1. Section 3.9 Part II Related Rates II
MAT 1234 Calculus I
Section 3.9 Part II
Related Rates II

2. Next.. WebAssign 3.9 II. Difficulty level *****
Please do not wait until Monday afternoon.
Be sure to do it ASAP. Tutors are available today after class.
Write down your solutions carefully!!!
Quiz 6: section 3.9 (30 min.)
One of these type of questions will be on the second exam.

3. Undecided …..
If you have not decided on your major and are doing well in this class, come talk to me. You may be a good candidate for math or applied math majors.
If you are doing well in this class and interested to know about math minor and/or applied math majors, talk to me.

4. Classwork Let you practice what you have just learned.
Most of you has been doing well.
Classwork should be completed within the class time.
There has been a grace period of an hour after class for the first 5 weeks.

5. Classwork - Starting Next Week …
Need to be done by 2:50-ish to be counted.
After 3:00, Elise can check it for you in the tutoring room but it will not be counted toward class participation.
We want to encourage you to take the classwork seriously and do it properly, and get it correct.

6. Recall: Related Rates
If 𝑥(𝑡) and 𝑦(𝑡) are related by an equation, their derivatives (rate of changes)
𝑥’(𝑡) and 𝑦’(𝑡)
are also related.
Note that the functions are time dependent
Extended Power Rule will be used frequently, e.g.

7. Example 2
A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

8. Example 2 Everyone, try step 1 and 2!
A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?
Everyone, try step 1 and 2!

9. Step 1 Draw a diagram
A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

10. Step 2: Define the variables
A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?
Remark: Do not define more variables than necessary.

11. Step 3: Write down all the information in terms of the variables defined
A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

12. Step 4: Set up a relation between the variables

13. Step 5: Use differentiation to find the related rate

14. Review: Similar Triangles
Two triangles are similar if and only if one of the following 2 conditions are satisfied 1. Their corresponding angles are the same. 2. The ratio of their corresponding sides are the same.

15. Review: Similar Triangles
In particular: If the corresponding angles are the same, then the ratio of their corresponding sides are the same.

16. Please wait…
We are going to walk through some of the main key points in your classwork.
Please do not start your classwork now, not even drawing the diagrams.

17. Example 3 (Classwork)
A street light is mounted at the top of a 12-ft-tall pole. A 6-ft-tall man walks away from the pole with a speed of 4ft/s along a straight path. How fast is the tip of his shadow moving when he is 35 ft from the pole?

18. Example 3
4 ft/s
Wall
12 ft
Man
6 ft
???? ft/s
@35 feet

19. Example 3 Remark: Do not define more variables than necessary.12 𝑥 6 𝑧
Remark: Do not define more variables than necessary.
For example, it is not necessary to define a variable for the length of the shadow.

20. Example 3 Remark: Do not define more variables than necessary.12 𝑥 6 𝑧
Remark: Do not define more variables than necessary.
For example, it is not necessary to define a variable for the length of the shadow.
𝑧−𝑥

21. Example 3 12 𝑥 6 𝑧
𝑧−𝑥

22. Hints Use similar triangles to find a relation between x and z.12 𝑥 6 𝑧
𝑧−𝑥
Use similar triangles to find a relation between x and z.
Solve z in terms of x.
Be sure to simplify before taking derivatives

23. The Answer
It turns out that in this problem, the answer is independent of the fact that x=35. This means that the tip of the shadow is moving at a constant rate.