This is a detailed lesson plan on Anti-derivative of a polynomial function using a developmental method.

1. LESSON PLAN FOR MATHEMATICS 11
I. INFORMATION
Subject Matter: Anti-derivative of a Polynomial Function
Grade Level: IX Time Allotment: 1 hour
Teacher/s: Elton John B. Embodo
Content Standard: The learner demonstrates understanding of antiderivatives and Riemann integral.
Performance Standard: The learner is able to formulate and solve accurately situational problems involving
population models.
Learning Competency: The learners compute the general antiderivative of polynomial, radical, exponential, and
trigonometric functions. STEM_BC11I-Iva-b-1
Objectives: At the end of the lesson, the students must have:
a. identified whether a polynomial function is an anti-derivative of another function.
b. computed the antiderivative of polynomial function.
c. discussed the relevance of anti-differentiation in other learning areas.
References: Egarguin, N., Fontail, L., & Lawas, V., (2016). Basic Calculus for Senior High School. C
& E Publishing, Inc.
Instructional Materials: PowerPoint, chalk
Skills: Analysis and Collaboration
Values: Unity, cooperation, camaraderie
Method: Developmental Method
II. LEARNING EXPERIENCES
Teacher’s Activity Students’ Response
A. Preparation
1. Prayer
2. Greetings
3. Reminders
4. Checking of Attendance
5. Classroom Rules - MATH
M - must come to class neat, clean, and prepared.
A - actively participate in the activities and pay attention to
the discussion.
T - talk appropriately and respectfully to your teacher and
classmates.
H - handle the learning materials with care.
Are my rules clear to you class?
a. Review
In our previous lesson class, you have learned much about the
process of differentiation or finding the derivatives of certain
mathematical expression using the different methods.
So, who can recall what differentiation is?
What is then a derivative?
To check, whether you have learned the process of finding
the derivatives of the functions or expressions, I have here a
short activity.
Yes Sir!
Differentiation is the process of finding the derivative of
function or expression.
Derivative is the rate of change of a function with
respect to a variable. It is also used to find the slope of a
line tangent to the curve.

2. Directions: Find the derivatives of the following expressions
1. 12
( )
f x x

2. 5
1
( )
f x
x

3. 4
xdx
4. 2
( 4 9 1)
x x dx
  
5.
5 3
3
4 2 5 7
2
x x x x
x
  
b. motivation
This time I am going to show to you 4 pictures, and all you
must do is to tell what specific concept the pictures depict.
What word you have thought to describe the four pictures
together?
You all have brilliant ideas. Those four pictures depict the
concept of inverse or reverse.
In every process of doing something class, there exists an
inverse or reverse process of it.
Just like finding the derivative of a function, there is also an
inverse process of doing it.
But how are we going to compute the inverse of the
derivatives?
Expected Answers
1. 11
12x
2. 6
5
x
 or 6
5x

3. 3
4
1
4x
4. 8 9
x
 
5. 7
3
2
5 35
4
4
x
x
x
 
I think the pictures sir depict the concept of opposite.
I think the pictures sir depict the concept of reverse.
I think the pictures sir depict the concept of inverse.

3. B. Presentation
So be with me this morning as we tackle on how to compute
the anti-derivative of a polynomial function. Everybody
read!
Listen to me attentively since you are expected to achieve
these objectives. Everybody read.
C. Development Proper
Let us first have the definition of antidifferentiation.
Everybody read!
The symbol is called integral sign while the function to be
integrated is called integrand.
For us to be guided on how to compute the anti-derivative of
a function. We must consider these four theorems.
Theorem 1
dx x C
 

Theorem 2 (The Power Rule)
1
1
n
n x
x dx C
n

 

 where 1
n  ¡
Theorem 3
adx a dx

  where a¡
Theorem 4 The Sum and Difference Rule
 
( ) ( ) ( ) ( )
f x g x f x dx g x dx
  
  
Giving and Discussing of Examples
1. 3dx

In this example, which theorem/s do you think we can use to
compute its anti-derivative?
That is right. So, by following the 3rd
theorem we can obtain,
3dx

= 3 dx

So, this time, how do we further simplify the obtained
expression?
“Anti-derivative of a Polynomial Function”
Antidifferentiation – is the inverse of finding
derivative of a function. It is also called integration.
( )
f x dx

I think the theorem that we can use sir is the 3rd
theorem because the given function suits to the pattern.
We can simplify the expression further sir by applying
the 1st
theorem.
Objectives:
a. identify whether a polynomial function
is an anti-derivative of another function.
b. compute the antiderivative of
polynomial functions.
c. discuss the relevance of anti-
differentiation in other learning areas.

4. Very good! So we have
3( )
x C
 or 3x C
 - answer
By the way class, we add C because it represents to all the
constants being eliminated in finding the derivative of a
function.
Do you understand class?
2.  
5
4
x x dx


With this example, how can we compute its anti-derivative?
What theorem do you think we can use?
All right! This time, we will apply these theorems so we can
compute its anti-derivative. So, by the 4th
theorem (sum and
difference rule), what do you think will happen to the
function?
Okay, since the terms involve exponents, we can now apply
the power rule. But before that, we must convert first the
given radical into its exponential form.
Here we have x , so what is the exponential form of the
square root of x?
So, we now have
1
5 2
4
x dx x dx

  . This time, we will now
apply the 2nd
theorem (power rule)
=
1
1
5 1 2
4
1
5 1 1
2
x x



 
=
3
6 2
4
3
6
2
x x

=
3
6
2
2
4
6 3
x
x
 
  
 
=
3
6 2
8
6 3
x x

=
6 3
8
6 3
x x

=
6
8
6 3
x x x
C
  - answer
Integration of Concept
Okay so we have just computed the anti-differentiation of
polynomial functions in our examples wherein we reversed
the process of computing for the derivative of a function
which also means the rate of change.
Yes sir!
I think sir we can use the 2nd
theorem (power rule) and
4th
theorem (sum and difference rule).
We can obtain, 5
4
x dx xdx

 
The exponential form of x is
1
2
x .

5. Class, apart from this course/subject itself, what other
learning area or field do you think this topic or lesson can be
relevant?
Yes, that is a brilliant idea. Apart from engineering courses
class, there are also other learning areas which integrate this
lesson just like, Biology, Chemistry, Physics, Computer
Science, Investment, and even Economics.
Economics as a subject integrates the concept of
differentiation or anti-differentiation specifically in
determining the minimum and maximum amount of
manufacturing goods which aims to optimize the production
in relation to the supply and demand.
If we look at this picture, how do you describe the
relationship of demand and supply?
Do you understand class?
Board Work Activity
Directions: This time, I need three volunteers to solve the
next examples that I am going to give you. And those who are
seated on their chairs, you are also going the solve the
examples and later we will explain your solutions and
answers.
3.   
1 1
x x dx
 

4.  
2
3
x dx


5.
3
3
x x
dx
x


(The teacher and the students discuss the solutions and
answers illustrated on board).
Do you have any clarification class?
I think sir that this lesson on computing the anti-
derivative of a polynomial function is also relevant in
Engineering courses or programs. Especially, that we are
students from STEM Strand, this lesson will help us
prepare for college.
I can see sir that the relationship of supply and demand
is inversely proportional. As the price, decreases, the
demand increases, and as the price increases, the
demand decreases.
Yes Sir!
Expected Answers of the Volunteers
3.
3
3
x
x C
 
4.
3
2
3 9
3
x
x x C
  
5.
3
3
3
x
x C
 

6. Collaborative Activity
All right, this time we will have a group activity. Please read
the mechanics.
The Polynomial Function
None, sir.
Expected outputs of the 3 groups
(Presentation and Checking of the outputs)
III. EVALUATION
Test I. IDENTIFICATION
Directions: Identify whether the given polynomial is the anti-derivative of another polynomial function. Write YES if
it is, otherwise, write NO in a ¼ sheet of paper. 2 points each
Expected Answers
1. 2
2
x dx x C
 
 1. No,
3
3
x
C

Mechanics of the Collaborative Activity
1. The class will be grouped into 3.
2. Each group will be given with the same
derivative of a polynomial function
and set of materials.
3. Each group will compute the anti-
derivative of the given function.
4. Each group will finish the task in 3
minutes.
5. Once the group is done, they must post
their output on the board.
6. The group which can finish first with
correct solution and answer will be the
winner.
 
7 4 6 3
16 5 3
x x x x dx

  

 
7 4 6 3
16 5 3
x x x x dx

  

=
1
7 4 6 3
16 5 3
x x x x

  
   
=
4
5 5 3
16 5 3
4
8 5 5
3
x x x

  
=
3
8 5
5
3 3
2
5 4
x x
x x C
x
   

7. Prepared:
ELTON JOHN B. EMBODO, MAED, LPT
Teacher Applicant
2. 4 3
5x dx x C
 
 2. No, 5
x C

3. 8 7
5 40
x dx x C
 
 3. No,
9
5
9
x
C

4. 3
2
1
2x dx C
x

  
 4. Yes
5.
3 2
2 2
(2 )
3 2
x x
x x dx C
   
 5. Yes
Test II. COMPUTATION
Directions: Compute for the anti-derivative of the following polynomial functions. Write your solutions and answers in
a ½ sheet of paper. 5 points each.
Expected Answers
1. 2
(3 2 1)
x x dx
 
 1. 3 2
x x x C
  
2. 4 2
( 5 3 3)
x x x dx
   
 2.
2
5 3
3
2
x
x x x C
    
3. 3
3(6 5)
x x dx
 
 3.
4 2
9 3
15
2
x x
x C

 
4.
2
4 3
4
3
8 11
x x dx
x

 
   
 
 
 4.
5
3
3
8 1
3 11
5
x
x x C
x
    
5. 2 3 4 5
1 1 1 1
dx
x x x x
 
  
 
 
 5. 2 3 4
1 1 1 1
2 3 4
C
x x x x
    
IV. ASSIGNMENT
Directions: Read in advance on how to compute for the anti-derivative of radical, exponential, and trigonometric
functions.