Antiderivatives in Basic Calculus with integration in Barkada kontra Droga

1. BAYUGAN NATIONAL COMPREHENSIVE HIGH SCHOOL
BAYUGAN CITY
DETAILED LESSON PLAN in TEACHING BASIC CALCULUS
with INTEGRATION in Barkda Kontra Droga (BKD)
Grade Level/section: 11/Riemann Semester: Second
Duration: 1 hour
Teaching Date: February 27, 2018 Time: 2:00 pm – 3:00 pm
I. OBJECTIVES:
A. Content Standard
The learners demonstrate an understanding of the antiderivatives of algebraic
functions.
B. Performance Standard
The learners shall be able to formulate the antiderivatives of algebraic functions.
C. Learning Competency
The learner shall be able to compute the general antiderivative of algebraic
functions. (STEM_BC11I-IVa-b-1)
Specific learning outcomes:
At the end of the lesson, the students shall be able to:
c.1. define what is antiderivatives;
c.3. formulate the antiderivatives of algebraic functions;
c.4. compute the antiderivatives of algebraic functions; and
c.2. appreciate the importance of antiderivatives of algebraic
functions in solving word problem involving population;
II. CONTENT
A. Topic: ANTIDERIVATIVES OF ALGEBRAIC FUNCTIONS
B. References
1. Teacher’s1111 Guide : pages 196-198
2. Learner’s Material: pages 227-229
3. Textbook: Basic calculus by John Gabriel P. Pelias; pages 178-181
4. Additional materials: retrieved from http://tutorial.math.lamar.edu/Classes/
C. Teaching Strategy: GUIDED-DISCOVERY LEARNING
III. PROCEDURES
A. Reviewing previous lesson or presenting new lesson
The lesson will begin by recalling the notations and terminologies of
antiderivatives.

2. Guide questions:
1. What do you mean by antiderivatives?
2. What symbol denotes the operation of antidifferentiation?
3. Why is it that we need to put +C at the end the antiderivatives?
B. Establishing a purpose for the lesson
The students will compute the antiderivatives of algebraic functions
Guide questions:
1. What if the given is only the derivatives, can we directly find its
antiderivatives?
2. Is there a formula in finding the antiderivatives of algebraic
functions?
C. Presenting examples/ instances of the new lesson
The discussion will begin by introducing a
mystery box found outside their classroom. But in
order open the box they need to read the attached
file. And this is their 1st Mission !
Directions: Identify the antiderivatives of the following derivatives.
Derivatives
1. 𝑓( 𝑥) = 10
2. 𝑓( 𝑥) = 15𝑥2 − 1
3. 𝑓( 𝑥) = −9𝑥2
Antiderivatives
a. 𝐹( 𝑥) = −3𝑥3 + 𝐶
b. 𝐹( 𝑥) = 10𝑥 + 𝐶
c. 𝐹( 𝑥) = 5𝑥3 − 𝑥 + 𝐶
READ ME FIRST:
In order to open the mystery box, you need to find the general rule in
finding the antiderivatives of algebraic functions. But once you accepted the
challenge you only have 5 minutes to formulate the rule otherwise we will
explode. Now to determine if it is the correct rule in finding the antiderivatives
you need to present an example that proves that it is the right formula. GOOD
LUCK!

3. D. Discussing new concepts and practicing new skills #1
Every group will choose a representative to share their ideas on how did they
come up with the answer. They will be rated according to the following criteria:
6 4 2
Content
Presentation had an
exceptional amount of
valuable material and was
extremely beneficial to
the class.
Presentation had a
good amount of
material and
benefited the class.
Presentation had
moments where
valuable material was
present but as a whole
content was lacking.
Presentation
The presenter was very
confident in delivery and
explained the material
clearly.
The presenter was
occasionally
confident in
delivery and
explained the
material.
The presenter was
little confident in
delivery and did not
explained the
material well.
Guide questions:
1. What value of n that the rule can be undefined?
2. Is -1 included of the value of the n?
Theorem 11: (Theorems on Antidifferentiation)
𝑎.) 𝐼𝑓 𝑛 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝑛 ≠ −1, 𝑡ℎ𝑒𝑛
∫ 𝑥 𝑛 𝑑𝑥 =
𝑥 𝑛+1
𝑛+1
+ 𝐶.
∫ 𝑥 𝑑𝑥 =
𝑥2
2
+ 𝐶
∫ 𝑥2
𝑑𝑥 =
𝑥3
3
+ 𝐶
∫ 𝑥3
𝑑𝑥 =
𝑥4
4
+ 𝐶
∫ 2 𝑑𝑥 = 2𝑥 + 𝐶
∫(3𝑥4
+ 10) 𝑑𝑥 =
3𝑥5
5
+ 10𝑥 + 𝐶
∫ 𝑥 𝑛
𝑑𝑥 =? ??

4. E. Discussing new concepts and practicing new skills #2 (Analysis)
The class will open their boxes and inside their boxes are envelopes and another
boxes. Inside the envelope is the information below.
At the back of the above text is the text below. And this is their 2nd mission.
F. Developing mastery
In a quiz bowl the students will answer the following questions.
Theorem 11: (Theorems on Antidifferentiation)
𝑎. 𝐼𝑓 𝑛 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝑛 ≠ −1, 𝑡ℎ𝑒𝑛
∫ 𝑥 𝑛 𝑑𝑥 =
𝑥 𝑛+1
𝑛+1
+ 𝐶.
𝑏. ∫ 𝑑𝑥 = 𝑥 + 𝐶
𝑐. 𝐼𝑓 𝑎 𝑖𝑠 𝑎𝑛𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 𝑓 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒𝑛
∫ 𝑎 𝑓( 𝑥) 𝑑𝑥 = 𝑎 ∫ 𝑓( 𝑥) 𝑑𝑥 .
𝑑. 𝐼𝑓 𝑓 𝑎𝑛𝑑 𝑔 𝑎𝑟𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙,
∫[ 𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥 = ∫ 𝑓( 𝑥) 𝑑𝑥 + ∫ 𝑔( 𝑥) 𝑑𝑥 .
Directions: Determine the following antiderivatives using the theorems. Each
correct answer merits 2 points. You will be given specific time to answer for each
questions.
1. ∫ 3 𝑑𝑥 = 3𝑥 + C
2. ∫ 20 𝑑𝑥 = 20𝑥 + 𝐶
3. ∫ 𝑥6 𝑑𝑥 =
𝑥7
7
+ 𝐶
4. ∫ 4𝑥 𝑑𝑥 = 2𝑥2 + 𝐶
5. ∫(12𝑥2 + 2𝑥) 𝑑𝑥 = 4𝑥3 + 𝑥2 + 𝐶
6. ∫(𝑥2 − 4𝑥 + 4 ) 𝑑𝑥 =
𝑥3
3
− 2𝑥2 + 4𝑥 + 𝐶
I can smell your eagerness what’s really inside the box. But before you open it, you
need to answer the questions I prepared for you.

5. G. Finding practical application of concept and skills in daily living
This activity is included in the quiz bowl.
Guide questions:
1. What is the rate of change?
2. Is this already the function that represents the population of drug
users in the Philippines?
3. And how do we find the function that represents the population of drug
users?
4. What is the function?
5. What is the Constant number of drug users in the Philippines?
6. Given this function what is the population of drug users after 5 years?
7. What have you observed on the number of drugs users in the
Philippines after 5 years?
8. As a student, what are you going to do to lessen the number of drug
users in our country?
9. What program does our school implemented towards this problem?
10.Do you think that this will give change to the problem we encountered
today?
H. Making generalizations and abstraction about the lesson
Guide questions:
1. What is antiderivatives?
2. How do we get the antiderivatives of algebraic functions?
3. What particular situation we can make use of the concept of
antiderivatives?
I. Evaluating learning
Inside the box is the text below. And this is their 3rd mission.
In 𝑥 years , the drug users in the Philippines will be changing at a rate of 12,500 +
100,000𝑥3
. If the current population of drug users is 1,800,000 , what is the number
of drug users 5 years from now? ( DDB’s 2015 Nationwide Survey)
Answer: 17,487,500
I can smell your eagerness in uncovering what’s really inside the box. But before you
can open it, you need to answer the questions I prepared for you.

6. In group activity (Riemann Math challenge 2018 ), they will answer the given
problem within the allotted time. Each correct answer merits two points. This is their
third mission.
J. Additional activities for remediation
I.Directions: Find the antiderivatives of the following functions.
1. ∫ 10 𝑑𝑥 6. ∫(3𝑥3
+ 4𝑥 + 27) 𝑑𝑥
2. ∫ 𝜋 𝑑𝑥 7. ∫(
1
4
𝑥3
− 𝑥2
+ 𝑥) 𝑑𝑥
3. ∫ 3𝑥 𝑑𝑥 8. ∫ (
9
𝑥3 +
5
𝑥
− 1 ) 𝑑𝑥
4. ∫ 2𝑥3
𝑑𝑥 9. ∫ 2√ 𝑥 𝑑𝑥
5. ∫(7𝑥8
− 𝑥3
) 𝑑𝑥 10. ∫
√𝑥45
3
𝑑𝑥
1. ∫(3𝑥3
+ 2𝑥2
) 𝑑𝑥 =
3𝑥4
4
+
2𝑥3
3
+ 𝐶 2. ∫(3𝑥2
− 𝑥 + 2) 𝑑𝑥 = 𝑥3
−
𝑥2
2
+ 2𝑥 + 𝐶
3. ∫(3𝑥4
+ 3𝑥2
+ 1) 𝑑𝑥 =
3𝑥5
5
+ 𝑥3
+ 𝑥 + 𝐶
1. ∫ (
1
3
𝑥3
−
2
5
𝑥2
− 3) 𝑑𝑥 =
1
12
𝑥4
−
2
15
𝑥3
− 3𝑥 + 𝐶
2. ∫(
𝑥2
3
+
𝑥3
4
+
𝑥4
5
) 𝑑𝑥 =
𝑥3
9
+
𝑥4
16
+
𝑥5
25
+ 𝐶
3. ∫(
1
𝑥5 −
1
𝑥6 )𝑑𝑥 = −
1
4𝑥4 −
1
5𝑥5 + 𝐶
30-second questions ( 3 points)
50-second questions (5 points )
Directions: Find the antiderivative of the function within the given time.
1. ∫ 1,000 𝑑𝑥 = 1000𝑥 + 𝐶 2. ∫ −20𝑥4
𝑑𝑥 = −4𝑥5
+ 𝐶
3. ∫ 15𝑥2
𝑑𝑥 = 5𝑥3
+ 𝐶 4. ∫ 𝑥4
𝑑𝑥 =
𝑥5
5
+ 𝐶
10-second questions (2 points)
II. Directions: Answer the problem and show your solution.
In 𝑥 years , the drug users in the Philippines will be changing at a rate of 11,000 +
400,000𝑥3
. If the current population of drug users is 2,900,000 , what is the number of
drug users 8 years from now?

7. V. REMARKS
The students understand directly the topic.
VI. REFLECTION
A. No. of learners who earned 80% on the formative assessment: None
B. No. of learners who require additional activities for remediation: None
C. Did the remedial lesson work? No. of learners who have caught up with the lesson:
None
D. No. of learners who continue to require remediation: None
E. Which of my teaching strategies worked well? Why did these work?
In my discussion, the students formulate the ruleby themselves and assess their own
learning.
Prepared by:
CLARIBEL C. AYANAN
Student teacher
Noted:
ANALYN S. LAUROWA-FEBRIO
Cooperating teacher