1. Table of Contents 18. Section 3.4 Rates of Change

2. Rates of Change Essential Question – What are some applications of rate of change?

3. When is finding tangent lines used? In mechanics and physics, the tangent line to a curve determines the direction of a body’s motion. In optics, the tangent determines the angle at which light enters a curved lens – this revolutionized glasses

4. Average rate of change

5. Instantaneous rate of change The slope of the curve y=f(x)

6. Example Let, the area of a circle. Find the average rate of change between r=2 and r=5. Then find instantaneous rate of change at r = 2 and r = 5. Notice this equals circumference!

7. Example – stopping distance The stopping distance of an car is given by. Calculate the stopping distance and the ROC of stopping distance when s = 30 mph and when s = 60 mph.

8. Example – Marginal Cost Marginal cost at production level x o is the cost of producing one additional unit. Airline data suggest that the total cost of a 1200 mile flight is where x is number of passengers. Estimate marginal cost of an additional passenger if flight already has 150 passengers. Is it more expensive to add a passenger when x = 150 or x = 200?

9. Example – Linear Motion Velocity is derivative of position. Velocity tells you speed and direction. A large truck enters the off-ramp of a freeway at t = 0. It position is How fast is the truck going at the moment it enters the off-ramp? Is the truck speeding up or slowing down?

10. Motion under influence of gravity Galileo discovered height s(t) and velocity v(t) of an object tossed vertically in air are given by where g is acceleration due to gravity 32 ft/s 2 or 9.8 m/s 2

11. Example – maximum height A slingshot launches a stone vertically with an initial velocity of 300 ft/s from an initial height of 6 ft. Find stone’s velocity at t = 2 and t =12. Explain change in sign. What is stone’s maximum height and when does it reach that height?

12. Example – Initial Conditions A bullet is fired vertically with an initial height of 0. What initial velocity is required for the bullet to reach a maximum height of 2 km?

13. Assignment Pg 92: #1-23 odd, 29-33 odd, 45