SABA Math Course Program B

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SABA Math Course

Program B

Version 2013

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The copyright of this learning material is held by the Math4All Foundation and the material is published under the creative commons licence.

The material is carefully selected and tested. However, the Math4All Foundation can not be held ac- countable for any inaccuracy or incompleteness and any personal damage that results from using of misusing this learning material.

This content is puplished under the Creative Commons Math4All-Non-Commercial 3.0 Dutch License (see http://creativecommons.org/licenses/by/3.0).

The learning material is open and free and is developed with the math educational learning objectives and competences for Dutch Secondary Education.

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Math4all Foundation October 3, 2013 page 1

Content

Preface 3

1 Working with formulas 5 1.1 Using formulas 6 1.2 Rewriting formulas 8

1.3 Formulas and the graphic calculator 11 1.4 Equations 13

2 Functions and graphs 15 2.1 Functions 16

2.2 Domain and range 18 2.3 Characteristics 20 2.4 Transformations 22 2.5 Inequalities 24

3 Linear relations 27 3.1 Linear functions 28 3.2 Linear relations 30 3.3 System of equations 32 3.4 Linear models 34

4 Exponential functions 37 4.1 Exponential growth 38 4.2 Real exponents 40

4.3 Exponents and powers 42 4.4 Exponential functions 44 4.5 More exponential functions 46

5 Logarithmic functions 49 5.1 Logarithms 50

5.2 Properties 52

5.3 Logarithmic scales 54 5.4 Logarithmic functions 56 5.5 Logarithmic equations 58

6 Power functions 61 6.1 Powers 62

6.2 Power functions 64 6.3 Quadratic functions 66 6.4 The quadratic formula 68 6.5 More power functions 70

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7 Periodic functions 73 7.1 Periodicity 74 7.2 Radians 76

7.3 The sine functions 78 7.4 The cosine functions 80

7.5 Trigonometric graphs and equations 83 7.6 More trigonometric graphs 85

8 Solids 87

8.1 Projections 88 8.2 Calculations 90 8.3 Views / Nets 93 8.4 Cross sections 96

8.5 Series of cross sections 99

9 Area and volume 103 9.1 Area 2D shapes 104 9.2 Area solids 106 9.3 Volume 109 9.4 Enlargements 112

10 Change 115 10.1 In graphs 116 10.2 Per step 118

10.3 Diﬀerence quotient 120 10.4 Derivative values 122 10.5 Derivative functions 124

11 Derivative functions 127 11.1 Derivative 128

11.2 Finding the derivative 130 11.3 Calculating extreme values 132 11.4 Points of inﬂection 134

12 Diﬀerentiation rules 137 12.1 Diﬀerentiation rules 138 12.2 Chain rule 140

12.3 Product rule 142 12.4 Quotient rule 144

12.5 Derivative of trigonometric functions 146 12.6 Using the derivative in modeling 148

Register 150

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Preface

The learning material in this book is based on the content you can ﬁnd at www.math4all.nl. In the text you may ﬁnd references to that website. The header on each page gives you the exact location.

A chapter consists of theory and exercises.

When you encounter references to worksheets or other resources you can ﬁnd these on the website.

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1 Working with formulas

Using formulas 6 Rewriting formulas 8

Formulas and the graphic calculator 11 Equations 13

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1.1 Using formulas

Theory

A formula is a sentence containing variables. Formulas often describe a relation between those variables, but not always.

Formulas often look like an equation, i.e. a sentence with an equals sign.

If a formula describes a relation between two variables, you can draw a corresponding graph. First, you make a table. Next you draw the points in a coordinate system and connect them.

In practice, formulas often describe relationships between quantities.

Those quantities are represented by a variable, a letter which corresponds to the name of the quantitiy used.

A quantity is measured in units.

> The formula 𝐴 = u�² is an equation deﬁning the relation between the variables 𝐴 and u�. You can make the corresponding table and plot the corresponding graph.

> The formula 2u�+40 = 300 gives information about the unknown u�. The solution to this equation is u� = 130, for 2 ⋅ 130 + 40 = 300.

> The formula (u� + 3)²= u�²+ 6u� + 9 is a calculation rule and is valid for all u�.

Exercises

Exercise 1

The volume of a cylindrical tin can is given by: 𝑉 = 𝜋 ⋅ u�²⋅ ℎ.

Here 𝑉 is the volume, u� the radius in centimeters and ℎ the height in centimeters.

a In what unit should 𝑉 be expressed?

b What is the volume of a tin can with a diameter of 80 millimeters and a height of 16 centimeters?

c What formula expresses the relationship between 𝑉 and u� for cans with a height of 16 centimeters?

d Draw a graph for the formula you found in c.

e For other cans the volume is known: 𝑉 = 1 L. What relationship is there between u� and ℎ? Draw a corresponding graph.

Exercise 2

Which of these formulas describes a relation between two variables? Draw a graph for the ones that do.

a (2 + u�) ⋅ u� = 2u� + u�u�

b volume(cube) = u�³ c u� = 400 − 5u�² d u�²+ u�²= u�²

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SABA PROGAM B > FUNCTIONS AND GRAPHS > WORKING WITH FORMULAS

Exercise 3

When you use electricity you pay a ﬁxed charge per year and a variable cost per kWh (kiloWatthour).

The total yearly cost depends on the number of kWh’s used. It is possible to convert the total cost to cost per kWh. This gives the formula:

𝐾 = 0,12 +³²_u�

Here u� is the number of used kWh’s and 𝐾 the cost per kWh (in euro’s).

a What is the ﬁxed cost per year?

b Draw a graph of 𝐾 as a function of u�. Why should 𝐾 be on the vertical axis?

c For what value of u� is the cost per kWh 16 eurocent?

Exercise 4

An electrical resistor is connected to a power source of 200 Volts.

An ammeter can be used to measure the electrical current. Ohm’s law is valid in this situation: 𝑈 = 𝐼 ⋅ 𝑅 where 𝑈 is the power voltage V (volt), 𝐼 the electrical current in A (ampère) and 𝑅 the resistance in (ohm).

a For a power voltage of 200 volts Ohm’s law gives the relation between 𝐼 and 𝑅. What is this formula? And what units belong to this formula?

b Draw the corresponding graph.

c What current is meaured when 𝑅 = 15 ?

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1.2 Rewriting formulas

Theory

Formulas like 2u� + 2u� = 60 and u� = 30 − u� describe the same relatonship, they are equivalent. You can reduce (or rewrite ) the formula 2u� + 2u� = 60:

2u� + 2u� = 60

beide zijden/2

u� + u� = 30

beide zijden−u�

u� = 30 − u�

u� has now been expressed in u�. The new formula contains fewer symbols and looks neater.

When rewriting formulas you use the following principles:

> you may add or subtract the same from both sides of an equals sign;

you may divide or multiply with the same number on both sides of an equals sign (except for multiplying or dividing by 0);

> removing brackets:

u� ⋅ (u� + u�) = u� ⋅ u� + u� ⋅ u�

and

(u� + u�) ⋅ (u� + u�) = u� ⋅ u� + u� ⋅ u� + u� ⋅ u� + u� ⋅ u�

> factorizing:

u� ⋅ u� + u� ⋅ u� = u� ⋅ (u� + u�) and

u�²+ u� ⋅ u� + u� = (u� + u�) ⋅ (u� + u�) with u� + u� = u� and u� ⋅ u� = u�

> adding/subtracting/multiplying and dividing fractions:

u�

u�±û�_u�=û�⋅u�_u�⋅u�±û�⋅u�_u�⋅u�= u�⋅u�±u�⋅u�

u�⋅u�

and

u�

u�⋅^u�_u�=_u�⋅u�^u�⋅u�

and

u�

u�/_u�û� =û�⋅u�_u�⋅u�/_u�⋅u�û�⋅u� =û�⋅u�_u�⋅u�

if

u� ≠ 0, u� ≠ 0(only when dividing) and u� ≠ 0

Exercises

Exercise 1

Rewrite these formulas into their simplest form:

a 4 ⋅ u� + 10 = 3 ⋅ u� − 2 ⋅ u�

b 2 ⋅ u� + 2 ⋅ u� ⋅ u� + 4 ⋅ u� = 6 ⋅ u�² c 4 ⋅ u� ⋅ ℎ + 2 ⋅ u�²= 100

d 𝑊 = u� ⋅ (650 − 2 ⋅ u�) − 20 ⋅ (650 − 2 ⋅ u�)

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Exercise 2

In these formulas express u� in u�. After that, write them in their simplest form.

a u� − 2u� = 10 b (u� + 2) ⋅ u� = 6 c u� = 4 − u�² d u� ⋅ u�²= 4

e 0.5u� + 1.5u� = 12 f (u� + u�)³= 8 g u�²− u�²= 25 h 2u�²+ 4u�u� = 100

Exercise 3

Remove the brackets:

a −2u� (u�²+ 6u�) b −2u� − (u�²+ 6u�) c (u� + 20) (u� − 5) d (u�²+ 1) (3u� − 2)

e (u� − 3) (u� + 3) f (6u� − 3)² g (u� −¹_u�)² h (u� − 2)³

Exercise 4 Factorize:

a u�²− 4u�

b −2u�²+ 18u�

c u�²+ 5u� − 6 d 12 − 4u� − u�²

e 4u�²− 16

f 2u�³− 2u�²− 24u�

g 16 − u�² h u�²− 10u� + 9

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Exercise 5

Write as one fraction (assume you never divide by 0):

a ³_u�+⁵_u�

b _u�−2³ −²_u�

c ²_u�/³_u�

d 2u� −_2u�¹

Exercise 6

A farmer has a rectangular plot of land with a length of twice its width. From a landscape management point of view he removes a 3 meter wide strip on both sides and replaces it with a narrow strip of woodland. He also makes a wider strip of woodland of 10 meters wide at one of the short ends of his plot. As a result the surface area of his land decreases with 2690 m².

a First make a sketch of the situation. The original width of the plot is u�(in meter). What is the surface area of this piece of land?

b What is the surface area after a part of it has been converted into woodland? (remember the brackets!) c Remove the brackets and calculate the width of the rectangular plot.

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1.3 Formulas and the graphic calculator

Theory

If a formula that decribes the relationship between u� and u� is given in the form u� = ... then u� is called a function of u�.

In the corresponding graph u� is always chosen on the vertical axis.

> In the formula u� = u�²+ 4 u� is a function of u�.

> In the formula 𝑃 = 0.052u�³ 𝑃 is a function of u�.

> The formula u� + 2u� = 6 can be written in two ways:

> u� = 6 − 2u�, where u� is a function of u�.

> u� = 3 − 0.5u�, where u� is a function of u�.

You can enter formulas given in the form u� = ... into the graphical calculator.

Exercises

Exercise 1

Plot the graphs of the following formulas. Remember to use brackets and appropriate window settings!

a u� = 250u� − 4.9u�² b u� = 0.04 +²⁰⁰_u�

c 4 ⋅ u� ⋅ ℎ + 2 ⋅ u�²= 100 d 𝑁 = _30+0.5u�⁶⁰ 2

Exercise 2

The volume of a cone is given by: 𝑉 = ¹₃𝐺ℎ, where 𝐺 is the area of the base and ℎ the height. The base is a circle with radius u�.

a What formula describes the relation between 𝑉, u� and ℎ?

Give the formula describing the relation between u� and ℎ for a cone with a volume of 1 liter from the given formula.

b Write the relation in such a way that u� is a function of ℎ.

c Determine the value of ℎ for u� = 10 cm. Round the answer to two decimals.

d Determine the value of u� for ℎ = 10 cm. Approximate with a value rounded to two decimals.

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Exercise 3

The monthly lease for a copying machine is € 200 to which an amount of 4 cent per copy is added. 𝐾 represents the total cost and u� is the (average) number of copies made in a month.

a Write down the formula for 𝐾 as a function of u�.

b Somebody using the copying machine pays 10 cent per copy. Write down the formula for the revenue 𝐼 as a function of u�.

c How many copies should be made in a month if you want the 10 cents per copy to cover the costs?

Exercise 4

Imagine a company that sells posters. In order to stand out the area of such a poster should be 1 m². The poster will be printed in such a manner that there is a white border of 10 cm on both sides and on the top. At the bottom that border is 15 cm. The management wonders what dimensions the poster can have now. They ﬁnd the formula:

(u� + 25) (u� + 20) = 10000.

a Show how they got this formula and what u� and u� mean.

b Plot the graph belonging to this formula.

c Check if all dimensions shown are feasible.

d In retrospect the management wants the printed part to be square.

What poster size would you recommend now?

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1.4 Equations

Theory

Formulas such as u� = 2u� + 3, or u� + 4u� − u� = 15, or 6u� + 10 = 2u� − 8 are called equations. You can look for values (or combinations of values) that satisfy the equation. This is called solving an equation.

Equations can be solved systematically by rewriting them. Especially for equations with one variable this is often done. You can use algebraic methods like:

> the balance method, in which, on both sides of the equals sign, you

> add or subtract the same amount;

> multiply or divide by the same number (but not by 0) on both sides of the equals sign.

> the reversing method, in which you undo calculations by doing the opposite:

> you undo addition by subtraction (and vice versa);

> you undo multiplication by division (and vice versa);

> you undo powers by extracting roots (and vice versa).

> factorization, where you use the fact that u� ⋅ u� = 0 is equivalent to u� = 0 ∨ u� = 0.

The ∨ sign means that you should read this expression as u� = 0 and/or u� = 0(so u� = 0 or u� = 0 or both).

When algebraic methods do not work, you can think of iteration: you search for an answer by decreas- ing the search area. Your graphic calculator has several built-in routines for this.

Exercises

Exercise 1

Solve the following equations algebraically.

a 4u� + 50 = 200 b 4u�²+ 50 = 200 c √u� + 4 = 20 d (2u� − 5)³= 125

e √u�²+ 4 − 20 = 0 f ¹²_u� = 400

g 2u�²− 2 = 12u� + 30

Exercise 2

Use iteration to solve the following equations with the help of your graphical calculator. Find all solutions.

a √u� = 6 − u�

b u�⁴= 2 + u�

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Exercise 3

The Empire State Building is a high sky scraper in New York. Imagine somebody dropping a small stone from the 381 m high building!

Under the inﬂuence of gravity the stone drops uniformly accelerated (ne- glect the air resistance). Physicists have devised a mathematical model for this. In this model the distance travelled is called u�(in meters) and the velocity u�(in m/s). Both depend on time u�(in seconds) according to the formulas u� = 4.9u�²and u� = 9.8u�.

a Give a formula for the height ℎ of the stone above the ground as a function of u�.

b Calculate the time at which the stone hits the ground.

c Calculate the velocity with which the stone hits the ground.

Give your answer in m/s and in km/h.

Exercise 4

For each of these formulas calculate the value of one variable if the other one is 0.

a 2u� − 3u� = 650

b 𝑊 = −0.25u� (0.5u� − 100) c u�²+ (u� + 2)²= 100 d u� =_600+0.2u�¹²⁰⁰ 2− 1

e (u�²− 4) (u�²− 9) = −36 f u�⁴+ 1 = _1+u�⁴2

Exercise 5

The township asks a farmer to surround a piece of land with a woodstrip of 4 m wide.

The piece of land is an exact square. At one side it already borders the woods.

‘I get to keep only half of my land’, the farmer complains.

If this is true, what is the surface area of the piece of land?

Solve this problem using an equation.

Exercise 6

Some candles have an almost perfect cylindrical shape. Imagine you want to make such a candle with a height of 20 cm. You take a wick with a 3 mm diameter and repeatedly dip it in a bath of molten wax. With each dip the diameter of the candle increases with 1 mm. The total volume of candle wax 𝑉 in the candle depends on the number of dips u� made.

a Give a formula for 𝑉 as a function of u�.

b Plot the graph of this function on your graphical calculator.

c How many dips do you need to make to give the candle an approximate volume of 106 cm³? First use your calculator to read out the answer from the graph, then ﬁnd the solution algebraically by solving the corresponding equation.