unit of work grade: 12 applied topic: integration 33 · 2018. 9. 6. · 1 12 applied mathematics...

1 12 Applied Mathematics – Integration

UNIT OF WORK

Grade: 12 Applied Topic: Integration No. of Lessons: 33

Indicator: 12Ap3 - Use and find the anti-derivative of a function and find the indefinite integral of a given function.

Content Suggested Expanded Content MoE Textbook Link

a. (3-1) Find the integral of a function in the form: f(x) = a f(x) = xn f(x)= axn f(x) = g(x) +/- h(x)

Explain the concept of an anti derivative and indefinite integrals

Pages 71 - 75 Exercises pgs 76 - 80

Find the indefinite integral for algebraic function

b. Find the definite integral of a function

Introduce a definite integral

Find the definite integral of a given function

c. (3-2) Find the properties of definite integrals.

Define the rules of a definite integral:

∫ ( )

0

∫ ( )

∫ ( )

∫ ( )

∫ ( )

∫ ( ) ( )

∫ ( )

∫ ( )

∫ ( )

∫ ( ) ∫ ( )

Pages 81 - 86 Exercises pg 87

Apply the rules of a definite integral

d. (3-3) Find the area between a curve and the x-axis

Use the rules of definite integrals to find the area between a curve and the x axis

A = ∫ ( )

,where f(x) , (a,b)

A = – ∫ ( )

,where f(x) , (a,b)

Page 88 – 93 Exercise pgs 94 – 95 General Exercises pg 96 - 97

Foundation Material:

This material is designed to provide the background skills and knowledge that students may not have been exposed to over the previous two years. This material should be covered immediately prior to the text book exercises in each section of the unit.

12Ap3 (a) Anti-derivative and indefinite integral (pages 3 – 9)

12Ap3 (b) Definite integral and its properties [exponential (index) notation] (pages 3 – 9)

12Ap3 (d) Area between curve and x – axis (pages 10 – 14)

Problem Solving:

Technology Investigation 1 – The horse breeders paddock (page 15)

Technology Investigation 2 – Combining Trapeziums (page 16)

Assessment:

NOTE: Some of these tasks may be suitable for continuous assessment purposes.

Investigation 1 – The Area Function (page 17)

Investigation 2 – ∫ ( )

and Areas (page 18)


Working Mathematically: Thinking Skills:

Students must be given opportunities to undertake the following 5 processes in order to work mathematically: 1. Questioning:

Students ask questions in relation to: mathematical situations mathematical experiences

2. Applying Strategies: Students develop, select and use a range of

strategies to explore and develop solutions in solving problems.

3. Communicating: Students develop and use appropriate language

and representations to formulate and express mathematical ideas.

4. Reasoning: Students develop and use processes for: exploring relationships checking solutions giving reasons to support their conclusions

5. Reflecting: Students reflect on their:

Experiences Critical understanding to make connections with, and generalisations about, existing knowledge and understanding

Creating Generating new ideas, products, or ways of viewing things

Designing, constructing, planning, producing, inventing. Now can you see why we must ….?

Now evaluate this…..?

Evaluating Justifying a decision or course of action

Checking, hypothesising, critiquing, experimenting, judging What have we learned from this …..?

Analysing

Breaking information into parts to explore understandings and relationships

Comparing, organising, deconstructing, interrogating, finding

Can you see why …..? What does this tell us ….?

Applying

Using information in another familiar situation Implementing, carrying out, using, executing

How do you evaluate …..? How can you find …..?

Understanding

Explaining ideas or concepts Interpreting, summarising, paraphrasing, classifying,

explaining What is the meaning of…..? What is the notation for ….? What should I do now ….?

Remembering

Recalling information Recognising, listing, describing, retrieving, naming, finding

What is the definition of …? What is the rule for ….?

Key Words:

English Arabic English Arabic

anticlockwise area

clockwise curve

antiderivative x-axis

integral y-axis

integer straight line

definite integral theorem

indefinite integral positive

limits negative

between


Grade 12 Applied Mathematics Foundation Material

12Ap3: Integration (Anti-derivatives) )القسم الثالث: التكامل)عكس المشتقة

(12Ap3 a, 3-1) Anti-derivative function and indefinite integral

(12Ap3 c, 3-2) Definite integral and its properties (12Ap3 d, 3-3) Area between a curve and the X-axis

(12Ap3 a, 3-1) الدالة المقابلة والتكامل غر المحدد (12Ap3 c, 3-2) التكامل المحدد وخواصه

(12Ap3 d) مساحة منطقة محصورة بن منحنى دالة ومحور السنات

12Ap3 (a, 3-1) and 12Ap3 (c, 3-2)

Working with surds and indices العمل مع الجذور واألسس

Exponential (Index) notation عالمة األ

Indices is the plural of index. An index can also be called a power or exponent.

والت تعن األسس index ه الجمع لكلمة Indices

The symbol tells us there are 3 factors of 4 multiplied together, or .

الرمز 4×4×4مضروبة ببعضها 4عن ان هناك ثالثة عوامل للرقم

Index or power or exponent االس

= = 64

Base القاعدة 3 factors ثالث عوامل basic numeral رقم عادي

The following table shows how the exponent form relates to the factorized number الجدول التال بن عالمة االس مع الرقم

المفكك الى العوامل:

Example 1 Write in exponent form حول التالة إلى أسس

* +

3وثالثة عوامل من 2من اربعة عوامل


No. Expression Explanation التوضيح

1

When multiplying, keep the base and add the powers.

عند الضرب احتفظ باألساس واجمع األسس

2

When dividing, keep the base and subtract the powers

عند القسمة احتفظ باألساس واطرح األسس

3 Any number to the power of zero equals to one ( )

1أي رقم أسه صفرا ساوي

4 ( ) Keep the base and multiply the powers احتفظ باألساس واضرب القوى المرفوعة

5 ( ) Each factor in the bracket is raised to the power of m

m كل عامل ف القوس مرفوع إلى القوة

6 (

)

Each number in the bracket is raised to the power of m

m كل رقم ف القوس مرفوع للقوة

7

i) √ =

ii) √

=

iii) √

iv) √

=

=

v) √

=

= = 125

8 i)

ii)

9

⁄

Use the following: Keep-change-flip Write the expression:

Keep change flip

1 x

Answer:

ر احفظ اقلب غ

1 x

Answer:

10

Simplify:

استخدم قانون األسس i) Use the index law

keep the base and add the powers. اجمع األسس

ii) Use

to

change the number with a negative index to one with a positive index. ر األس غ السالب إلى موجب


Worked Examples: أمثلة محلولة

بسط

خطوات

أضرب البسط مع البسط

والمقام مع المقام

األرقامقسم

الت تحتوي على نفس القاعدة) إطرح األسس(

ر من قسمة إلى ضرب) إقلب الكسر الثان( غ

بسط االرقام ف األقواس

اضرب البسط بالبسط والمقام بالمقام

اختصر

قسم األرقام الت تحوي نفس القاعدة

بسط األرقام ف األقواسبالمقاماضرب البسط بالبسط والمقام

قسم االرقام الت تحوي نفس القاعدة


Student Activity نشاط طالبي

حول إلى جذور

حول إلى أس

ما هو الناتج بدون استخدام اآللة الحاسبة



ف ابسط صورة

بسط كال مما ل


اختر اإلجابة الصححة

بسط كال مما ل:


بسط كال مما ل

بسط كال مما ل:

بسط ما ل، وضع اجابتك بالجذور الموجبة

بسط ما ل، وضع اجابتك بالجذور الموجبة


12Ap3 (d, 3-3)

Area between the curve and the -axis The formulae for calculating the area of the most commonly occurring shapes should be familiar to you and are given

below:

المستطيل

المثلث

متوازي األضالع

شبه المنحرف

الدائرة

القطاع الدائري


مساحة المقطع ه نسبة كسرة من مساحة الدائرة

اوجد مساحة االشكال التالة

المساحة االرتفاع القاعدة المساحة

اوجد المساحة الغر مظللة ف االشكال التالة

المساحة المساحة مساحة المثلث -مساحة الصغيرة مساحة شبه المنحرف –مساحة شبه الدائرة الكبيرة


سم ، أوجد الزاوة المحصورة ف المركز 6ونصف قطر 2سم 25المقطع عنده مساحة

قاس الزاوة

أوجد صغة أو قانون المساحة ل

المساحة = مساحة شبه الدائرة + مساحة المثلث

قطر شبه الدائرة هو a وحدة


Investigation

أوجد مساحة األوراق بعد المربعات

عد 1المربع الكامل

1اكثر من نصف مربع يعد مربع كامل أي 0اقل من نصف مربع يعد

1. By using the area formulae, find the Area under the curve (dotted lines) for each of the following:

بإستخدام القانون للمساحة: أوجد المساحة تحت الخط المستقيم) الخط المنقط ( للرسومات التالية:

2.1


2.2

2.3


Technology Investigation 1

12Ap3 Integration Learning Objective: To use Excel to investigate area and prove this with calculus


Technology Investigation 2

12Ap3 Integration Learning Objective: To use Excel to investigate area between a curve and the x-axis.


Investigation 1

12Ap3 Integration Learning Objective: To use integration to find the area between a straight line and the x- axis.


Investigation 2

12Ap3 Integration

Learning Objective: To use integration to find areas

unit of work grade: 12 applied topic: integration 33 · 2018. 9. 6. · 1 12 applied mathematics...

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