Application of Calculus

1. REVIEW OF THE
PREVIOUS
LESSON

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

5. AVERAGE RATE OF CHANGE OF A
FUNCTION

6. INSTANTANEOUS RATE OF
CHANGE

7. RELATED RATES

8. OBJECTIVES
Define related rates
Solve situational problems involving related rates

9. RELATED RATES
Definition

10. POLYA’S FOUR STEP RULE
IN PROBLEM SOLVING

11. Example 1: If 𝑥2
+ 𝑦2
= 25 and
𝒅𝒙
𝒅𝒕
= 𝟕, find
𝒅𝒚
𝒅𝒕
when 𝒙 = 𝟑.

12. Example 2: Assume that a point is moving along the graph
𝒙𝟐
+ 𝟐𝒚𝟐
= 𝟏𝟐. When the point is at (−𝟐, 𝟐), its 𝑥-coordinate
is increasing at the rate of 0.4 unit per second. How fast is the
y-coordinate changing at that moment? (Note: See Figure 1)

13. Example 3: Given two variables 𝒂 and 𝒃 which
are both differentiable functions of 𝒕. They are
related by the equation 𝒃 = 𝒂𝟐 − 𝟕. Given that
𝒅𝒂
𝒅𝒕
= 𝟐, find
𝒅𝒃
𝒅𝒕
when 𝒂 = 𝟒.

14. Example 4: Air is being pumped into a spherical balloon
at a rate of 𝟓 𝒄𝒎𝟑
/𝒎𝒊𝒏. Determine the rate at which the
radius of the balloon is increasing when the diameter of the
balloon is 𝟐𝟎 𝒄𝒎.

15. Example 5. A 17 ft-ladder is leaning against the building. The
foot of the ladder is 8 ft from the base of the building and it’s
sliding away from the building at 3 ft/s.
a. How fast is the top of the ladder sliding down the wall of
the building?
b. How fast is the area formed by the ladder changing at this
instant?
c. Find the rate at which the angle between the ladder and the
ground is changing at this instant.

16. Example 5. A 17 ft-ladder is leaning against the building. The
foot of the ladder is 8 ft from the base of the building and it’s
sliding away from the building at 3 ft/s.
a.
𝑑𝑦
𝑑𝑡
= −
8
5
𝑓𝑡/𝑠𝑒𝑐
b.
𝑑𝐴
𝑑𝑡
=
161
10
𝑓𝑡2
/𝑠𝑒𝑐
c.
𝑑𝜃
𝑑𝑡
= −
1
5
𝑟𝑎𝑑/𝑠𝑒𝑐

17. GROUP ACTIVITY
JIGSAW PUZZLE PRESENTATION

18. Instruction:
The class will be divided into 6 groups. Then, each group
will be assigned to a problem. The group will solve that
problem and write the solution in a Manila paper or
cartolina. The presentation will follow a Jigsaw Puzzle
approach format.