1. Calculus warm-up Find

2. xf(x)g(x)f’(x)g’(x) 318-3-5 63-245 834 12-650 For each expression below, use the table above to find the value of the derivative at x = 3 Calculus Warm-up

3. Olympic National Park, Washington Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 2.6 Related Rates 2014

4. Olympic National Park, Washington Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 2.6 Related Rates

5. Olympic National Park, Washington Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 2.6 Related Rates

6. We have seen how the Chain Rule can be used to find dy/dx implicitly. Another important use of the Chain Rule is to find the rates of change of two or more related variables that are changing with respect to time. Examples: The rate of change in the radius of a balloon being inflated. The rate of movement of a rotating spotlight moving across a wall. The approach rate of two vehicles moving towards an intersection.

7. Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation that relates the variables. 5. Differentiate both sides with respect to time. 6. Solve for one of the rates.

8. Three Important Rules for Related Rates Problems: 1.Use Geometry to establish the relationship between the variables. 3. Wait until after you have differentiated to substitute any values for the variables. 2.Only changing quantities get variables.

9. A simple one: Find: dy/dt when x=1, given that dx/dt =2 when x=1.

10. Consider a sphere of radius 10cm. (Possibly a soap bubble or a balloon.) Suppose that the radius of the sphere is changing at an instantaneous rate of 0.1 cm/sec. At what rate is the sphere changing when the radius is 10 cm.? The sphere is growing at a rate of. when the radius is 10 cm. Sphere Problem:

11. A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of the outer ripple is Increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? The area is changing at a rate of when the radius is 4. Ripples in The Pond: Do work

12. Water is draining from a cylindrical tank at a constant rate of 3000 cubic centimeters/second. How fast is the surface dropping? Find ( r is a constant.) (We need a formula to relate V and h. ) Cylindrical Tank Problem:

13. Hot Air Balloon Problem: Given: How fast is the balloon rising at the instant when ? Find

14. Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

15. The formula for the volume of a cone is Find the rate of change of the volume if is 2 inches per minute and h = 3r when r = 6 inches.

16. Cube Problem: All edges of a cube are expanding at a rate of 3 cm. per second. How fast is the volume changing when each edge is a) 1 cm. and b) 10cm.?

17. BC Homework 2.6 pg.154 13-23 odd,31,33,35,43

18. AB Homework 2.6 pg.154 13-23 odd

19. AB Homework 2.6 Day 2 pg.154 27,31,33,35,43,45

20. RELATED RATES – DAY 2

21. B A Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later?

22. B A Truck Problem: How fast is the distance between the trucks changing 6 minutes later? Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. 

23. Lamppost Problem: A man 6 feet tall walks at a rate of 5 feet per second towards a lamppost that is 20 feet above the ground. When he is 10 feet from the base of the light, a)At what rate is the tip of his shadow moving? b) At what rate is the length of his shadow changing?

24. 2.6 Related Rates

26. 2.6 Related Rates (#27) A 25 ft. ladder is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 ft. per second. a)How fast is the top of the ladder moving down the wall when its base is 7 ft.? a)Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall b)Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

27. Homework MMM pgs. 68-73