1. 3.11 Related Rates Tues Nov 10 Do Now Differentiate implicitly in terms of t 1) 2)

2. Related Rates When we use implicit differentiation, we obtain dy/dx, or the change of y in terms of x. In many real life situations, each quantity in an equation changes with time (or another variable) In this case, any derivative we find is called a related rate, since each rate in the derivative is related to each other

3. Related Rates Steps 1) Make a simple sketch, if possible 2) Identify what rate you are looking for 3) Set up an equation relating ALL of the relevant quantities 4) Differentiate both sides of the equation in terms of the variable you want –if you want dv/dt, you differentiate in terms of t 5) Substitute in values we know 6) Solve for the remaining rate

4. Ex 1 A 5-meter ladder leans against a wall. The bottom of the ladder is 1.5 m from the wall at time t=0 and slides away from the wall at a rate of 0.8m/s. Find the velocity of the top of the ladder at time t=1

5. Ex 2 Water pours into a fish tank at a rate of 0.3 m^3 / min. How fast is the water level rising if the base of the tank is a rectangle of dimensions 2 x 3 meters?

6. Ex 3 Water pours into a conical tank of height 10 m and radius 4 m at a rate of 6 m^3/min A) At what rate is the water level rising when the level is 5 m high? B) As time passes what happens to the rate at which the water level rises?

7. Ex 4 A spy uses a telescope to track a rocket launched vertically from a launching pad 6km away. At a certain moment, the angle between the telescope and ground is equal to pi/3 and is changing at a rate of 0.9 radians/min. What is the rocket’s velocity at that moment?

8. Ex 5 See book

9. Closure At what rate is the diagonal of a square increasing if its sides are increasing at a rate of 2 cm/s? HW: p.199 #4 9 13 17 22 25 35 39 41 Ch 3 Test Fri

10. 3.11 Related Rates Cont’d Wed Nov 11 Do Now Air is being pumped into a spherical balloon at a rate of 5 cm 3 /min. Determine the rate at which the radius of the balloon is increasing when the radius of the balloon is 10 cm. (hint: Volume = 4/3 pi x r^3)

11. HW Review p.199 Probably all of them

12. More practice worksheet

13. Closure Hand in: A 15 foot ladder is resting against the wall. The bottom is initially x feet away from the wall and is being pushed towards the wall at a rate of 0.5 ft/sec. How fast is the top of the ladder moving up the wall when the bottom of the ladder is 4 feet from the wall?? (Hint: Use Pythagorean Theorem) HW: finish packet Test Friday

14. 4.7 4.8 3.11 Review Thurs Nov 12 Do Now Farmer Al needs to fence in 800 square yards, with one wall being made of stone which costs $24 per yard, and the other three sides being wire mesh which costs $8 per yard. What dimensions will minimize the cost?

15. HW Review

16. Test Review 4.7 Optimization 3.11 Related Rates 4.8 Newton’s Method