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Lesson 12: Functions. The Last Joke (III)

June 24, 2012

 

First some notes to appreciate the joke:

Log means  tronco but it also represents the abbreviation of the logarithmic function to the base 10 you will study in your fourth year.

Timber  means viga madero para construcción.

To hesitate means vacilar, dudar

To aid and abet somebody  means instigar y secundar a alguien (en la comisión de un delito)

This is the joke:

A math student is pestered by a classmate who wants to copy his homework assignment. The student hesitates, not only because he thinks it’s wrong, but also because he doesn’t want to be sanctioned for aiding and abetting.
His classmate calms him down: “Nobody will be able to trace my homework to you: I’ll be changing the names of all the constants and variables: a to b, x to y, and so on.”
Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying.
After the deadline, the student asks: “Did you really change the names of all the variables?”
“Sure!” the classmate replies. “When you called a function f, I called it g; when you called a variable x, I renamed it to y; and when you were writing about the log of x+1, I called it the timber of x+1…”

Enjoy your Summer!!

 

 

 

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Lesson 12: Functions (II)

June 22, 2012

Now we are going to define one of the more important concepts in Maths

2. Definition of function

A function is a relation between two numeric variables that assigns to each input number  EXACTLY ONE output number .

We name  independent variable (we choose it is fixed by us; and we name  dependent variable because it depends from the value of  we choose.

If we represent the pairs of values related by the function  on the Cartesian plane we obtain the graph of the function.

 Vertical line test

If no vertical line can be drawn so that it intersects a graph more than once, then it is a graph of a function.

Think about it, if a vertical line intersects a graph in more than one place, then the x value (input) would associate with more than one y value (output), and you know what that means.  The relation is not a function.

The next two examples illustrate this concept.

We are going to use the vertical line test to determine whether each graph is a graph of a function.

 
For  we can find two points in the graph (3,4)  and (3,6). So to 3 corresponds to two values of , 4 and 6. It is not a function
 
To each value of  x corresponds only one value of y.This graph would pass the vertical line test, because there would not be any place on it that we could draw a vertical line and it would intersect it in more than one place.Therefore, this is a graph of a function.

Function Notation

  • We use for functions lower case letters like f, g, h.
  • We write f(x) = y .
  • f(x)  read “f of x”.

This a very good video about how to use the vertical test:

3. Different ways of displaying a function

A function can be displaying through a table of values, an algebraic expression, or words.

3.1. Representing a function with a table of values

To plot a function expressed by a table of values we show the pairs of corresponding values of the variables independent and dependent as if they were the coordinates of points in the Cartesian plane. Take into account that:

  • The input values, the values of the independent variable, are represented on the x–axis.
  • The output values, the values of the dependent variable, are represented on the y–axis.

Example

1. A shop sells mobile phones with a price of 45€ pee unit. Write the function that relates the number of mobiles bought and the price in €.

 If I buy 1 mobile I will pay 45€, if I buy 2 mobiles I will pay 90€, if I buy 3 mobiles I will pay 135€..

Num of mobiles 1 2 3 4 5 6
Price in € 45 90 135 180 225 265

 We plot the coordinates of these points in the Cartesian plane and we obtain the graph of the function (“the picture”).

The number of mobiles is represented on the x–axis (abscissas).

The price is represented on the y–axis (ordinates).

 

We can’t join the points because we can’t buy 1.5 mobiles or 2.3 mobiles. This type of variables is named discrete variables.

3.2. Representing a function through an equation

The algebraic expression of a function is written y = f(x) and it is named function equation.

  • Recall the function notation f(x)  read “f of x”.
  • f is the function name.
  • Output values are also called functional values. Note that you can use any letter to represent a function name, fis a very common one used.
  • xis your input variable.
  • y is your output variable.

Think of functional notation as a fancy assignment statement.   When you need to evaluate the function for a given value of x, you simply replace x with that given value and simplify.  Just like you do when you are finding the value of an expression when you are given a number for your variable as shown in Lesson 5: Algebraic Expressions. For example, if we are looking for f(0), we would plug in 0 for the value of x in our function f.

Example

2. Find the functional values f(-2), f(-1),  f(0), f(1) and f(2) for the function f(x) = 2x – 1. Draw the graph of the function.

Again, think of functional notation as a fancy assignment statement.  For example, when we are looking for f(0), we are going to plug in 0 for the value of x in our function f and so forth.

Values of the independent variable Values of y
 f(x) = 2x – 1  Plug in -2 for xf(x) = 2 · (-2)- 1 Evaluate: f(-2) = -5
 f(x) = 2x – 1  Plug in -1 for x f(x) = 2 · (-1)- 1 Evaluate: f(-1) = -3
 f(x) = 2x – 1  Plug in 0 for x f(x) = 2 · 0 – 1 Evaluate: f(0) = -1 
 f(x) = 2x – 1  Plug in  1 for x f(x) = 2 · 1 – 1 Evaluate: f(1) = 1
 f(x) = 2x – 1  Plug in  2 for xf(x) = 2 · 2 – 1 Evaluate: f(2) = 3 

 We can build a table of values

 -2  -1  0  1  2
y  -5  -3  -1  1  3

 Representing the points we obtain the graph of the function.

In this case we can join the points because you can always work out triple x minus one unit, even if the number is not an integer.

In the links below you can practice how to evaluate a function

Evaluate a linear function (this is a function whose graph is a line)

Complete a function table  (of a linear function, this is a function whose graph is a line)

Evaluate a nonlinear function

In the following link you can practice how to get an equation (only a linear expression as, for instance,  y = 45 · x or y = 45 · x +1)  for a table.

Write a rule for a function table  

3.3. Representing a function using words

This situation is usually found in word problem where you are asked for the graph and the equation of the function. In case of having to find out the equation of a function given in words the process is the same as when solving word problems with equations. This is you have to translate words into algebra.

Let’s return to example 1.

Example

3.A shop sells mobile phones with a price of 45€ per unit. Write the function that relates the number of mobiles bought and the price in €. Find out the equation of this function.

If I buy 1 mobile I will pay 45€, if I buy 2 mobiles I will pay 90€, if I buy 3 mobiles I will pay 135€.. if I buy x mobiles I will pay 45 · x €. Therefore the equation of the function will be

 y = 45 · x

We use to write it  y = 45  x.

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Lesson 12: Functions (I)

June 22, 2012

 Vocabulary on Functions

Lesson 12: Functions (Notes)

Exercises for Functions

Linear Functions Word Problems

I Have to Know by the End of this Lesson

 

A little of History

The French philosopher and mathematician, René Descartes (1596-1650) developed his philosophy on the idea of needing a starting point on which all other knowledge can be based: I think, therefore I am. In the world of mathematics he is the founder of analytic geometry, which uses a pair of perpendicular lines that cross at a point called the origin as its basis, and is also referred to as the Cartesian coordinates system.

First recall some concepts about Cartesian coordinates on the plane.

1. The Cartesian plane

We will begin with vocabulary.

First, a coordinate. A coordinate is a number  that labels a point on a line.

The coordinate 0 is called the origin of coordinates.

Points to the right of 0 are labelled with positive numbers: 1, 2, 3, etc.  Points to the left of the origin are labelled with negative numbers: −1, −2, −3, etc. Those coordinates are the “addresses” of those points.

A coordinate axis is a line with coordinates.

Now, to label the points in a plane, we will need more than one coordinate axis, and we place a second at right angles to the first.  

Points above the origin have positive coordinates; points below have negative coordinates.

Those lines are called rectangular coordinate axes, because they are at right angles to one another; the coordinates on them are called rectangular coordinates.

Hence we have the name coordinate geometry or, as it is often called, analytic geometry.

The rectangular coordinates of a point are an ordered pair, (x, y).

The coordinates of the origin O are (0, 0). We don’t move right or left and we don’t move up or down. We will see that 0 is an extremely important coordinate.  It means that the point is on one of the axes.

The horizontal axis is called the X-axis or axis of abscissas, the vertical axis is called the Y-axis or axis of ordinates  and the fixed point O is called the origin.

This system of reference can be used to determine the position of any point on a set of axes by an ordered pair of numbers, which are usually written in brackets and separated by a comma.

The numbers in each ordered pair of numbers are called the coordinates of the corresponding point. The first number is called the x-coordinate (or abscissa) and the second the y-coordinate (or ordinate).

We always write (x, y); x is named abscissa and y ordinate

Quadrants

To make it easy to talk about where on the coordinate plane a point is, we divide the coordinate plane into four sections called quadrants, labelled counter-clockwise: The first, the second, the third, and the fourth

   

       In Roman numbers 

   Arabic numbers

                                                

Points in Quadrant 1 have positive x and positive y coordinates.

Points in Quadrant 2 have negative x but positive y coordinates

Points in Quadrant 3 have negative x and negative y coordinates.

Points in Quadrant 4 have positive x but negative y coordinates.

In the following links you can practice several skills:

1. Points on coordinate graphs

2. Quadrants and axes

3. Coordinate graphs as maps

4. Distance between two points

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Lesson 11: Volume of Solids (II)

June 6, 2012

3. Volume of a cuboid

Volume of a cuboid whose side lengths are l, w and h is:

Click below and you  will practice how to calculate the volume of a cuboid.

 3.1. Cavallieri’s  principle

In English we name cross sections to the shape made when a solid is cut through parallel to the base.

Note: don’t draw the rest of the object, just the shape made when you cut through.

cross-section of a pyramid

Example:

The cross-section of a rectangular pyramid is a rectangle

cross-section of a cylinder

Example:

The cross-section of a circular cylinder is a circle

Compare the figures below.

That the stacks have equal volume is made clearer with a simple transformation of the stack on the right.If we cut both figures before through parallel to the base, the cross sections are the same.

The theorem bellow is known as Cavallieri’s Principle.

If, in two solids of equal altitude, the areas of cross sections at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.

Cavalieri’s principle reveals its power when applied to more complicated shapes (or when the rectangles are allowed to become infinitely thin). We will apply it to find many volumes of solids we studied in lesson 10.

4. Volume of prisms and cylinders

 4.1. Volume of a prism

Applying Cavallieri’s Principle, the volume of a cuboid and the volume of a prism with the same height and whose bases have the same area, are the same. If we take into account that , where  is the base area we can conclude:

The volume of a prism with height h and base area AB is

V = AB · h

For instance, if we have a triangular prism AB is the area of a triangle (base · times height of the triangle divides by two).

Click below if you want to see a video on this topic:

You can practice below:

Volume of prisms and pyramids

4.2. Volume of a cylinder

 Notice that any cross-section of a cylinder is equal to the base. Applying Cavallieri’s Principle again, the volume of a cuboid and the volume of a cylinder with the same height and whose bases have the same area are the same.

The volume of a cylinderwith height and radius  is V = AB · h =πr2 · h

 Click below if you want to see a video on this topic:

5. Volume of pyramids and cones

5.1. Volume of a pyramid

 

The prism and the pyramid on the right have the same height and the same base.

 

 

If we fill up the pyramid with sand, we can see we need three times the sand contained in the pyramid to fill up the prism. So, we can deduce that the volume of the pyramid is a third of the volume of the prism.

The volume of a pyramid with height and base area is

V = 1/3 · (AB · h)  

Note:If we have a cone and a pyramid with the same height  and the same base area , by applying Cavallieri’s principle we can conclude that they have the same volume.

 Click below if you want to see a video on this topic:

 You can practice on the following link:

Volume of prisms and cylinders

5.2. Volume of a cone

The cylinder and the cone below have the same height and the same base. 

If we fill up the cone with sand, we can see we need three times the sand contained in the cone to fill up the cylinder. So we can deduce that the volume of the cone is a third of the volume of the cylinder.

The volume of a cone with height  and radius is

V = 1/3 · (πr2 · h) 

Click beñow if you want to see a video on this topic:

You can practice on the link below:

Volume of pyramids and cones

6. Volume of a sphere

 Consider the cylinder circumscribed to a sphere with radius . This cylinder has height 2r and radius r.

 If we fill up the semi sphere with sand, we need three times the sand that contains the semi sphere to fill up the cylinder. 

 

So the volume of the sphere, that is twice the volume of the semi sphere, is

V =  4/3 π r³. 

See a video on this topic:

You can practice on the following link:

Volume and surface area of spheres

 

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Lesson 11: Volume of Solids (I)

May 31, 2012

Vocabulary on Volume of Solids

Lesson 11: Volume of Solids 

Volume Worksheet

Volume Word Problems

 I Have to Know by the End of this Lesson

 

1. Volume of a solid

1.1. Volume units

Volume is length by length by length, so the basic unit of volume is a cube with edge length one metre. Its volume is 1 metre × 1 metre × 1 metre, which is written m3 (cubic metre).

The basic unit of volume is the cubic metre, which is written m3.

Multiples and submultiples of cubic metre are the following:

  unit it is the volume of a it equals symbol
multiples cubic kilometre cube whose edges measure one kilometre 1000000000 m3 km3
cubic hectometre cube whose edges measure one hectometre 1000000 m3 hm3
cubic decametre cube whose edges measure one decametre 1000 m3 dam3
base unit cubic metre cube whose edges measure one metre   m3
submultiples cubic decimetre cube whose edges measure one decimetre 0.001 m3 dm3
cubic centimetre cube whose edges measure one centimetre 0.000001 m3 cm3
cubic millimetre  cube whose edges measure one millimetre 0.000000001 m3 mm3

 You have to bear in mind that volume units are cubes and three-dimensional. The conversion of a unit into another one is done by dividing or multiplying both the length, the height and the width, that is dividing and multiplying by ten three times; in other words, dividing and multiplying by 1000.

The volume units go up by a factor of 1000 at the time.

Each jump to a smaller unit is equivalent to multiply by 1000, you have to move the decimal point three places to the right. Each jump to a larger unit is equivalent to divide by 1000, you have to move the decimal point three places to the left. Here you are a sketch:

 

Now you can practice on the following links:

1 – Volume unit conversion. (with hints)

2 – Volume unit conversion.

3 – Match the equal volumes.

 

 1.2 Volume of an object

 Volume of an object is the amount of space it occupies.

 We will begin by a basic solid, the cube.

2. Relationships among volume, capacity and mass

2.1 Volume and capacity

The capacity of a container is known as the volume of the liquid or gas that it can hold.
Capacity and volume have equivalent meanings. Establishing a special unit to measure volume isn’t necessary, so using the cubic metre would be enough, but for practical use the litre was established as a unit. If you pour one litre in a cube with an edge of 1 dm it will fit in the cube exactly.

The litre is identical to the cubic decimetre (dm³), although it wasn’t always so . Recall

The litre is the capacity of a cubic decimetre.

We know 1 dm is 0·1 m or 10 cm, so 1 dm³ is 10×10×10 cm:

1 dm3, 10 cm along each side, 1000 cm3

This, of course, means that there are 1,000 cm³ in a litre, or that 1 cm³ is equal to 1 mL.

The following table shows the equivalence between volume units and capacity units.

Volume units m3     dm3     cm3
Capacity units kl hl dal l dl cl ml


2.2. Volume, mass and capacity

A recipient contains a litre of pure water, which occupies 1 dm3. We weight it and it weights 1 kilogram exactly.

One kilogram is the weight of 1 dm3  of pure water.

If we weigh a container with 1 ml of pure water with, which occupies 1 cm3, it weights 1 gram.

One gram is the weight of 1 cm3 of water

 The following table shows the equivalence among volume units, capacity units and mass units for pure water.

Volume units m3     dm3     cm3
Capacity units kl hl dal l dl cl ml
Mass units t q mag kg dag hg g

1 l =1 dm3 = 1 kg of pure water

 NOTE: If we have other substance different from pure water one litre doesn’t weight one kilo.

In the following video you can find a long explanations about these topics:

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Lesson 10: Solids. Surface Areas. An Extra Question with a Prize (IV)

May 23, 2012


Are you able to prove it?

Think of the net of a cone.



If the base of the cone has radius r and the cone has generatrix g , I told you the lateral area (the area of the circle sector) was πrg

Could you prove it?

There is an extra 0.5 in this lesson if you give the right proof. Anyway your interest and effort will be prized in the global mark of this term.

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Lesson 10: Solids. Surface Areas. Revolution solids (III)

May 22, 2012

5. Solids of revolution

A solid of revolution is a solid figure obtained by rotating a flat figure around a line that lies on the same plane. The axis of revolution is the line about which the revolution takes place.

5.1. Cylinder

A cylinder is a solid of revolution generated by a rectangle that rotates around one of its sides.

Elements of a cylinder

Surface area of a cylinder

The net of a cylinder is formed by:

  • A rectangle whose length is the length of the circumference of the bases and its width is the height of the cylinder.
  • Two equal circles that are its bases.

The surface area of a cylinder can be calculated from its net. So it will be the sum of the lateral area and the areas of the two equal bases:

  • Lateral area AL: It is the area of the rectangle whose length is the length of the circumference of the bases with radius r, 2πr and its width is the height of the cylinder,h.

AL = 2πr · h

  • Area of the bases AB: This is the sum of the areas of the bases with radius . 

AB = πr2

The surface area of a cylinder is:

AT = AL + 2·AB = 2πr · h + 2· πr2 = 2πr (h + r)

5.2. Cone

The cone is a solid of revolution generated by a right triangle that rotates around one of its legs.

Elements of a cone

Calculation of the generatrix of a cone

 By the Pythagorean Theorem, the slant height or generatrix (or slant height, s) of the cone is equal to:

Surface area of a cone

 The net of a cone is formed by:

  • Circular sector of length2πr · h, being r the radius of the base, and radius g the generatrix of the cone.
  • A circle that is the base

As in other cases we work out the surface area from its net. So it will be the sum of the lateral area and the area of the base:

  • Lateral area AL: It is the area of the circular sector whose length is 2πr and its radius is,g.

AL = πr · g

  • Area of the base AB: This is the area of the base with radius . 

AB = πr2

The area of a cone is:

AT = AL + AB = πr · g + · πr2 = πr (g + r)

You can get the net of  a cone and a cylinder: http://www.senteacher.org/wk/3dshape.php

5.3. Sphere

A spherical surface is the surface generated by rotating a circle about its diameter.

A sphere is the region of the space that is inside a spherical surface.

A sphere can also be seen as the region of the space obtained by rotating a semicircle around its diameter.

 Elements of a sphere

The centre is the interior point that is equidistant to all points on the surface of the sphere.

Radius is the distance from the centre to any point on the surface of the sphere, .

The diameter is the distance from one point through the centre to another point . A diameter is twice the radius.

Surface area of a sphere

The surface area of sphere of radius is :

A = 4 πr2

In the following links you will find exercises about all the 3-D shapes we studied:

Finally, the following video will show you how to find the areas of a cylinder, a cone and a sphere. Improve your listening!