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Unit 1: Fundamentals of Trigonometry
My goals
IntroductionTrigonometry studies relationship involving lengths and angle of a triangle.
The techniques in trigonometry are used for finding relevance in navigation particularly satellite systems and
astronomy, naval and aviation industries, oceanography, land surveying, and in cartography (creation of maps).
Now those are the scientific applications of the concepts in trigonometry, but most of the mathematics we study would seem (on the surface) to have little real-life application. Trigonometry is really relevant in our day to day activities. The following pictures show us the areas where this science finds use in our daily activities.In this unit we will see how we can use this to resolve problems we might encounter.
Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4Required outcomes
After completing this unit, the learners should be able to:
»» Define sine, cosine, and tangent (cosecant, secant and
cotangent) of any angle – know special values.
»» Convert radians to degree and vice versa.
»» Use trigonometric identities.
»» Apply trigonometric formulae in real world problems.1. Trigonometric concepts
Trigonometry is the study of how the sides and angles of a triangle are related to each other. A rotation angle is formed by rotating an initial side through an angle, about a fixed point called vertex, to terminal position called terminal side.It is drawn in what is called standard position if the initial side is on the positive x-axis and the vertex of the angle is at the origin.
Angles in standard position that have a common terminal side are called co-terminal angles; the measure of smallest positive rotation angle is called principal angle. Angle is positive if rotated in a counterclockwise direction and negative when rotated clockwise. Angles are named according to where their terminal side lies, for instance, the x-axis and y-axis divide a plane into four quadrants as follow
Example 2
Draw each of the following angles in standard position and which of these angles is co-terminal to 300 ?
a) 300 b) 1200 c) -2300 d) 7500 e) -3300Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4
Example 3
Draw each of the following angles in standard position and indicate in which quadrant the terminal side is.
Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4
Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4
Notice: Unit circle
A unit circle is a circle of radius
one centered at the origin (0,0) in
the Cartesian coordinate system
in the Euclidean plane.
In the unit circle, the coordinate
axes delimit fourquadrants that are numbered in an anticlockwise direction. Each quadrant measures 90 degrees, meaning that the entire circle measures 360 degrees or 2π radians.Exercise 2
Trigonometric ratios of acute angles
Activity 3
Construct two right angled triangles, one of which is an enlargement of the other.How is the side opposite to the right angle (or the longest side)
called?
For both triangles, consider an angle and compute the following
ratios.
• Opposite side to the considered angle and hypotenuse.
• Adjacent side and hypotenuse.
• Opposite side to the considered angle and adjacent side.
How can you conclude?Consider the following circle with radius r.
In this triangle, we define the following
six ratios:
• The three primary trigonometric values. The ratio x/r is called cosine of the angle
α , noted cos α .• The ratio y/r is called sine of the angle α ,
noted sin α• The ratio y/x is called tangent of the angle
α , noted tan α .
Example 10
Calculate the six trigonometric values for the diagram.
Solution
Use adjascent = 3,opposite = 4 and hypotenuse = 5
Example 11
For each angle, calculate
the reciprocal trigonometric
values.
Example 12
Exercise 3
For each angle, calculate the reciprocal trigonometric values
Trigonometric Number of special Angles 300, 450, 600
From the figure, we have;
CAST RuleActivity 5
Copy and complete the table to summarise the signs of the trigonometric values in each quadrant.
The following diagram shows which primary trigonometric values are positive in each quadrant. This is called the CAST rule.
Sine is positive in first and second quadrant but negative in third and fourth quadrant. Cosine is postive in the first and fouth quadrant but negative in second and third quadrant Tangent is negative in in the first and third quadrants but negative in second and fourth quadrant.
Trigonometric identities
Activity 6Here is a right triangle.
Basic Rules
Exercise 6
2. Triangle and applications
Solving triangle
Solving a triangle is to find the length of its sides and measures of its angles. There are two methods for solving a triangle: cosine law and sine law.Cosine law
Exercise 7
Sine law
CX is perpendicular to side AB and BY is perpendicular to side AC. Let AB =c, AC=b, BC=a, CX=h and BY=k
1. In triangle BCX find sin B. In triangle AXC find sin A. Deduce the relationship between side a and side c.
2. In triangle ABY find sin A. In triangle BCY find sine C.Deduce the relationship between side b and side a.
3. Deduce relationship between three sides.
The sine law (or sine formula or sine rule) is an equation relating the lengths of the sides of a triangle to the sine of its angles.
If a,b,c are the lengths of the sides of a triangle and A, B,C are the opposite angles respectively, then thesine law is
Exercise 8
Applications
Many real situations involve right triangles. Using angles and trigonometric functions, we can solve problems
involving right triangle. We have already seen how to solve a triangle.
If B is east of A then the bearing is 90o.Similarly, if B is south of A then the bearing is 180o, and if B is west of A then the bearing is 270o. The bearing can be any number between 0 and 360, because there are 360 degrees in a circle. We can also use right triangles to find distances using angles given as bearings. In navigation, a bearing is the direction from one object to another. Further, angles in navigation and surveying may also be given in terms of north, east, south, and west. For example, N70 E refers to an angle from the north, towards
the east, while N70 0W W refers to an angle from the north, towards the west.
Example 19A ship travels on a N500E course. The ship travels until it is due north of a port which is 10 kilometers due east of the port from which the ship originated. How far did the ship travel?
solution
Example 20
An airplane flies on a course of S300E , for 150 km. How far south is the plane from where it originated?
Solution
Using known information, consider the following figure:
2. Angle of elevation and angle of depression
You can use right triangles to find distances, if you know an angle of elevation or an angle of depression. The figure below shows each of these kinds of angles.
Suppose that an observer is standing at the top of a building and looking straight ahead at the birds (horizontal line). The observer must lower his/her eyes to see the car parked (slanting line). The angle formed between the two lines is called the angle of depression.
Suppose that an observer is standing at the top of a building and looking straight ahead at the birds (horizontal line). The observer must raise his/her eyes to see the airplane (slanting line). The angle formed between the two lines is called the angle of elevation.
Example 21
The angle of elevation of the top of a pole measures 48 from a point on the ground 18 metres away from its base. Find the height of the flagpole.
Solution
Example 22
An airplane is flying at a height of 2 kilometres above the level ground. The angle of depression from the plane to the foot of a tree is 15o. Find the distance that the air plane must fly to be directly above the tree.
Solution
Let x be the distance the airplane must fly to be directly above the tree. The level ground and the horizontal are parallel, so the alternate interior angles are equal in measure.
So, the airplane must fly about 7.46 kilometres to be directly above the tree.
3. Inclined plane
An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. On the inclined plane the weight of the object causes the object to push into and, the object slides, to rub against the surface of the incline. Also the weight causes the object to be pulled down the slant of the incline. The component that pushes the object into the surface is
Example 23
An object with a mass of 2.5 kg is placed on an inclined plane. The angle of the inclined is 20 degrees. What are the parallel force and the perpendicular force? (g=9.8m/s2)
Solution
Unit summary
1. Trigonometry is the study of how the sides and angles of a triangle are related to each other. A rotation angle is formed by rotating an initial side through an angle, about a fixed point called vertex, to terminal position called terminal side. Angle is positive if rotated in a counterclockwise direction and negative when rotated clockwise.
2. The amount we rotate the angle is called the measure of the angle and is measured in: degree, grade or radian.
4. Unit circle is a circle of radius one centered at the origin (0,0) in the Cartesian coordinate system in the Euclidian plane. In the unit circle, the coordinate axes delimit four quadrants that are numbered in an anticlockwise direction.
Each quadrant measures 90 degrees, means that the entire circle measures 360 degrees or radians.
9. Applications
Many real situations involve right triangle. Using angles and trigonometric functions, we can solve problems involving right triangle like:
• Bearings and air navigation
• Angles of elevation and angle of depression
• Inclined pla
File: 1UNIT 2 : Set IR of real numbers
My goals
Introduction
Real numbers are all Rational and Irrational numbers. They can also be positive, negative or zero. A simple way to think about the Real Numbers is: any point anywhere on the number line. Powers (convenient way of writing multiplications that have many repeated terms), radicals (an expression that has a square root, cube root, …), a logarithm (the power to which a number must be raised in order to get some other number) of a number .... are all real numbers.
If you have bacteria that divide every 40 minutes and are currently taking up 0.1% of the petri dish, you can use logarithms to estimates how long it will take them to fill up the entire dish. The same goes for 2000 francs in an account, logarithms will tell you when you will have 4000 francs. In finance and business logarithms can be useful for calculating compound interest.
Required outcomesAfter completing this unit, the learners should be able to:
» Define absolute value of a real number and solve simple equations involving absolute value.
» Define powers and their properties.
» Define radicals and their properties.
» Define decimal logarithm of a real number and solve simple logarithmic equations.
1. Absolute value and its properties
Activity 1
Draw a number line and state the number of units that are between;
Absolute value of a number is the distance of that number from the original (zero point) on a number line. The symbol
|| is used to denote the absolute value.
Example 1
7 is at 7 units from zero, thus the absolute value of 7 is 7 or |7| = 7. Also -7 is at 7 units from zero, thus the absolute value
Solution
Solution
Exercise 1
Find the value(s) of x
Activity 2
Evaluate the following operations and compare your results in eache case
Properties of the Absolute Value
Opposite numbers have equal absolute value.
Example 4
The absolute value of a product is equal to the product of the absolute values of the factors.
Example 5
The absolute value of a sum is less than or equal to the sum of the absolute values of the addends.
Example 6
Exercise 2
Simplify:
2. Powers and radicals
Powers in IR
Activity 3
Peter suggested that his allowance be changed. He wanted $2 the first week, with his allowance to be doubled each week. His parent investigated the suggestion using this table
1. Complete the table to find how many dollars Peter would be paid each of the first five weeks.
2. How much would Peter be paid the seventh week? The tenth week?
3. Do you think his parent will agree with his suggestion? Explain.
Example 7
Notice
Properties of powers
These properties help us to simplify some powers. There is no general way to simplify the sum of powers, even when the powers have the same base. For instance,
is not an integer power of 2. But some products or ratios of powers can be simplified using repeated multiplication model of n a.Example 8
Exercise 3
Simplify
Radicals in real numbers
Activity 4
Evaluate the following powers
Example 9
Example 10
Properties of radicals
Example 11
Simplify:
Solution
Exercise 4
Simplify:
Operations on radicals
When adding or subtracting the radicals, we may need to simplify if we have similar radicals. Similar radicals are the radicals with the same indices and same bases.
Activity 5
Simplify the following and keep the answer in radical sign
Addition and subtraction
When adding or subtracting the radicals we may need to simplify if we have similar radicals. Similar radicals are the radicals with the same indices and same bases.
Example 12
Exercise 5
Simplify
Rationalizing
Activity 6
Make the denominator of each of the following rational;
Rationalizing is to convert a fraction with an irrational denominator to a fraction with rational denominator. To do this, if the denominator involves radicals, we multiply the numerator and denominator by the conjugate of the denominator.
Example 13
Exercise 6
Rationalize the denominator;
3. Decimal logarithms and properties
Activity 7
What is the real number for which 10 must be raised to obtain
1. 1 2. 10
3. 100 4. 1000
5. 10000 6. 100000
The decimal logarithm of a positive real number x is defined to be a real number y for which 10 must be raised to obtain x.
Example 14
log (100) = ?
We are required to find the power to which 10 must be raised to obtain 100
So log(100) = 2
y= log x means 10y = x
Be Careful!
Example 15
Example 16
Example 17
Solution
Example 18
Solution
Co-logarithm
Co-logarithm, sometimes shortened to colog, of a number is the logarithm of the reciprocal of that number, equal to the negative of the logarithm of the number itself,
Example 19
Exercise 7
Unit summary
UNIT 3 : Linear, quadratic equations and inequalities
My goals
Introduction
An equation is a statement that the values of two mathematical expressions are equal while an inequality is a statement that the values of two mathematical expressions that are not equal.
Lots of students are interested in aviation. Would you like to be a private pilot and fly your own small plane? You can’t pass the test to get a pilot’s license unless you can work out equations and calculate how to load the plane correctly with people and baggage. You should already understand what it means to make a profit - you sell an item for more than it cost to make it. So in very simple terms, companies use this equation all the time: Profit = selling price - cost. There are many financial decisions we make everyday based on how much we earn, less all our expenses, then we say we can spend up to x on a new purchase or you put x in a saving account, or something else. These are all inequalities.
A quadratic equation is an equation of degree 2, meaning that the highest exponent is 2. In daily life, say that you want to throw a ball into the air and have your friend catch it, but you want to give her/him the precise time it will take the ball to arrive.
To do this, you would use the velocity equation, which calculates the height of the ball based on a parabolic (quadratic) equation. Quadratics are equations and inequalities are useful in calculating areas, figuring out profits, finding speeds, athletics,…
Required outcomes
After completing this unit, the learners should be able to:
» Solve equation of the first degree and second degree.
» Solve inequality of the first degree and second degree.
» Solve a system of linear equations.
» Use equations and inequalities to solve word problems.
» Apply equations and inequalities in real life problems.
1. Equations and inequalities in one unknown
Equations
Activity 1
Find the value of x such that the following statements are true;
To do this, rearrange the given equation such that variables will be in the same side and constants in the other side and then find the value of the variable.
Example 1
Solve in set of real numbers:
Solution
Exercise 1
Solve in set of real numbers:
Equations products / quotients
Activity 2
State the method you can use to solve the following equations
Example 2
Solve in set of real numbers
Solution
Exercise 2
Solve in set of real numbers:
Inequalities
Activity 3
Find the value(s) of x such that the following statements are true
Recall that
• When the same real number is added or subtracted from each side of the inequality, the direction of the inequality is not changed.
• The direction of the inequality is not changed if both sides are multiplied or divided by the same positive real number and is reversed if both sides are multiplied or divided by the same negative real number.
Example 3
Solve in set of real numbers:
Solution
Exercise 3
Solve the following inequalities
Inequalities products / quotients
Activity 4
State the method you can use to solve the following inequalities
Example 4
Solve in set of real numbers;
Solution
Exercise 4
Solve the following inequalities:
Inequalities involving absolute value
Activity 5
State the set of all real numbers whose number of units from zero, on number line, are
1. greater than 4
2. less than 6
Hint Draw a number line
Example 5
Solution
Example 6
Solution
Example 7
Solution
Example 8
A technician measures an electric current which is 0.036 A with a possible error of ±0.002 A. Write this current, i, as an inequality with absolute values.
Solution
The possible error of ±0.002 A means that the difference between the actual current and the value of 0.036 A cannot be more than 0.002 A.
So the values of i we have can be expressed as:
Exercise 5
Solve the following inequalities:
Equations and inequalities in real life problems
Activity 6
How can you do the following?
1. A father is 30 years older than his son. 5 years ago he was four times as old as his son. What is the son’s age?
2. Betty spent one fifth of her money on food. Then she spent half of what was left for a haircut. She bought a present for 7,000 francs. When she got home, she had 13,000 francs left. How much did Betty have originally?
Equations can be used to solve real life problems.
To solve real life problems, follow the following steps:
a) Identify the variable and assign symbol to it.
b) Write down the equation.
c) Solve the equation.
d) Interpret the result. There may be some restrictions on the variable.
Example 9
Kalisa is four times as old as his son, and his daughter is 5 years younger than his brother. If their combined ages amount to 73 years, find the age of each person.
Solution
Example 10
Concrete is a mixture of cement, sand and aggregate. If 4 kg of sand and 6kg of aggregate are used with each kg of cement, how many kg of each are required to make 1,210 kg?
Solution
Example 11
John has 1,260,000 Francs in an account with his bank. If he deposits 30,000 Francs each week into the account, how many weeks will he need to have more than 1,820,000 Francs on his account?
Solution
Example 12
The yield from 50 acres of wheat was more than 1,250 tones. What was the yield per acre?
Solution
Exercise 6
1. The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the numbers.
2. The difference between the two numbers is 48. The ratio of the two numbers is 7:3. What are the two numbers?
3. The length of a rectangle is twice its breadth. If the perimeter is 72 metre, find the length and breadth of the rectangle.
4. Aaron is 5 years younger than Ron. Four years later, Ron will be twice as old as Aaron. Find their present ages.
5. Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored?
6. Joe enters a race where he has to cycle and run. He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed. Joe completes the race in less than 2½ hours, what can we say about his average speeds?
2. Simultaneous equations in two unknowns
Consider the following system
Combination (or addition or elimination) method
Activity 7
For each of the following, find two numbers to be multiplied to the equations such that one variable will be eliminated;
We try to combine the two equations such that we will remain with one equation with one unknown. We find two numbers to be multiplied on each equation and then add up such that one unknown is cancelled.
Exercise 7
Use elimination method to solve;
Substitution method
Activity 8
In each of the following systems find the value of one variable from one equation and substitute it in the second.
We find the value of one unknown in one equation and put it in another equation to find the value of the remaining unknown.
Example 17
Solve the following system:
solution
Example 18
solution
Example 19
Solve the following system:
solution
Example 20
Solve the following system:
This system is a dependent system. Thus, there is an infinity solutions.
Exercise 8
Use elimination method to solve;
Cramer’s rule (determinants method)
Activity 9
Find the following determinants. Find the following determinants.
In order to use Cramer’s rule, x’s must be in the same position and y’s in the same position.
Consider the following system.
Example 21
Solve the following system:
solution
Example 22
Solve the following system:
solution
First rearrange the system such that x’s will be in the same position and y’s will be in the same position.
Example 23
Solve the following system:
solution
This system is inconsistent. Thus, there is no solution.
Example 24
solution
Solve the following system:
This system is a dependent system. Thus, there is an infinity solutions.
Exercise 9
Use Cramer’s rule to solve;
Graphical method
Activity 10
Given the system
1. For each equation, choose any two values of x and use them to find values of y; this gives you two points in the form (x,y).
2. Plot the obtained points in xy plane and join these points to obtain the lines. Two points for each equation give one line.
3. What is the point of intersection for two lines?
Some systems of linear equations can be solved graphically. To do this, follow the following steps:
1. Find at least two points for each equation.
2. Plot the obtained points in xy plane and join these points to obtain the lines. Two points for each equation give one line.
3. The point of intersection for two lines is the solution for the given system.
Example 25
Solve the following system by graphical method
Solution
The two lines intersect at point (3,1). Therefore the solution is S={(3,1)}.
Example 26
Solve the following equations graphically if possible
Solution
We see that the two lines are parallel and do not intersect. Therefore there is no solution. Note that the gradients of the two lines are the same.
Example 27
Solve the following equations graphically if possible
Solution
We see that the two lines coincide as a single line. In such case there is infinite number of solutions.
Exercise 10
Solve the following system by graphical method
Solving word problems using simultaneous equations
Activity 11
How can you do the following question?
Margie is responsible for buying a week’s supply of food and medication for the dogs and cats at a local shelter. The food and medication for each dog costs twice as much as those supplies for a cat. She needs to feed 164 cats and 24 dogs. Her budget is $4240. How much can Margie spend on each dog for food and medication?
To solve word problems, follow the following steps:
a) Identify the variables and assign symbol to them.
b) Express all the relationships, among the variables using equations.
c) Solve the simultaneous equations.
d) Interpret the result. There may be some restrictions on the variables.
Example 28
Peter has 23 coins in his pocket. Some of them are 5 Frw coins and the rest are 10 Frw coins. The total value of coins is 205 Frw . Find the number of 10 Frw coins and the number of 5 Frw coins.
Solution
Let x be the number of 10 Frw coins and y be the number of 5 Frw coins. Then,
In second equation,
Thus, there are 18 coins of 10 Frw and 5 coins of 5 Frw .
Example 29
Cinema tickets for 2 adults and 3 children cost 1,200 Frw . The cost for 3 adults and 5 children is 1,900 Frw. Find the cost of an adult ticket and the cost of a child ticket.
Solution
Let x be the cost of an adult ticket and y be the cost of a child ticket, then
Thus, the cost of an adult ticket is 300 Frw and the cost of a child ticket is 200 Frw.
Exercise 11
1. A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test?
2. Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?
3. The state fair is a popular field trip destination. This year, the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 8 vans and 8 buses with 240 students. High School B rented and filled 4 vans and 1 bus with 54 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus.
4. The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number?
5. A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current?
3. Quadratic equations and inequalities
Quadratic equations by factorizing or finding square roots
Activity 12
The method of solving quadratic equations by factorization should only be used if is readily factorized by inspection. The method of solving quadratic equations by factorization should only be used if is readily factorized by inspection.
Example 30
Solution
Example 31
Solution
Exercise 12
Solve in set of real numbers the following equations by factorization
Quadratic equations by completing the square
Activity 13
Before solving quadratic equations by completing the square, let’s look at some examples of expanding a binomial by squaring it.
add a constant to one side of the equation, be sure to add the same constant to the other side of equation.
Example 32
Solution
Example 33
Solution
Exercise 13
Solve in set of real numbers the following equations by completing the square
Quadratic equations by the formula
Activity 14
Think about two numbers a and b (a can be equal to b) such that
Example 34
Solution
Example 35
Solution
Example 36
Solution
Example 37
Solution
Exercise 14
Solve in set of real numbers;
Notice:
Factor form of a quadratic expression
Activity 15
In each of the following, remove brackets and discuss about the result and original form.
Exercise 15
Find the factor form of
Equations Reducible to Quadratic Form
a) Biquadratic equations
Activity 16
In each of the following rewrite the given equation letting
Example 41
Solution
So, the basic process is to check that the equation is reducible to a quadratic form, then make a quick substitution to turn it into a quadratic equation. In most cases, to make the check that it’s reducible to quadratic form, all we really need to do is to check that one of the exponents is twice the other.
Exercise 16
Solve in set of real numbers
b) Nested radicals
Activity 17
Example 42
Solution
Example 43
Solution
Exercise 17
Solve in set of real numbers the following equations
c) Irrational equations
Activity 18
Consider the following equation
1. Square both sides of the equation
2. Solve the obtained equation
3. Verify that the obtained solutions are solution of the original equation and then give the solution set of the original equation (given equation)
Irrational equation is the equation involving radical sign. We will see the case the radical sign is a square root.
To solve an irrational equation, follow these steps:
a) Isolate a radical in one of the two members and pass it to another member of the other terms which are also radical.
b) Square both members.
c) Solve the equation obtained.
d) Check if the solutions obtained verify the initial equation.
e) If the equation has several radicals, repeat the first two steps of the process to remove all of them.
Example 44
Solve in set of real numbers
Solution
Example 45
Solve in set of real numbers
Solution
Exercise 18
Solve in set of real numbers the following equations
Quadratic inequalities
Activity 19
Find the range where
Example 46
Solution
Example 47
Solution
Example 48
Solution
Example 49
Solution
Example 50
Solution
Exercise 19
Solve in set of real numbers
4. Applications
Activity 20
1. Explain how linear equations can be used in daily life1.
2. Give three examples of where you think quadratic equations are useful in daily life
a) Supply and demand analysis
Market equilibrium is when the amount of product produced is equal to the amount of quantity demanded. We can see equilibrium on a graph when the supply function and the demand function intersect, like shown on the graph below. Max can then figure out how to price his new lemonade products based on market equilibrium.
Example 51
Assume that in a competitive market the demand schedule is and the supply schedule is p=60+0.4q (p=price, q=quantity). If the market is in equilibrium then the equilibrium price and quantity will be where the demand and supply schedules intersect. As this will correspond to a point which is on both the demand schedule and the supply schedule the equilibrium values of p and q will be such that both equations hold.
b) Linear motion
Linear motion is a motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration.
Example 52
Some examples of linear motion are given below:
1. An athlete running 100m along a straight track
2. Parade of the soldiers
3. Car moving at constant speed
4. A bullet targeted from the pistol
5. A man swimming in the straight lane
6. Train moving in a straight track
7. Object dropped from a certain height
8. Balancing equation
c) Balancing equation
In chemistry, to balance the chemical equation we set the reactants and products equal to each other.
Example 53
d) Calculating Areas
People frequently need to calculate the area of things like rooms, boxes or plots of land.
Example 54
An example might involve building a rectangular box where one side must be twice the length of the other side. So long as this ratio is satisfied, the box will be as a person wants it. Say, however, that a person has only 4 square meters of wood to use for the bottom of the box. Knowing this, the person needs to create an equation for the area of the box using the ratio of the two sides. This means the area (the length times the width) in terms of x would equals , or . This equation must be less than or equal to 4 for the person to successfully make a box using his constraints. Once solved, he now knows that one side of the box must be meters long, and the other must be meters long.
e) Figuring out a profit
Sometimes calculating a business’ profit requires using a quadratic function. If you want to sell something (even something as simple as lemonade) you need to decide how many things to produce so that you’ll make a profit.
Example 55
Let us say that you’re selling glasses of lemonade, and you want to make 12 glasses. You know, however, that you’ll sell a different number of glasses depending on how you set your price. At 100 francs per glass, you are not likely to sell any, but at 10 francs per glass, you will probably sell 12 glasses in less than a minute. So, to decide where to set your price, use P as a variable. Let’s say you estimate the demand for glasses of lemonade to be at 12 - P. Your revenue, therefore, will be the price times the number of glasses sold: P(12 - P), or 12P–P2. Using however much your lemonade costs to produce, you can set this equation equal to that amount and choose a price from there.
f) Quadratics in Athletics
In athletic events that involve throwing things, quadratic equations are highly useful.
Example 56
Say, for example, you want to throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take the ball to arrive.
To do this, you would use the velocity equation, which calculates the height of the ball based on a parabolic (quadratic) equation. So, say you begin by throwing the ball at 3 meters, where your hands are. Also assume that you can throw the ball upward at 14 meters per second, and that the earth’s gravity is reducing the ball’s speed at a rate of 5 meters per second squared. This means that we can calculate the height, using the variable t for time, in the form of h=3+14t –t2 . If your friend’s hands are also at 3 metres in height, how many seconds will it take the ball to reach her? To answer this, set the equation equal to 3 = h, and solve for t. The answer is approximately 2.8 seconds.
g) Finding a Speed
Quadratic equations are also useful in calculating speeds. Avid kayakers, for example, use quadratic equations to estimate their speed when going up and down a river.
Example 57
Assume a kayaker is going up a river, and the river moves at 2 km/hr. Say he goes upstream -- against the current -- at 15 km, and the trip takes him 3 hours to go there and return. Remember that time = distance / speed. Let v = the kayak’s speed relative to land, and let x = the kayak’s speed in the water. So, we know that, while traveling upstream, the kayak’s speed is v = x - 2 (subtract 2 for the resistance from the river current), and while going downstream, the kayak’s speed is v = x + 2.
Unit summary
Balancing equation In chemistry, to balance the chemical equation we set the reactants and products equal to each other.
Calculating Areas People frequently need to calculate the area of things like rooms, boxes or plots of land.
Figuring Out a Profit Sometimes calculating a business’ profit requires using a quadratic function. If you want to sell something (even something as simple as lemonade) you need to decide how many things to produce so that you’ll make a profit.
Quadratics in Athletics In athletic events that involve throwing things, quadratic equations are highly useful.
Finding a Speed Quadratic equations are also useful in calculating speeds. Avid kayakers, for example, use quadratic equations to estimate their speed when going up and down a river.
UNIT 4 : Polynomial, Rational and irrational functions
My goals
Introduction
A polynomial is an expression that can have constants, variables and exponents, that can be combined using addition, subtraction, multiplication and division, but:
• no division by a variable.
• a variable’s exponents can only be 0,1,2,3,... etc.
• it can’t have an infinite number of terms.
Polynomials are used to describe curves of various types; people use them in the real world to graph curves.
For example, roller coaster designers may use polynomials to describe the curves in their rides. Polynomials can be used to figure how much of a garden’s surface area can be covered with a certain amount of soil. The same method applies to many flat-surface projects, including driveway, sidewalk and patio construction. Polynomials can also be used to model different situations, like in the stock market to see how prices will vary over time. Business people also use polynomials to model markets, as in to see how raising the price of a good will affect its sales.
Functions are important in calculating medicine, building structures (houses, businesses,…), vehicle design, designing games, to build computers (formulas that are used to plug to computer programs), knowing how much change you should receive when making a purchase, driving (amount of gas needed for travel).
Required outcomes
After completing this unit, the learners should be able to:
» Demonstrate an understanding of operations on polynomials, rational and irrational functions, and find the composite of two functions.
» Identify a function as a rule and recognize rules that are not functions.
» Determine the domain and range of a function
» Find whether a function is even , odd , or neither
» Construct composition of functions
1. Generalities on numerical functions
Activity 1
In the following arrow diagram, for each of the elements of set A, state which element of B is mapped to it.
A function is a rule that assigns to each element in a set A one and only one element in set B) We can even define a function as any relationship which takes one element of one set and assigns to it one and only one element of second set. The second set is called a co-domain. The set A is called the domain, denoted by Domf.
If x is an element in the domain of a function f, then the element that f associates with x is denoted by the symbol
Example 1
Four children, Ann, Bob, Card and David, are given a spelling test which is market out of 5; their marks for the test are shown in the arrow diagram:
Functions for which each element of the domain is associated onto a different element of the range are said to be one-toone. Relationships which are one-to-many can occur, but from our preceding definition, they are not functions.
Example 2
Exercise 1
Classification of functions
Activity 2
State which of the following functions is a polynomial, rational or irrational function
a) Constant function
A function that assigns the same value to every member of its domain is called a constant function C.
Example 6
Remark:
The constant function that assigns the value c to each real number is sometimes called the constant function c.
Example 7
b) Monomial
A function of the form cxn , where c is constant and n a nonnegative integer is called a monomial in x.
Example 8
c) Polynomial
A function that is expressible as the sum of finitely many monomials in x is called polynomial in x.
Example 9
d) Rational function
A function that is expressible as a ratio of two polynomials is called rational function. It has the form
Example 10
e) Irrational function
A function that is expressed as root extractions is called
Example 11
Finding domain of definition
Activity 3
For which value(s) the following functions are not defined:
Case 1: The given function is a polynomial
Case 2: The given function is a rational function
Exercise 2
Find the domain of definition for each of the following functions:
Case 3: The given function is an irrational function
Activity 4
For each of the following functions, give a range of values of the variable x for which the function is not defined.
Example 12
Example 13
Solution
Example 14
Solution
Exercise 3
Find the domain of definition for each of the following functions;
Operations on functions
Activity 5
Just as numbers can be added, subtracted, multiplied and divided to produce other numbers, there is a useful way of adding, subtracting, multiplying and dividing functions to produce other functions. These operations are defined as follows:
Example 19
Solution
Example 20
Solution
Example 21
Example 22
Solution
Example 23
Solution
To find the domain of this function, we need to divide it into other functions.
Exercise 4
Further considerations Odd and even functions
Activity 6
Even Function
Example 24
Odd function
The graph of such a function looks the same when rotated through half a revolution about 0. This is called rotational symmetry.
Example 25
Example 26
Exercise 5
Study the parity of the following functions:
Composite functions
Activity 7
Example 27
Solution
Example 28
Example 29
Solution
Example 30
Solution
Exercise 6
The inverse of a function
Activity 8
Find the value of x in function of y if
Only one-to-one functions can have an inverse function. To find the inverse of one-to-one functions, we change the subject of a formula.
Example 31
Solution
Example 32
Find the inverse of the function f (x)=3x–1
Solution
Example 33
Solution
Another method is to write the separate operations of the function as follow chart. We, then, reverse the flow chart, writing the inverse of each operation.
Example 34
Solution
Example 35
Solution
Example 36
Solution
Exercise 7
Find the inverse of the following functions
3. Applications
Activity 9
Give three examples of where you think functions can be used in daily life.
Polynomials are used to describe curves of various types; people use them in the real world to graph curves.
For example, roller coaster designers may use polynomials to describe the curves in their rides. Polynomials can be used to figure how much of a garden’s surface area can be covered with a certain amount of soil. The same method applies to many flat-surface projects, including driveway, sidewalk and patio construction. Polynomials can also be used to model different situations, like in the stock market to see how prices will vary over time. Business people also use polynomials to model markets, as in to see how raising the price of a good will affect its sales. For people who work in industries that deal with physical phenomena or modeling situations for the future, polynomials come in handy every day. These include everyone from engineers to businessmen. For the rest of us, they are less apparent but we still probably use them to predict how changing one factor in our lives may affect another--without even realizing it.
Functions are important in calculating medicine, building structures (houses, businesses,…), vehicle design, designing games, to build computers (formulas that are used to plug to computer programs), knowing how much change you should receive when making a purchase, driving (amount of gas needed for travel).
Example 37
Solution
Unit summary
12. Polynomials are used to describe curves of various types; people use them in the real world to graph curves. Functions are important in calculating medicine, building structures (houses, businesses,…), vehicle design, designing games, to build computers (formulas that are used to plug to computer programs), knowing how much change you should receive when making a purchase, driving (amount of gas needed for travel).
Revision exercise
Unit 5:Limits of polynomial, rational and irrational functions
My goals
By the end of this unit, I will explain:
. Concepts of limits.
. Indeterminate cases.
. Applications.
Introduction
The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular point.
Suppose that a person is walking over a landscape represented by the graph of the function f (x). His/her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. His/ her altitude is given by the coordinate y. He/she is walking towards the horizontal position given by x = p. As he/ she gets closer and closer to it, he/she notices that his/her altitude approaches L. If asked about the altitude of x = p, he/she would then answer L. It means that her/his altitude gets nearer and nearer to L except for a possible small error in accuracy.
The limit of a function f (x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f (x) remain within the target distance.
Limits are also used to find the velocity and acceleration of a moving particle.
Required outcomes
After completing this unit, the learners should be able to:
» Calculate limits of certain elementary functions.
» Apply informal methods to explore the concept of a limit including one sided limits.
» Solve problems involving continuity.
» Use the concepts of limits to determine the asymptotes to the rational and polynomial functions.
» Develop calculus reasoning.
1. Concepts of limits
Neighbourhood of a real number
Activity 1
Study the following political map of Lesotho, Swaziland and South Africa. What can you say about the boundaries of Lesotho and Swaziland?
Exercise 1
1. Apart from The Kingdom of Lesotho, give two examples of countries or Cities in the world that are surrounded by a single country or city.
2. Give three examples of intervals that are neighbourhoods of -5?
3. Is a circle a neighborhood of each of its points? Why?
4. Draw any plane and show three points on that plane for which the plane is their neighborhood.
Note:
A deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function.
Limit of a function
Activity 2
d) Division:
A really large (positive or negative) number divided by any number that is not too large, is still a really large number and we have to be careful with signs.
Beware!
So, we have dealt with almost every basic algebraic operation involving infinity. There are three cases that we have not dealt with yet. These are
To find a limit graphically, we must understand each component of the limit to ensure the graph is used properly to evaluate the limit.
UNIT 6 : Differentiation of polynomial, rational and irrational functions
My goals
Introduction
Calculus is concerned with things that do not change at a constant rate. The values of the function called the derivative will be that varying rate of change.
1. It is used economics a lot, calculus is also a base of economics.
2. It is used in history, for predicting the life of a stone.
3. It is used in geography, which is used to study the gases present in the atmosphere.
4. It is mainly used in daily life by pilots to measure the pressure in the air.
Differentiation can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. Cost, strength, amount of material used in a building, profit, loss, etc.). Derivatives are met in many engineering and science problems, especially when modeling the behavior of moving objects.
Required outcomes
After completing this unit, the learners should be able to:
» Use properties of derivatives to differentiate polynomial, rational ad irrational functions
» Use first principles to determine the gradient of the tangent line to a curve at a point.
» Apply the concepts of and techniques of differentiation to model, analyze and solve rates or optimization problems in different situations.
» Use the derivative to find the equation of a line tangent or normal to a curve at a given point.
1. Concepts of derivative of a function
Activity 1
Consider the following figure
provided that the limit exists.
Exercise 1
Find the derivative of
Right-hand and left-hand derivatives
Activity 2
1. Consider the function
Example 3
Exercise 2
Notation
Example 4
Geometric interpretation of derivative
2. Rules of differentiation
a) Constant function and Powers
Activity 3
Derivative of a constant function
From activity 1
Derivative of a power
If n is any real number, then
Particular case
Exercise 3
Find the derivative of the following functions
b) Multiplication by a scalar and product of two functions
Activity 4
Multiplication by a scalar
From activity 2
If f is a differentiable function of x, and c is a constant, then
Derivative of a product
From activity 2:
Exercise 4
Find the derivative of the following functions
c) Sum (difference)of functions
Activity 5
Example 14
Solution
Exercise 5
Find the derivative of the following functions
d) Reciprocal function and quotient
Activity 6
Derivative of the reciprocal function
From activity 4,
From activity 4,
Exercise 6
Find the derivative of the following functions
e) Composite function
Activity 7
Derivative of a composite function: Chain rule
Exercise 7
Successive derivatives
Activity 8
The successive derivatives of a function f are higher order derivatives of the same function.
We denote higher order derivatives of the same function as follows:
Exercise 8
3. Applications of differentiation
Equation of tangent line and normal line
Activity 9
Tangent line
Remark
Normal line
Example 24
a) Find the point where the tangent line is parallel to the bisector of the first quadrant.
b) Find the tangent line to the curve of this function at point
Solution
The tangent line is
Normal line
Example 25
Rates of change
The purpose here is to remind ourselves one of the more important applications of derivatives. That is the fact that
Example 26
Example 27
Critical points
Example 28
Let us determine all the critical points for the function
Example 29
Let us determine all the critical points for the function
Example 30
Let us determine all the critical points of the function
Now, we have two issues to deal with. First the derivative will not exist if there is division by zero in the denominator.
So, we can see from this that the derivative will not exist at t = 3 and t =−2. However, these are not critical points since the function will also not exist at these points. Recall that in order for a point to be a critical point the function must actually exist at that point.
At this point, we have to be careful. The numerator does not factor, but that does not mean that there are not any critical points where the derivative is zero. We can use the quadratic formula on the numerator to determine if the fraction as a whole is ever zero.
Finding absolute extrema
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