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) • • • • • • 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 • • • • • 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 • • • • • 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 • • • • • 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? •Read pg 213 Use the chain rule to find f ’(X) •f(X) = (3X+1)⁴ The chain rule Read pg 215 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 • Read page 220 • 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 •Read page 224 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? • • • • 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 • • • • • 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 Read pages 7 and 8 of your textbook • 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) In your notebook 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) In your note books 1a). Define the following terms and give two examples of each. (Use google) (i) Extensive property (ii) Intensive property FORMATIVE GRADE: Classroom materials DEADLINE: 25th Sept 2019 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. • • • • 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 https://www.youtube.com/watch?v=bkYqqJa5P8w Mixtures can be separated using the following methods. • • • • 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.
