# IB MATH SL NOTES ```IB MATH SL
G12
7.2 The tangent line
Objectives
• Find the gradient of a secant
• Write the expression for the gradient of a secant line using the difference
quotient.
• Use the derivative formula to find the gradient of a tangent line at a given
value of x
• Use the definition of derivative to find the derivatives of X2, X3, X4 and
make a conjecture about the derivative of Xn (power rule)
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Meaning of convergent sequence and write its notation
Meaning of secant line and Tangent line
Investigation (done in pairs)
Isaac Newton’s concept about the gradient of a curve at a given point
Expression for difference quotient
Ex 7C # 1
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Function that gives the gradient of f and x
Derivative of f notations
Ex 7D #1
Investigation of f(x)=Xn
Ex 7E
Bell ringer
•Exercise 7E # 6 Pg 202
7.2 More rules for derivatives
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Constant rule pg 204
Constant multiple rule
Sum or difference rule
Example 6
Exercise 7F Pg 205
Equations of tangent and normal lines
• Example 7 pg 205
• Exercise 7G pg 207
Bell ringer (5min)
• If f(x) = ¼X³ - 3X, Find f ’(4)
More rules for derivatives pg 208
• If f(x) = ¼X³ - 3X, Use a GDC to find f ’(4)
(i) nDeriv under MATH-8
(ii) dy/dx under CALC
Investigate the derivatives of exponential
functions
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Investigation pg 209
f(x) = lnx [Y1=, 2ndCalc#1, x=2 Enter]
f ’(x) = [2ndCalc#6, x=2 Enter]
Make a conjecture about the derivative of f(x) = lnx
Example 8 pg 209
Bell ringer
Find the derivative of each function.
1. f(X) = (3X + 1)(X²- 1)
2. f(X) = 2X/X³
3. f(X) = (3X+1)²
Product rule (pg211)
Use the product rule to find the derivative
of the following
f(X) = (3X + 1)(X²- 1)
The quotient rule (pg 211)
• Use the quotient rule to find the
derivative of each function.
1. f(X) = 2X/X³
When to use the rules?
Use the chain rule to find f ’(X)
•f(X) = (3X+1)⁴
The chain rule
Example 12
Formula booklet
•The product, quotient and
chain rules are all given in the
formula booklet.
CW
Exercise 7J page 214 #1, 3, 4, 14, 17, 18
Exercise 7K page 217 #1, 3, 7, 9, 10
TEST REMINDER
• DATE: 16TH OCT 2019
• CHAPTERS 7
• PERIOD B
Classwork
•Exercise 7I page 212 #1-10
Bell ringer
•Exercise 7L page 219 #5
Higher order derivatives
• Example 15 page 220
• Exercise 7M # 1-6 (work in pairs)
CW
• Ex 7H Page 209 #8-12
Rates of change and motion in a line
Objectives:
1. Calculate average rate of change e.g. average
velocity (within an interval of time)- Use slope of
secant line
2. Calculate instantaneous rates of change e.g.
velocity at a given time. – Use slope of tangent line
Examples
•Example 16 page 221
•Example 17 page 222
Exercise 7N page 223
• #1 (All)
• #2 (pair)
• #3 CW
• #4 HW
Motion in a line
Vector and scalar
• A vector is a quantity with both magnitude
and direction. For example displacement,
velocity, acceleration.
• A scalar is a quantity with only magnitude. For
example distance, speed
Displacement, s(t)
If
• S(t) ˂ 0; object is moving to the left or
below origin
• S(t) ˃ 0; object is moving to the right or
above origin
Velocity, V(t)
• Velocity is the instantaneous rate of change of
displacement. V(t) = S’(t)
If
• V(t) ˂ 0; object is moving left or down
• V(t) ˃ 0; object is moving right or up
• V(t) = 0; object is at rest ; V(0) is initial velocity
Activity
•Example 18 page 224
•Exercise 7O page 225 # 1-3
Acceleration, a(t)
• Acceleration is the instantaneous rate of change of
velocity, V’(t)
• For a(t) = 0 ; velocity is constant
• a(t) ˂ 0; velocity of object is decreasing
• a(t) ˃ 0; velocity of object is increasing
Speeding up(speed is increasing)
V(t) and a(t) have the same signs
+ and +
Or
- and -
Slowing down (speed is decreasing)
• V(t) and a(t) have the opposite signs
• + and –
• Or
• - and +
Activities
• Investigating v(t), a(t) and speed pg 227
• Example 20 pg 228
• Exercise 7P; page 229 # 1-4
8.1 Univariate analysis
Objectives
Population, sample, random sample, discrete and continuous data
Presentation of data: frequency distributions (tables); frequency
histograms with equal class intervals; box and whisker plots;
outliers; grouped data
Use of mid-interval values for calculations; Interval width,
boundaries and modal class.
Statistics
Statistics is a set of tools used to
organize and analyze data.
Statistics is concerned with;
• Designing experiments and other data collection
• Representing and analyzing information to aid
understanding
• Drawing conclusions from the data
• Estimating the present and predicting the future
Calculating median, mean, Q1, Q3 (TI-84)
• Stat1: Edit (enter data)
• Stat → CALC1: 1-Var stats –Enter-Enter-Enter
Graphing a frequency histogram
1. Entering the data
Stat , Enter [L1 Enter the boundary numbers|L2 Enter the frequency]
2. Setting the graph features
2nd, Y1=, Enter, Enter- select the histogram icon [↓2nd, 1 Enters L1 |↓2nd 2
Enter L2]
3. Graph the histogram: Zoom ↓ Zoom stat, Enter |Window ↓ Xscl = [type in
class width] -Graph
Calculating mean using frequency
• Vars 5: Statistics → ∑
• Find ∑XY and ∑Y then divide
Objectives
• Define Chemistry
• List examples of the branches of chemistry
• Distinguish between the physical and chemical properties
What is chemistry?
• Chemistry is the study of the composition, structure and properties of
matter and the changes matter undergoes .
• Chemistry deals with questions such as: What is a material’s make up? How
does a material change when heated, cooled or mixed with other materials
and why does this behavior occur? Chemists answer these kinds of questions
during their work.
Branches of chemistry
1. Organic chemistry- the study of most carbon-containing compounds.
2. Inorganic chemistry- the study of non-organic substances.
3. Physical chemistry- the study of the properties and changes of matter and their
relation to energy.
4. Analytical chemistry- the identification of the components and composition of
materials.
5. Biochemistry- the study of substances and processes occurring in living things.
6. Theoretical chemistry- the use of mathematics and computers to understand the
principles behind observed chemical behavior.
Matter and its properties
• All things are made up of matter.
What is matter?
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Matter is anything that has mass and takes up space.
Mass is a measure of the amount of matter.
Mass is measured using a balance.
All matter has volume (Volume is the three-dimensional space an object
occupies .)
Building blocks of matter
• Atoms and molecules are the building blocks of matter.
• These particles make up elements and compounds.
• An atom is the smallest unit of an element that maintains the chemical
identity of that element.
Element
• An element is a pure substance that cannot be broken down into simpler,
stable substances and is made of one type of atom. E.g carbon is an element
and contains one kind of atoms.
Q1
Name any five examples of elements
1………………………………….
2…………………………………..
3…………………………………….
4…………………………………..
5……………………………………
Compound
• A compound is a pure substance that can be broken down into simple stable
substances. Each compound is made from the atoms of two or more
elements that are chemically bonded.
• For example water is a compound made of two elements- Hydrogen and
oxygen.
Oxygen atom
Hydrogen atom
Q2
• State the two types of pure substances.
a)……………………………………..
b)……………………………………
Q3
Identify each of the following as either an element or compound
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CO
K
Mg
N2
HCl
Physical properties and physical changes
• A physical property is a characteristic that can be observed or measured
without changing the identity of the substance.
• Examples of physical properties include; melting point, boiling point,
density, odor, color.
• Physical change is the change in a substance that does not involve a change
in the identity of the substance.
• Examples of physical changes include; grinding, cutting, melting and boiling
a material.
CW
• Complete the table below.
Physical property
Physical Change
Definition
Examples (i)
(i)
(ii)
(ii)
(iii)
(iii)
(iv)
(iv)
Characteristic property
• The characteristic properties of a substance are always the same whether the
sample being observed is large or small.
• Examples of characteristic properties include freezing/melting point,
boiling/condensing point, density, viscosity and solubility
Bell ringer (5min)
1. Give two examples of physical property.
2. Give two examples of physical change
Objectives
• Classify changes in matter as physical or chemical
Chemical properties
• A chemical property relates to a substance’s ability to undergo changes that
transform it into different substances.
• Examples of chemical properties include;
• - Ability of charcoal(C) to burn in air to form carbon dioxide gas
• - Ability of hydrogen gas to burn in air to form water
Chemical changes
• A chemical change or chemical reaction is a change in which one or more
substances are converted into different substances.
• The substances that react in a chemical change are called reactants.
• The substances that are formed by the chemical change are called the
products.
• In case of burning charcoal, carbon and oxygen are the reactants in a
combustion or burning reaction. Carbon dioxide gas and ashes are the
products.
Guided practice
Quizlet
Is it Physical or chemical change?
Bell ringer (5 min)
1. What are the three states of matter?
2. What are the 3 states of matter made up of ?
Objectives
• Describe the motion of particles in matter (solid, liquid and gas) according
to the kinetic molecular theory
• Distinguish between the two types of solids.
Kinetic Molecular Theory of Matter
• Matter is made up of atoms and molecules. These atoms and molecules act
like tiny particles that are always in motion.
• All particles of matter possess kinetic energy
- The higher the temperature of the substance, the faster the particles move.
• The kinetic energy of the particles increase in the order : Solids, Liquids and
gases.
• The kinetic theory helps to explain the difference between the 3 common
states of matter: solid, liquid and gas.
• You can classify matter as a solid, a liquid or gas by determining whether the
shape and volume are definite or variable.
Three States of Matter
Types of Solids
There are two main categories of solids: crystalline and amorphous
• Crystalline Solid:
Crystalline solids are the most common type of solid. They are characterized by a regular crystalline
organization of atoms that confer a long-range order.
Examples: Diamonds, metals, salts, etc.
• Amorphous Solid:
A solid with no defined shape (not a crystal)
A solid that lacks an ordered internal structure
Examples: Clay, Rubber, Glass, Plastic, Asphalt
Bell ringer (10min)
1a). Define the following terms and give two
(i) Extensive property
(ii) Intensive property
1.
2.
3.
4.
5.
6.
Portfolio with 40-60 pockets
Colored pencils, highlighters
Subject notebook
Signed copy of classroom rules and expectations.
Pen (red, blue, black)
Eraser, sharpener and ruler.
Objectives
• Discuss the change in particle movement, energy
and space between particles
• Describe the process of boiling, freezing, melting,
sublimation and deposition.
What kind of energy do all particles of matter
have?
• Because they are in motion, all particles of matter have kinetic energy.
- Energy is the capacity to do work.
• Temperature is a measure of average kinetic energy.
• Thermal energy depends on particle speed and number of particles.
- Thermal energy is the total kinetic energy of a substance’s atoms.
Kinetic Energy and States of Matter
Give the state of matter with the given
characteristic
1. Particles have the highest kinetic energy………………..
2. Particles slide over each other………………………….
3. Has the strongest force of attraction between particles...
4. Particles vibrate in a fixed position………………….
5. Particles have kinetic energy……………………
6. Particles very loosely packed………………………………….
7. Has a definite volume but no definite shape………………..
8. It takes up the volume of any container in which it is put………..
9. It has a fixed shape and fixed volume…………………………..
10. It has a fixed volume but variable shape ……………………….
Changes of state
• Define the following terms and give an example of each.
1. Melting…………………………………………………………………………
2. Boiling…………………………………………………………………………..
3. Freezing…………………………………………………………………………
4. Condensation…………………………………………………………………..
5. Sublimation…………………………………………………………………….
6. Deposition……………………………………………………………………….
Bell ringer
1. State the difference between an element and
a compound.
2. Give two examples of elements and two
examples of compounds.
TEST (26/09/2019)
• Physical and Chemical properties
• Physical and Chemical changes
• Kinetic Molecular theory of matter
• States of Matter
Objectives
• Distinguish between Heterogeneous and homogeneous mixtures
• List three different solute-solvent - and combination
Mixture and Pure substance
• A mixture is a blend of two or more kinds of matter, each of which retains
its own identity and properties.
• Pure substance has a fixed composition. The composition of a pure
substance is the same throughout and does not vary from sample to sample.
For example, pure water is always 11.2% hydrogen and 88.8% oxygen by mass.
• A pure substance can be an element or compound.
Homogeneous and heterogeneous mixtures
• Mixtures that are uniform in composition are said to be
homogeneous(solution). For example a salt-water solution, air, metallic alloy
(e.g stainless steel)
• Mixtures that are not uniform throughout are said to be heterogeneous.
For example blood, granite, wood, milk
Solution
• A solution consists of solute and solvent. A solvent is that component which
is present in larger amount by mass.
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Examples of solutions
Salt-water solution: Solvent is………..and solute is………………….
Air: Solvent is ……………………and solute is oxygen and other gases
Stainless steel: Solvent is ……………..and solute is…………………..
Suspensions and colloids
• Suspensions and colloids are heterogeneous mixtures. A suspension is
identifiable because its particles are large and settle out of the dispersing
medium due to the effects of gravity. The dispersed particles of a colloid are
intermediate in size between those of a solution and a suspension.
CW
• STATES OF MATTER WORKSHEET
Objectives
• Discuss separation methods for mixtures
Separation of mixtures
Mixtures can be separated using the following methods.
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Filtration. Separating an insoluble solid from a liquid.
Separating funnel: separating immiscible liquids e.g mixture of oil and water
Using a centrifuge – separation of solid-liquid mixtures such as those in blood.
Chromatography –separation of mixture of dyes or pigments because the different substances move at
different rates on the paper.
• Evaporation to dryness removes a liquid from a solution to leave a solid material. E.g separating salt-water
solution.
• Distillation –separating a mixture of liquids with different boiling points e.g ………..
1. Evaporate the liquid by heating. 2. Condense the vapour by cooling
Objectives
• Explain the difference between evaporation and distillation
Distillation
Separating pure substances (compounds)
1. Electrolysis: Passing electrical current through water causes the compound
to break down into the elements hydrogen and oxygen.
2. Decomposition by heating: Example- when sucrose (C12H22O11) is
heated to high enough temperature, it breaks down completely into carbon
and water.
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