocity and the angular revolution velocity of

the Earth in rad/s.b) The linear rotation velocity of Saskatoon in m/s, if the angle

between the Earth’s axis of rotation and a direct line between the Earth’s core and Saskatoon is 30 degrees.

c) The linear revolution velocity in m/s.

Velocity

Velocity is a vector expressing displacement over time: v = d / t

GE111 velocity problems involve velocity acting in multiple directions, and typically require trigonometry to solve.

The TWO most important things about Velocity are:

Drawing an Accurate Picture of the problemand understanding that Velocity is Trigonometry

Velocity Example

(2013 midterm)An pilot is operating a plane with malfunctioning equipment. He believes that the plane is flying with a speed of 128 km/h when it takes off from Maple Creek with a heading of 30 degrees, with a 20 km/h wind blowing from West to East, and intends to fly for 2.5 hours before reaching Saskatoon.a) Assume the pilots calculations are correct. What distance should the

airplane fly, between Maple Creek and Saskatoon?b) The real weather is actually a breeze from South to North at 30 km/h.

How far from Maple Creek will the airplane be after 2.5 hours?c) In case b), how far will the plane be from Saskatoon after 2.5 hours?

Material Balance

Material Balance problems deal with solving for changes in the material contents of a system.

Mass Out = Mass In ± Change in Storage

Mass In Mass Out

Change

The TWO most important things about Material Balance are:

Drawing an Accurate Picture and Setting Up Equations

Material Balance Example

(midterm 2009)Fresh orange juice contains 10 % solids and the balance water. Initially, a single evaporation process was used for the concentration, but volatile constituents of the juice escaped with water and left the concentrate with a flat taste. In order to overcome this problem, the present process bypasses the evaporator with a fraction of the fresh juice and then mixed with the juice that went through the evaporator. Fresh orange juice is fed to the process at the rate of 100 kg/hr. The juice that enters the evaporator is concentrated to 50 % solids and the desired final concentration of solids is 40 %. a)Calculate the flow rate of concentrated juice product. b)Calculate the flow rate of water removed from the evaporator. c)Calculate the percentage of fresh orange juice that bypasses the evaporator

Additional Problems

Which subjects need the most review?

• Exponentials & Logarithms• Linear Data Graphing• Non-Linear Data Graphing• Trigonometry & Rotation• Velocity• Material Balance

Logarithm & Exponential 1

(2013 midterm)

When log a and log b are two solutions of x^2 + 4x -2 = 0, find

logab + logba. Don’t use a calculator, or we will find you and burn down your family’s house. (not the exact wording of the question)

Logarithm & Exponential 2

Solve these simultaneous equations for p & q:

(log3p)2 = log3(p2)log3(p+q) = log3((p2+qp)/q)

Graphing Non-Linear Data 1

(2013 midterm)Plot the following acoustic attenuation data on a log-log scale, draw a line of best fit and use the method of selected points to find the equation. Clearly show the resultant equation, along with required calculations.

y x

200200 100

565600 200

7296500 1100

50590000 4000

Trigonometry & Rotation 1

Two pulleys of different sizes are connected by a belt. The larger pulley’s diameter is 10m and its angular velocity is 1.2 rad/s. The smaller pulley has an angular velocity of 3 rad/s. The centres of the 2 pulleys are 12m apart.

1) Determine the diameter of the smaller pulley2) How long would it take for a point on the belt to return to its

starting position if the pulleys are rotating in the same direction?3) Ditto, but pulleys are rotating in opposite directions.

Velocity 1

A medical emergency just occurred on a boat 10km west and 7km north of a harbour, and the boat needs to go to the harbour immediately to get help. The ocean current is towards the west at 10 km/hr, and the boat speed relative to water is 80 km/hr.1) What heading and bearing will get the boat back to dock?2) How long (in seconds) does the boat take to get to the harbour?

Material Balance 1

(2013 midterm)An oil refinery receives 35 rail cars of crude from Dakota Bakken and 40 cars from Great West each day. Each car holds 130m³. The refinery receives both streams at once, mixes them, and fractionates it into 3 outflow streams; gas, light oil, and heavy oil. Dakota Bakken (with a density of 0.80 tonnes/m³) fractionates into 15% gas, 65% light oil, and the remainder heavy oil. Great West (0.90 tonnes/m³) fractionates into 10% gas, 40% light oil, and the remainder heavy oil. All percentages are on a mass basis.1) Draw and label a material flow diagram using known values2) For a one day period, what is the mass percentage (to the nearest 0.1%) of the heavy

oil outflow stream of the three outflow streams.3) The light oil must form at least 60% of the outflow streams and 35 tank cars of Dakota

Bakken must be processed. How many whole tank cars of Great West are used per day and what is the final percentage light oil composition of the three outflow streams?

Matrices – Linear Equations

Linear equations have the form:

a1x1 + a2x2 … = b

And do NOT contain:-Products or Roots of Variables-Trig Functions of Variables-Log Functions of Variables-Exponents of Variables

If b=0, the equation is called homogeneous!

Dependent vs. Independent Equations- An equation is Linearly Independent if it cannot be generated by a linear combination of other equations

3 Elementary Row Operations:1. Multiply by a Constant2. Add or subtract another Equation3. Switch 2 Rows

Matrices A matrix is a useful way to arrange numbers or coefficients in order to perform operations on them

Augmented Matrix: contains the coefficients and solutions to a set of linear equations

Identity Matrix: contains 1’s in its PRINCIPAL DIAGONAL, and zeroes. Row-Echelon Form

Reduced Non-Reduced

Matrices- Solving!Gauss Elimination Gauss-Jordan Elimination

-row echelon form -reduced row echelon form

-Both methods use the 3 Elementary Row Operations!1. Multiply by a Constant2. Add or subtract another Equation3. Switch 2 Rows

Solve these Matrices

Matrix Basics

All matrices consist of elements indicated by a subscript The subscript is of the form “row, column”

Matrix Properties

Trace: The sum of the principal diagonal

Rank: The number of linearly independent rows

Determinant: The sum of ‘ \ ‘ diagonal products minus the sum of the ‘ / ‘ diagonal products

Find the Trace, Rank and Determinant of this Matrix!

Matrix Operations – Addition & Subtraction

Matrices must be the same size.

Matrix Operations – Addition & Subtraction

Matrix Operations: Multiplication

Matrix Multiplication for Linear Systems

Matrices are often used to express systems of linear equations, where the form Ax=b is used.

A= coefficient matrixx=variable matrixb=solutions matrix

A x b

Matrix Operations- Transpose

A matrix transpose, indicated with a superscript ‘T’ is found by swapping the matrix’s rows with the columns

Find the transpose of this matrix:

Matrix Operations- Inverse

Inverse of a 2x2 Matrix:

The inverse of a matrix is defined as:

This is an important property because:

Invert this matrix!

3x3 - Gauss-Jordan Inverse Strategy

Use Elementary Matrix Properties to move the Identity matrix from the right to the left side of the matrix.

Invert this matrix!

Inverse – Adjoint Method Definitions

Minor: Determinant of a smaller square matrix

Cofactor: special minor matrix

Adjoint: Transpose of Cofactor Matrix

Inverse – Adjoint Method

Invert this matrix!

Determinant- Cofactor Method

Find the determinant of the following matrix by this method:

Cramer’s Rule- Method to Solve Matrices1. Find Determinant of Coefficient Matrix2. Substitute Solution column for first Coefficient column3. Calculate Determinant of new matrix4. Repeat for all columns of coefficient matrix5. Each variable is then found by the ratio of itssubstituted determinant to the actual determinant

Let’s do an example!

Eigenvalues & Eigenvectors

λ represents the eigenvalues of a given matrix if it fulfills the above equations.This is not as scary as it seems!

Find the Eigenvalues and Eigenvectors!

Complex NumbersThe imaginary number ‘i’ is the primary facet of complex number theory

All numbers are said to have a real and an “imaginary” component,and can be graphed 2-dimensionally in the complex plane

Addition, subtraction, and multiplicationoccur as you might expect. Let’s do someexamples.

Conjugate & Modulus, for:

Conjugate Modulus

Complex Numbers – Polar Form

Polar form is an alternate way of expressing the valueof a complex number using the complex plane

Complex Numbers – Euler’s Formula

“If you are afraid, do not do it. If you are doing it, do not be afraid.”

Good Luck!