# 5.4: Further Topics in Functions (Exercises) - Mathematics

## 5.1: Function Composition

subsection{Exercises}

In Exercises ef{funccompeval1first} - ef{funccompeval1last}, use the given pair of functions to find the following values if they exist.

egin{multicols}{3}

egin{itemize}

item $(gcirc f)(0)$

item $(fcirc g)(-1)$

item $(f circ f)(2)$

end{itemize}

end{multicols}

egin{multicols}{3}

egin{itemize}

item $(gcirc f)(-3)$

item $(fcirc g)left(frac{1}{2} ight)$

item $(f circ f)(-2)$

end{itemize}

end{multicols}

egin{multicols}{2}

egin{enumerate}

item $f(x) = x^2$, $g(x) = 2x+1$ label{funccompeval1first}

item $f(x) = 4-x$, $g(x) = 1-x^2$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = 4-3x$, $g(x) = |x|$

item $f(x) = |x-1|$, $g(x) = x^2-5$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = 4x+5$, $g(x) = sqrt{x}$

item $f(x) = sqrt{3-x}$, $g(x) = x^2+1$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = 6-x-x^2$, $g(x) = xsqrt{x+10}$

item $f(x) = sqrt[3]{x+1}$, $g(x) = 4x^2-x$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = dfrac{3}{1-x}$, $g(x) = dfrac{4x}{x^2+1}$

item $f(x) = dfrac{x}{x+5}$, $g(x) = dfrac{2}{7-x^2}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = dfrac{2x}{5-x^2}$, $g(x) = sqrt{4x+1}$

item $f(x) =sqrt{2x+5}$, $g(x) = dfrac{10x}{x^2+1}$ label{funccompeval1last}

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

In Exercises ef{funccompexp1first} - ef{funccompexp1last}, use the given pair of functions to find and simplify expressions for the following functions and state the domain of each using interval notation.

egin{multicols}{3}

egin{itemize}

item $(g circ f)(x)$

item $(f circ g)(x)$

item $(f circ f)(x)$

end{itemize}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = 2x+3$, $g(x) = x^2-9$ label{funccompexp1first}

item $f(x) = x^2 -x+1$, $g(x) = 3x-5$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = x^2-4$, $g(x) = |x|$

item $f(x) = 3x-5$, $g(x) = sqrt{x}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = |x+1|$, $g(x) = sqrt{x}$

item $f(x) = 3-x^2$, $g(x) = sqrt{x+1}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = |x|$, $g(x) = sqrt{4-x}$

item $f(x) = x^2-x-1$, $g(x) = sqrt{x-5}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = 3x-1$, $g(x) = dfrac{1}{x+3}$

item $f(x) = dfrac{3x}{x-1}$, $g(x) =dfrac{x}{x-3}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(x) = dfrac{x}{2x+1}$, $g(x) = dfrac{2x+1}{x}$

item $f(x) = dfrac{2x}{x^2-4}$, $g(x) =sqrt{1-x}$

label{funccompexp1last}

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

pagebreak

In Exercises ef{threefunccompfirst} - ef{threefunccomplast}, use $f(x) = -2x$, $g(x) = sqrt{x}$ and $h(x) = |x|$ to find and simplify expressions for the following functions and state the domain of each using interval notation.

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(hcirc g circ f)(x)$ label{threefunccompfirst}

item $(hcirc f circ g)(x)$

item $(gcirc f circ h)(x)$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(gcirc h circ f)(x)$

item $(fcirc h circ g)(x)$

item $(fcirc g circ h)(x)$ label{threefunccomplast}

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

In Exercises ef{breakdowncompexfirst} - ef{breakdownxomexlast}, write the given function as a composition of two or more non-identity functions. (There are several correct answers, so check your answer using function composition.)

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $p(x) = (2x+3)^3$ label{breakdowncompexfirst}

item $P(x) = left(x^2-x+1 ight)^5$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $h(x) = sqrt{2x-1}$

item $H(x) = |7-3x|$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $r(x) = dfrac{2}{5x+1}$

item $R(x) = dfrac{7}{x^2-1}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $q(x) = dfrac{|x|+1}{|x|-1}$

item $Q(x) = dfrac{2x^3+1}{x^3-1}$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{2}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $v(x) = dfrac{2x+1}{3-4x}$

item $w(x) = dfrac{x^2}{x^4+1}$ label{breakdownxomexlast}

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{enumerate}

setcounter{enumi}{value{HW}}

item Write the function $F(x) = sqrt{dfrac{x^{3} + 6}{x^{3} - 9}}$ as a composition of three or more non-identity functions.

item Let $g(x) = -x, , h(x) = x + 2, , j(x) = 3x$ and $k(x) = x - 4$. In what order must these functions be composed with $f(x) = sqrt{x}$ to create $F(x) = 3sqrt{-x + 2} - 4$?

item What linear functions could be used to transform $f(x) = x^{3}$ into $F(x) = -frac{1}{2}(2x - 7)^{3} + 1$? What is the proper order of composition?

setcounter{HW}{value{enumi}}

end{enumerate}

In Exercises ef{pointcompexfirst} - ef{pointcompexlast}, let $f$ be the function defined by [f = {(-3, 4), (-2, 2), (-1, 0), (0, 1), (1, 3), (2, 4), (3, -1)}] and let $g$ be the function defined [g = {(-3, -2), (-2, 0), (-1, -4), (0, 0), (1, -3), (2, 1), (3, 2)}]. Find the value if it exists.

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(f circ g)(3)$ label{pointcompexfirst}

item $f(g(-1))$

item $(f circ f)(0)$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(f circ g)(-3)$

item $(g circ f)(3)$

item $g(f(-3))$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(g circ g)(-2)$

item $(g circ f)(-2)$

item $g(f(g(0)))$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $f(f(f(-1)))$

item $f(f(f(f(f(1)))))$

item $underbrace{(g circ g circ cdots circ g)}_{mbox{$n$times}}(0)$ label{pointcompexlast}

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

%pagebreak

In Exercises ef{twofuncgraphcompfirst} - ef{twofuncgraphcomplast}, use the graphs of $y=f(x)$ and $y=g(x)$ below to find the function value.

egin{center}

egin{tabular}{cc}

egin{mfpic}[20]{-1}{5}{-1}{5}

axes

label[cc](5,-0.5){scriptsize $x$}

label[cc](0.5,5){scriptsize $y$}

xmarks{1,2,3,4}

ymarks{1,2,3,4}

lpointsep{5pt}

scriptsize

axislabels {x}{{$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

axislabels {y}{{$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

polyline{(0,4), (1,2), (2,3), (3,3), (4,0)}

point[3pt]{(0,4), (1,2), (2,3), (3,3), (4,0)}

ormalsize

caption{$y = f(x)$}

end{mfpic}

&

hspace{1in}

egin{mfpic}[20]{-1}{5}{-1}{5}

axes

label[cc](5,-0.5){scriptsize $x$}

label[cc](0.5,5){scriptsize $y$}

xmarks{1,2,3,4}

ymarks{1,2,3,4}

lpointsep{5pt}

scriptsize

axislabels {x}{{$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

axislabels {y}{{$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

polyline{(0,0), (1,3), (2,3), (3,0), (4,4)}

point[3pt]{(0,0), (1,3), (2,3), (3,0), (4,4)}

ormalsize

caption{$y = g(x)$}

end{mfpic}

end{tabular}

end{center}

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(gcirc f)(1)$ label{twofuncgraphcompfirst}

item $(f circ g)(3)$

item $(gcirc f)(2)$

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{multicols}{3}

egin{enumerate}

setcounter{enumi}{value{HW}}

item $(fcirc g)(0)$

item $(fcirc f)(1)$

item $(g circ g)(1)$ label{twofuncgraphcomplast}

setcounter{HW}{value{enumi}}

end{enumerate}

end{multicols}

egin{enumerate}

setcounter{enumi}{value{HW}}

item The volume $V$ of a cube is a function of its side length $x$. Let's assume that $x = t + 1$ is also a function of time $t$, where $x$ is measured in inches and $t$ is measured in minutes. Find a formula for $V$ as a function of $t$.

## Malaysia Students

Please take note that the first eight topics (Paper 1) of Maths T and Maths S are the same. Besides, Maths T and Maths S are mutually exclusive. In other words, a STPM candidate cannot take both subjects at the same time. Maths T is taken by most science stream stuedents whereas Maths S is taken by some art stream students. Meanwhile, Further Maths is taken as the optional fifth subject by some science stream students.

Numbers and Sets
Real numbers
Exponents and logarithms
Complex numbers
Sets

Hi, can you change the template, I prefer the previous format. It is harder to read your post with this new look

Hello anonymous, are you a regular reader of this blog? Or are you one of the contributors? Do you mind to leave your contact details like email address or blog url?

Thanks for your suggestions. However, I think that the current template is cleaner and the fonts are larger than the previous template. Moreover, I would like to keep the blog's freshness by changing the template once in a two months. You might notice that it is the third template used on Malaysia Students blog if you have visited this blog since March this year.

Besides, I have learned that this month, the average pages viewed by a reader has increased approximately 23% compared to the last month. I believe blog template is a factor for this improvement.

I agree that this template has its demerits too! I think when you said "It is harder to read your post with this new look", you refer to the text ( fonts ) right? Please be patient while I figure out how to change the fonts.

Update: The fonts in this blog have been changed to the fonts used in the previous blog template.

## Sets, Functions, and Logic : An Introduction to Abstract Mathematics, Third Edition

Keith Devlin. You know him. You've read his columns in MAA Online, you've heard him on the radio, and you've seen his popular mathematics books. In between all those activities and his own research, he's been hard at work revising Sets, Functions and Logic, his standard-setting text that has smoothed the road to pure mathematics for legions of undergraduate students.

Now in its third edition, Devlin has fully reworked the book to reflect a new generation. The narrative is more lively and less textbook-like. Remarks and asides link the topics presented to the real world of students' experience. The chapter on complex numbers and the discussion of formal symbolic logic are gone in favor of more exercises, and a new introductory chapter on the nature of mathematics--one that motivates readers and sets the stage for the challenges that lie ahead.

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Dr. Devlin is also one of the world's leading popularizers of mathematics. Known as "The Math Guy" on NPR's Weekend Edition, he is a frequent contributor to other local and national radio and TV shows in the US and Britain, writes a monthly column for the Web journal MAA Online, and regularly writes on mathematics and computers for the British newspaper The Guardian.

## 5.4 Right Triangle Trigonometry

We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:

In this section, we will see another way to define trigonometric functions using properties of right triangles .

### Using Right Triangles to Evaluate Trigonometric Functions

In earlier sections, we used a unit circle to define the trigonometric functions . In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of t t is its value at t t radians. First, we need to create our right triangle. Figure 1 shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point ( x , y ) ( x , y ) to the x-axis, we have a right triangle whose vertical side has length y y and whose horizontal side has length x . x . We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.

#### Understanding Right Triangle Relationships

Given a right triangle with an acute angle of t , t ,

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”

### How To

Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.

1. Find the sine as the ratio of the opposite side to the hypotenuse.
2. Find the cosine as the ratio of the adjacent side to the hypotenuse.
3. Find the tangent as the ratio of the opposite side to the adjacent side.

### Example 1

#### Evaluating a Trigonometric Function of a Right Triangle

Given the triangle shown in Figure 3, find the value of cos α . cos α .

#### Solution

The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:

### Try It #1

Given the triangle shown in Figure 4, find the value of sin t . sin t .

#### Relating Angles and Their Functions

When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in Figure 5. The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.

### How To

Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.

1. If needed, draw the right triangle and label the angle provided.
2. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
3. Find the required function:
• sine as the ratio of the opposite side to the hypotenuse
• cosine as the ratio of the adjacent side to the hypotenuse
• tangent as the ratio of the opposite side to the adjacent side
• secant as the ratio of the hypotenuse to the adjacent side
• cosecant as the ratio of the hypotenuse to the opposite side
• cotangent as the ratio of the adjacent side to the opposite side

### Try It #2

#### Finding Trigonometric Functions of Special Angles Using Side Lengths

We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of 30° , 30° , 60° , 60° , and 45° , 45° , however, remember that when dealing with right triangles, we are limited to angles between 0° and 90° . 0° and 90° .

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

### How To

Given trigonometric functions of a special angle, evaluate using side lengths.

1. Use the side lengths shown in Figure 8 for the special angle you wish to evaluate.
2. Use the ratio of side lengths appropriate to the function you wish to evaluate.

### Example 3

#### Evaluating Trigonometric Functions of Special Angles Using Side Lengths

Find the exact value of the trigonometric functions of π 3 , π 3 , using side lengths.

### Try It #3

Find the exact value of the trigonometric functions of π 4 , π 4 , using side lengths.

#### Using Equal Cofunction of Complements

If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of π 6 π 6 and π 3 , π 3 , we see that the sine of π 3 , π 3 , namely 3 2 , 3 2 , is also the cosine of π 6 , π 6 , while the sine of π 6 , π 6 , namely 1 2 , 1 2 , is also the cosine of π 3 . π 3 .

### Cofunction Identities

The cofunction identities in radians are listed in Table 1.

### How To

Given the sine and cosine of an angle, find the sine or cosine of its complement.

1. To find the sine of the complementary angle, find the cosine of the original angle.
2. To find the cosine of the complementary angle, find the sine of the original angle.

### Example 4

#### Solution

According to the cofunction identities for sine and cosine,

### Try It #4

#### Using Trigonometric Functions

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

### How To

Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.

1. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
3. Using the value of the trigonometric function and the known side length, solve for the missing side length.

### Example 5

#### Finding Missing Side Lengths Using Trigonometric Ratios

Find the unknown sides of the triangle in Figure 11.

#### Solution

We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

We rearrange to solve for a . a .

We can use the sine to find the hypotenuse.

Again, we rearrange to solve for c . c .

### Try It #5

#### Using Right Triangle Trigonometry to Solve Applied Problems

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See Figure 12.

### How To

Given a tall object, measure its height indirectly.

1. Make a sketch of the problem situation to keep track of known and unknown information.
2. Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
3. At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
4. Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
5. Solve the equation for the unknown height.

### Example 6

#### Measuring a Distance Indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of 57° 57° between a line of sight to the top of the tree and the ground, as shown in Figure 13. Find the height of the tree.

#### Solution

The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of 57° , 57° , letting h h be the unknown height.

The tree is approximately 46 feet tall.

### Try It #6

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of 5 π 12 5 π 12 with the ground? Round to the nearest foot.

### Media

Access these online resources for additional instruction and practice with right triangle trigonometry.

### 5.4 Section Exercises

#### Verbal

For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the x- and y-coordinates?

The tangent of an angle compares which sides of the right triangle?

What is the relationship between the two acute angles in a right triangle?

The mathematics content taught at Grade 11 and 12 do in the mainstream schools and distance learning modes are similar. There is not much difference. Students can use the past Advanced and General mathematics exam questions and answers as revision questions.

See below how you can get the past maths exam questions.

Students who wish to get practise maths questions and answers can download them here. Get the question booklet: Grade 12 AM and GM past exam papers, PDF. Though the papers are from mainstream schools, they are also the best questions to practise for your mathematics examination.

The ‘latest’ Grade 12 mathematics exam papers are password-protected. We will release them password before the Grade 12 mathematics examination. Check them out.

## Description

### Features

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## Ethiopian Grade 12 Mathematics Teacher Guide

Mathematics study at Grade 12 level is mainly aimed at exposing students to higher mathematical knowledge and competencies necessary to enable them to peruse with their higher education. The first part, which is common to both natural science and social science streams students is an introduction to calculus where the basic concepts of differential and integral calculus are introduced with intuitive explanations and examples followed by formal definitions.

Number of Periods

• 1.1 Sequences
• 1.2 Arithmetic sequences and geometric sequences
• 1.3 The sigma notation and partial sums
• 1.4 Infinities series
• 1.5 Applications of arithmetic progressions and geometric progressions
• 2.1 Limits of sequences of numbers
• 2.2 Limits of functions
• 2.3 Continuity of a function
• 2.4 Exercises on applications of limits
• 12
• 6
• 5
• 3
• 2
• 3.1 Introduction to Derivatives
 understanding rates of change
 Graphical definition of the derivative
 Formal definition (Differentiability at a point)
 Differentiability over an interval
• 3.2 Derivatives of different functions.
 Differentiation of power, simple trigonometric, exponential
and logarithmic functions.
• 3.3 Derivatives of combinations and compositions of functions
• 3.4 Miscellaneous exercise
• 10
• 4.1 Extreme values of functions
• 4.2 Minimization and maximization problems
• 4.3 Rate of change
• 13
• 6
• 6
• 5.1 Integration as inverse process of differentiation
 Integral of:
– Constant
– Power
– Trigonometric
– Exponential and logarithmic functions
• 5.2 Techniques of integration
 Elementary substitution
 Partial fractions
 Integration by parts
• 5.3 Definite integrals, area and fundamental theorem of calculus
• 7
• 6.1 Coordinate axes and coordinate planes in space
• 6.2 Coordinates of a point in space
• 6.3 Distance between two points in space
• 6.4 Mid-point of a line segment in space
• 6.5 Equation of sphere
• 6.6 Vectors in space
• 2
• 2
• 2
• 1
• 2
• 8
• 7.1 Revision on logic
• 7.2 Different types of proofs
• 7.3 Principle and application of mathematical induction
• 5
• 4
• 4
• 2
• 8.1 Sampling techniques
• 8.2 Representation of data
• 8.3 Construction of graphs and interpretation
• 8.4 Measures of central tendency and variation of a set of data,
including grouped data. (Mean, Median, Mode, Range, Inter
quartile rang and Standard deviation from the data itself or
from given totals)
• 8.5 Analysis of frequency distributions with equal means but
different variances (coefficients of variation).
• 8.6 Use of cumulative frequency graph to estimate
• 3
• 2
• 6
• 5
• 9.2 Percent increase and percent decrease
• 9.3 Real estate expenses
• 9.4 Wages
• 3
• 4
• 4
• 4

Units 1, 2, 3, 4, 5, 6 and 7 of Grades 11 and units 1, 2, 3, 4 and 5 of Grades 12 are common to both natural science and social science stream students, While units 7 and 8 of Grades 11 and units 6 and 7 of Grades 12 are to be offered only to natural science stream students and units 9 and 10 of Grades 11 and units 8 and 9 of Grades 12 are only for social science stream students.

## Two-point methods

### 2.5 Kung-Traub’s multipoint methods

A fundamental and one of the most influential papers in the topic of multipoint methods for solving nonlinear equations is certainly Kung-Traub’s paper ( Kung and Traub, 1974 ). Aside from the famous conjecture on the upper bound of the order of convergence of multipoint methods with fixed number of F.E. (see Section 1.3), this paper presents two optimal multipoint families of iterative methods of arbitrary order. These general families will be studied in later chapters together with their convergence analysis. However, frequent use and citation of these families, called K-T family for brevity, imposes their short introduction. We present Kung-Traub’s families in the form given in Kung and Traub (1974) .

K-T (2.109) : For any m , define iteration function p j ( f ) ( j = 0 , … , m ) as follows: p 0 ( f ) ( x ) = x and for m > 0 ,

for j = 1 , … , m - 1 , where R j ( y ) is the inverse interpolating polynomial of degree at most j such that

Let us note that the family K-T (2.109) requires no evaluation of derivatives of f. The order of convergence of the family K-T (2.109) , consisting of m - 1 steps, is 2 m - 1 ( m ⩾ 2 ) .

K-T (2.110) : For any m, define iteration function q j ( f ) ( j = 1 , … , m ) as follows: q 1 ( f ) ( x ) = x , and for m > 1 ,

for j = 2 , … , m - 1 , where S j ( y ) is the inverse interpolating polynomial of degree at most j such that

The order of convergence of the family K-T (2.110) , consisting of m - 1 steps, is 2 m - 1 ( m ⩾ 2 ) .

p 1 in (2.109) and q 1 in (2.110) are only initializing steps and they do not make the first step of the described iterations.

In this chapter we study two-point methods so that it is of interest to present two-point methods obtained as special cases of the Kung-Traub families (2.109) and (2.110) . First, for m = 3 we obtain from (2.109) the derivative free two-point method

The two-point method (2.111) is of fourth order and requires three F.E. so that it belongs to the class Ψ 4 of optimal methods.

The iterative scheme (2.111) can be rewritten in the form

where t k = f ( y k ) / f ( x k ) and s k = f ( y k ) / f ( x k + γ f ( x k ) ) . Comparing (2.112) to (2.91) with h given by (2.98) , we observe that the family (2.91) is a generalization of the Kung-Traub two-point method (2.111) .

Taking m = 3 in (2.110) , we obtain Kung-Traub’s two-point method of fourth order,

Note that the family (2.74) is a generalization of Kung-Traub’s method (2.113) , which follows from this family taking r = - 2 in (2.80) or a = 1 in (2.83) .

We have applied some of the presented two-point methods of fourth order to the function

Table 2.3 . Example 2.6 – f ( x ) = ( x - 2 ) ( x 10 + x + 1 ) e - x - 1 , α = 2

Two-point methods | x 1 - α | | x 2 - α | | x 3 - α | | x 4 - α |
Ostrowski’s IM (2.47) 1.72(−3)3.13(−10)3.49(−37)5.43(−145)
Maheshwari’s IM (2.85) 5.27(−3)1.59(−7)1.45(−25)9.97(−98)
(2.91) h = 1 + s + t , γ = 0.01 1.01(−3)7.84(−11)2.93(-39)5.68(−153)
(2.91) h = 1 + s 1 - t , γ = 0.01 3.29(−4)3.66(−13)5.59(−49)3.04(−192)
Ren-Wu-Bi’s IM (2.104) , a = 02.66(−2)2.09(−3)1.26(−6)2.53(−19)
Kung-Traub’s IM (2.112) γ = 0.01 7.56(−3)6.80(−7)4.88(−23)1.29(−87)
Kung-Traub’s IM (2.113) 3.45(−3)1.36(−8)3.38(−30)1.31(−116)

## 5.4: Further Topics in Functions (Exercises) - Mathematics

### Everyday Mathematics and the Common Core State Standards for Mathematical Practice

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