1. Chapter 4 Additional Derivative Topics Section 6 Related Rates

2. 2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objective for Section 4.6 Related Rates ■ The student will be able to solve related rate problems and applications.

3. 3 Real-life Scenarios  Workers are concerned that the rate at which their wages are increasing isn’t keeping up with the rate of increase in the company’s profits.  An automobile dealer wants to predict how badly an increase in interest rates will decrease his rate of sales.  An investor is studying the relationship between the rate of increase in the Dow Jones average and the rate of increase in the gross domestic product over the past 50 years. Barnett/Ziegler/Byleen Business Calculus 12e

4. 4 Related Rates  In each of the scenarios, there are two quantities that are changing with respect to time. Wages and profits Interest rates and sales Dow Jones average and gross domestic product  These are examples of related rates. Barnett/Ziegler/Byleen Business Calculus 12e

5. 5 Related Rates - The Process  Remember the mnemonic DREDS: D = Diagram R = Rates E = Equation D = Derivative S = Substitute  Each step will be described through examples. Barnett/Ziegler/Byleen Business Calculus 12e

6. 6 Notation Barnett/Ziegler/Byleen Business Calculus 12e

7. 7 Related Rates Example 1 Barnett/Ziegler/Byleen Business Calculus 12e

8. 8 Related Rates Example 2 Barnett/Ziegler/Byleen Business Calculus 12e

9. 9 Example 2 Continued Barnett/Ziegler/Byleen Business Calculus 12e

10. 10 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 A weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high?

11. 11 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 A weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high? D: Diagram 300 What is changing with respect to time? x = distance from observer to balloon y = height of balloon x y

12. 12 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 (continued) y 300 x A weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high?

13. 13 Barnett/Ziegler/Byleen Business Calculus 12e E: Equation - We need an equation showing the relationship between x and y: Pythagorean Theorem 300 2 + y 2 = x 2 Example 1 (continued) y 300 x D: Derivative – Find the implicit derivative

14. 14 Barnett/Ziegler/Byleen Business Calculus 12e S: Substitute Example 1 (continued) y 300 x A weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high? When the balloon is 400 m high, the rate at which the distance between the observer and balloon is changing is 4 m/sec.

15. 15 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet?

16. 16 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (continued) A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet? R D: Diagram What is changing with respect to time? R = radius of the ripple A = area of the ripple

17. 17 Barnett/Ziegler/Byleen Business Calculus 12e A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet? R Example 2 (continued)

18. 18 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet? R E: Equation - We need an equation showing the relationship between R and A: A =  R 2 D: Derivative – Find the implicit derivative

19. 19 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet? R S: Substitute When the radius is 10 feet, the rate at which the Area is changing is 40  ft/sec.

20. 20 Video Resources  Video Tutorials PatrickJMT.com oSelect “Calculus” oSelect topics with “related rates” Mathtv.com oSelect “Videos by Topic” oSelect “Calculus” oSelect “Applications of derivatives” https://www.khanacademy.org/ oSearch “related rates” Barnett/Ziegler/Byleen Business Calculus 12e

21. 21 Homework Barnett/Ziegler/Byleen Business Calculus 12e