application of calculus

1. REVIEW OF THE
PREVIOUS
LESSON

2. What was the previous
discussion last meeting
all about?

3. RATE OF CHANGE OF A
FUNCTION
It is defined as the rate at which one quantity is
changing with respect to another quantity.

4. AVERAGE RATE OF CHANGE OF A
FUNCTION

5. INSTANTANEOUS RATE OF
CHANGE

6. ACTIVITY

7. GUIDE QUESTIONS
1.What was the video all about?
1.What was the video all about?
1.What was the video all about?
2. Describe the motion of the vehicle in the
video.
3. What can be measured in a moving object?

8. RECTILINEAR
MOTION

9. OBJECTIVES
Define concepts involving rectilinear motion
(average velocity, instantaneous velocity,
speed and acceleration)
Solve problems involving the concept of
differentiation (rectilinear motion)

10. KINEMATICS
Kinematics is the study of motion of a system of
bodies without directly considering the forces or
potential fields affecting the motion. In other
words, kinematics examines how the momentum
and energy are shared among interacting bodies.

11. RECTILINEAR MOTION
Rectilinear motion is another name for
straight-line motion (vertical or horizontal
orientation). This type of motion describes
the movement of a particle or a body.

12. GUIDE QUESTIONS
1. Does the video showcase a rectilinear motion?
2. What are some examples of rectilinear motion?

13. QUANTITIES
Scalar Quantities Vector Quantities
Distance Displacement
Speed Velocity
Time Acceleration

14. SCALAR QUANTITIES
Scalar quantity is defined as the physical
quantity with magnitude and no direction.
Example: Distance, Speed, Time

15. REVIEW OF
CONCEPTS
IN PHYSICS

16. SCALAR QUANTITIES
Time is the progression of events from the past
to the present into the future.
Units: Seconds ,Minutes, Hours, Day, Week,
Month

17. SCALAR QUANTITIES
Distance is a scalar quantity that refers to
"how much ground an object or a particle has
covered" during its motion.
Units: Meter, Kilometer, Miles, etc.

18. SCALAR QUANTITIES
Speed is a scalar quantity that refers to "how
fast an object is moving." Speed can be thought
of as the rate at which an object covers
distance.
𝒔 =
𝒅
𝒕
Units: m/s, km/hr., mi/s, mi./min, etc.

19. VECTOR QUANTITIES
Vector quantity is defined as the physical
quantity that has both magnitude and direction.
Example: Velocity, Acceleration

20. VECTOR QUANTITIES
Velocity is a vector quantity that refers to "the
rate at which an object changes its position."
Average Velocity Instantaneous Velocity

21. VECTOR QUANTITIES
The instantaneous velocity is
the specific rate of change of
position (or displacement) with
respect to time at a single
point.

22. VECTOR QUANTITIES
Average velocity is the average rate of change of position
(or displacement) with respect to time over an interval.

23. VECTOR QUANTITIES
Formula of Velocity
𝑣 =
∆𝑑
∆𝑡
Units: m/s, km/hr., mi/s, mi./min, etc.

24. VECTOR QUANTITIES
Acceleration is a vector quantity that is defined as
the rate at which an object changes its velocity. An
object is accelerating if it is changing its velocity.
𝑎 =
∆𝑣
∆𝑡
Units: 𝑚/𝑠2
, 𝑘𝑚/ℎ𝑟.2
, 𝑚𝑖/𝑠2
, 𝑚𝑖./𝑚𝑖𝑛.2
, 𝑒𝑡𝑐.

25. Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
i. The average velocity ∆𝒗 of P at time 𝑡1 to 𝑡2 is ∆𝒗 =
∆𝒔 𝒕
∆𝒕
.
The average velocity of a particle in a rectilinear motion
is the average rate of change of its position function.
RELATIONSHIP OF
POSITION/DISTANCE, SPEED
VELOCITY AND ACCELERATION
OF A PARTICLE IN A
RECTILINEAR MOTION

26. Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
ii. The instantaneous velocity 𝒗(𝒕) of P at time t is
𝒗 𝒕 = 𝒔′ 𝒕 =
𝒅𝒔
𝒅𝒕
.
• The instantaneous velocity of a particle in a rectilinear motion is the
instantaneous rate of change of its position function.
• The first derivative of the position function is the instantaneous
velocity of P.
RELATIONSHIP OF
POSITION/DISTANCE, SPEED
VELOCITY AND ACCELERATION
OF A PARTICLE IN A
RECTILINEAR MOTION

27. Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
iii. The speed of P at time t is 𝒗 𝒕 =
𝒅𝒔
𝒅𝒕
.
The speed of P is the absolute value of first
derivative of the position function which is the
velocity of P.
RELATIONSHIP OF
POSITION/DISTANCE, SPEED
VELOCITY AND ACCELERATION
OF A PARTICLE IN A
RECTILINEAR MOTION

28. Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
iv. The acceleration 𝒂 𝒕 of P at time t is
𝒂 𝒕 = 𝒔′′ 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
.
The acceleration of P is the second derivative of
the position function. It is also the first derivative
of the velocity function of P.
RELATIONSHIP OF
POSITION/DISTANCE, SPEED
VELOCITY AND ACCELERATION
OF A PARTICLE IN A
RECTILINEAR MOTION

29. Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
i. Average velocity ∆𝒗 =
∆𝒔 𝒕
∆𝒕
ii. Instantaneous velocity 𝒗 𝒕 = 𝒔′ 𝒕 =
𝒅𝒔
𝒅𝒕
iii. Speed 𝒗 𝒕 =
𝒅𝒔
𝒅𝒕
iv. Acceleration 𝒂 𝒕 = 𝒔′′ 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
RELATIONSHIP OF
POSITION/DISTANCE, SPEED
VELOCITY AND ACCELERATION OF A
PARTICLE IN A RECTILINEAR
MOTION

30. EXAMPLE
A particle moves along a line so that its position at any time 𝒕 > 𝟎 is
given by the function 𝒔 𝒕 = 𝒕𝟒 − 𝟑𝒕 + 𝟐 where s is measured in
meters and t is measured in seconds.
a.Find the average velocity during the first 5 seconds.
b.Find the distance traveled by the particle after 7 seconds.
c.Find the instantaneous velocity when 𝒕 = 𝟒.
d.Find the acceleration of the particle when 𝒕 = 𝟒.

31. RELATIONSHIP OF
POSITION/DISTANCE, VELOCITY AND
ACCELERATION OF A PARTICLE IN A
RECTILINEAR MOTION
Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
i. Average velocity ∆𝒗 =
∆𝒔 𝒕
∆𝒕
ii. Instantaneous velocity 𝒗 𝒕 = 𝒔′ 𝒕 =
𝒅𝒔
𝒅𝒕
.
iii. Speed 𝒗 𝒕 =
𝒅𝒔
𝒅𝒕
iv. Acceleration 𝒂 𝒕 = 𝒔′′ 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
.

32. SOLUTION
a.To find the average velocity of the particle during first 5 seconds,
we use the formula ∆𝒗 =
∆𝒔 𝒕
∆𝒕
=
𝑠 𝑡2 −𝑠 𝑡1
𝒕𝟏−𝒕𝟐
at 𝑡1 = 0 and 𝑡2 = 5.
To find the value of 𝑠 𝑡1 and 𝑠 𝑡2 , substitute 𝑡1 = 0 and 𝑡2 = 5 to the
given function
𝒔 𝒕 = 𝒕𝟒
− 𝟑𝒕 + 𝟐 when 𝒕𝟏 = 𝟎
𝒔 𝟎 = 𝟎 𝟒
− 𝟑 𝟎 + 𝟐
𝒔(𝟎) = 𝟎 − 𝟎 + 𝟐
𝒔(𝟎) = 𝟐
𝒔 𝒕 = 𝒕𝟐
− 𝟑𝒕 + 𝟐 when 𝒕𝟐 = 𝟓
𝒔 𝟓 = 𝟓 𝟒
− 𝟑 𝟓 + 𝟐
𝒔 (𝟓) = 𝟔𝟐𝟓 − 𝟏𝟓 + 𝟐
𝒔(𝟓) = 𝟔𝟏𝟐

33. SOLUTION
To find the average velocity, substitute 𝒔(𝟎) = 𝟐, 𝒔(𝟓) = 𝟏𝟐, 𝑡1 =
0 and 𝑡2 = 5 to the formula:
∆𝒗 =
∆𝒔 𝒕
∆𝒕
=
𝑠 𝑡2 − 𝑠 𝑡1
𝒕𝟏 − 𝒕𝟐
∆𝒗 =
612 − 2
𝟓 − 𝟎
∆𝒗 =
610
𝟓
∆𝒗 = 𝟏𝟐𝟐 meters per second

34. RELATIONSHIP OF
POSITION/DISTANCE, VELOCITY AND
ACCELERATION OF A PARTICLE IN A
RECTILINEAR MOTION
Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
i. Average velocity ∆𝒗 =
∆𝒔 𝒕
∆𝒕
ii. Instantaneous velocity 𝒗 𝒕 = 𝒔′ 𝒕 =
𝒅𝒔
𝒅𝒕
.
iii. Speed 𝒗 𝒕 =
𝒅𝒔
𝒅𝒕
iv. Acceleration 𝒂 𝒕 = 𝒔′′ 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
.

35. SOLUTION
b. To find the distance traveled by the particle after 7
seconds, substitute 𝒕 = 𝟕 to the position function.
𝒔 𝒕 = 𝒕𝟒
− 𝟑𝒕 + 𝟐 when 𝒕 = 𝟕
𝒔 𝟕 = 𝟕 𝟒
− 𝟑 𝟕 + 𝟐
𝒔(𝟓) = 𝟐𝟒𝟎𝟏 − 𝟐𝟏 + 𝟐
𝒔(𝟓) = 𝟐𝟑𝟖𝟐 𝒌𝒎

36. SOLUTION
c. To find the instantaneous velocity of the particle
when 𝒕 = 𝟒, we differentiate 𝒔 𝒕 = 𝒕𝟐
− 𝟑𝒕 + 𝟐 with
respect to time (𝒕).
𝒗 𝒕 = 𝒔′(𝒕) =
𝒅𝒔
𝒅𝒕
=
𝒅
𝒅𝒕
𝒕𝟒
− 𝟑𝒕 + 𝟐
𝒗 𝒕 = 𝒔′(𝒕) =
𝒅𝒔
𝒅𝒕
=
𝒅
𝒅𝒕
𝒕𝟒
−
𝒅
𝒅𝒕
𝟑𝒕 +
𝒅
𝒅𝒕
𝟐
𝒗 𝒕 = 𝟒𝒕𝟑
− 𝟑

37. SOLUTION
Substitute 𝒕 = 𝟒 to the velocity function,
𝒗 𝟒 = 𝟒𝒕𝟑
− 𝟑
𝒗 𝒕 = 𝟒 𝟒 𝟑
− 𝟑
𝒗 𝒕 = 𝟐𝟓𝟔 − 𝟑
𝒗 𝒕 = 𝟐𝟓𝟑 meters per second

38. RELATIONSHIP OF
POSITION/DISTANCE, VELOCITY AND
ACCELERATION OF A PARTICLE IN A
RECTILINEAR MOTION
Let the position of a particle (P) (or position) on a straight
line at time 𝒕 be 𝒔 𝒕 . Note that 𝒔 𝒕 is the position function.
i. Average velocity ∆𝒗 =
∆𝒔 𝒕
∆𝒕
ii. Instantaneous velocity 𝒗 𝒕 = 𝒔′ 𝒕 =
𝒅𝒔
𝒅𝒕
.
iii. Speed 𝒗 𝒕 =
𝒅𝒔
𝒅𝒕
iv. Acceleration 𝒂 𝒕 = 𝒔′′ 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
.

39. SOLUTION
d. To find the acceleration of the particle when 𝒕 = 𝟒, compute the second
derivative of the position function 𝒔 𝒕 = 𝒕𝟒 − 𝟑𝒕 + 𝟐 or compute the first
derivative of the velocity function 𝒗 𝒕 = 𝟐𝒕 − 𝟑.
𝒂 𝒕 = 𝒔′′ 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
𝒗 𝒕 = 𝒔′(𝒕) =
𝒅𝒔
𝒅𝒕
=
𝒅
𝒅𝒕
𝒕𝟒
− 𝟑𝒕 + 𝟐
𝒗 𝒕 = 𝒔′(𝒕) =
𝒅𝒔
𝒅𝒕
=
𝒅
𝒅𝒕
𝒕𝟒
−
𝒅
𝒅𝒕
𝟑𝒕 +
𝒅
𝒅𝒕
𝟐
𝒗 𝒕 = 𝟒𝒕𝟑 − 𝟑

40. SOLUTION
𝒂 𝒕 = 𝒗′ 𝒕 =
𝒅
𝒅𝒕
𝟒𝒕𝟑 − 𝟑
𝒂 𝒕 = 𝒗′ 𝒕 =
𝒅𝒗
𝒅𝒕
=
𝒅
𝒅𝒕
𝟒𝒕𝟑 −
𝒅
𝒅𝒕
𝟑
𝒂 𝒕 = 𝒗′ 𝒕 = 𝟏𝟐𝒕𝟐
Substitute 𝒕 = 𝟒 to the acceleration function,
𝒂 𝒕 = 𝒗′ 𝒕 = 𝟏𝟐 𝟒 𝟐
𝒂 𝒕 = 𝒗′ 𝒕 = 𝟏𝟗𝟐 meters per second squared 𝒎/𝒔𝟐

41. GROUP ACTIVITY

42. INSTRUCTION
1.The class will be divided into four groups. Each group will be
given a problem to solve.
2.The group will write their answers in a Manila Paper.
3.After ten minutes, each group will choose a representative to
present their outputs. Additionally, a rubric will be provided
for each group.

43. GROUP 1 & 3
The height s at time t of a one-peso coin dropped from the top of a
building is given by 𝑠 = −4.9𝑡2
+ 45 where s is measured in meters
and t is measured in seconds.
a.Find the average velocity on the interval 1, 2 .
b.Find the instantaneous velocity when t=1 and t=2.
c.How long will it take for one-peso coin hit the ground?
d.Find the velocity of the one-peso coin when it hits the ground.

44. GROUP 2 & 4
A projectile is fired straight upward. Its distance above the ground
after t seconds is given by 𝑠 𝑡 = −4.9𝑡2 + 300𝑡.
a.Find the time at which the projectile hits the ground.
b.Find the velocity at which the projectile hits the ground.
c.What is the maximum altitude achieved by the projectile?
d.What is the acceleration at any time t?

45. ASSIGNMENT NO. 2
Solve the problem. Then, show the solution.
A projectile is fired directly upward. Its height s(t) in meters
above the ground after t seconds is given by 𝒔(𝒕) = 𝟐𝒕𝟐 − 𝟒. 𝟗𝒕.
• What are the velocity and acceleration after t
seconds? After 10 seconds?
• What is the maximum height?
• When does it strike the ground?

46. PERFORMANCE TASK:
VIDEO TUTORIAL
PROJECT
Instruction:
1. This project is a group project. Each group will have a maximum of four
members.
2. The group will collaboratively formulate an original problem involving
rectilinear motion or other application of differentiation (such as rate of change,
related rates, etc.). Also, the group will solve their formulated problem.
3. To present the problem and its solution, the group will create a video tutorial.
The video must be less than seven minutes.
4. The group must submit a mp4 format of their output thru e-mail.
e-mail: john.arlando@deped.gov.ph