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
- Velocity
- Material Balance
- Matrices
- Complex Numbers
30%
70%
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 = x
log3243 = 5
The ONE most important thing about
Exponentials and Logarithms is:
Knowing the RULES
Exponential Rules
•Product Rule:
•Quotient Rule:
x
b
y
b =
x+y
b
*
bx / by = bx-y
•Addition/Subtraction: bx + bx = 2 bx
x
y
xy
•Multiplication:
(b ) = b
•Zero-th Power:
b0 = 1
Logarithm Rules
•Product Rule:
•Quotient Rule:
logb(x*y) = logbx + logby
logb(x / y) = logbx - logby
•Power Rule:
•Zero-th Power Rule:
•Change of Base Rule:
logb(xc)= c*logbx
logb1=0
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
y (dependent)
axis
(ordinate)
b
• The linear form is y = mx+b
y is the dependent variable
x is the independent variable
m is the slope, or the rise/run relationship
b is the constant, or y-intercept of the relationship
x (independent) axis
(abscissa)
The TWO most important things about
Graphing Linear Data are:
Formatting a Graph
Finding your Variables
Proper Graph Format
Axis Label, Symbol (units)
Descriptive Graph Title
Appropriate Scale
Proper Graph Paper
Axis Label, Symbol (units)
Variable Search Strategies
Slope, m: found by dividing rise/run, b = Δy / Δx
y-intercept, b: found by setting x = 0, or reading from chart, or
substituting in a pair of x and y variables with a known slope
y & x: found by substitution into y = mx + b formula
Linear Graphing Example
The following data represent the time/velocity relationship
for a skier travelling down a hill.
Time, t(s)
Velocity, v (m/s)
5
50
10
95
15
140
20
185
25
230
a) Plot this data on a rectilinear graph
b) Determine the equation of the relationship
c) Estimate the average acceleration
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 form
Converting 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
SOH CAH TOA
sin(α) = Opposite/Hypotenuse
cos(α) = Adjacent/Hypotenuse
tan(α) = Opposite/Adjacent
Opposite
α
Area = base * height/2
Adjacent
Pythagoras’ Theorem
x2 + y2 = z2
Basic Rotation
S = α*r
α
(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
c
B
A
a
C
b
Rotation Equations
θ = s/r
ω = θ/t
f = 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 problem
and 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 pulley
2) 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 values
2) 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 Constant
2. Add or subtract another Equation
3. 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 Constant
2. Add or subtract another Equation
3. 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 matrix
x=variable matrix
b=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
The inverse of a matrix is defined as:
Invert this matrix!
This is an important property because:
Inverse of a 2x2 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 Matrices
1. Find Determinant of Coefficient Matrix
2. Substitute Solution column for first Coefficient column
3. Calculate Determinant of new matrix
4. Repeat for all columns of coefficient matrix
5. Each variable is then found by the ratio of its
substituted 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 Numbers
The 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 multiplication
occur as you might expect. Let’s do some
examples.
Conjugate & Modulus, for:
Conjugate
Modulus
Complex Numbers – Polar Form
Polar form is an alternate way of expressing the value
of 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!