Derivatives 2.2

1. Derivatives 2.2 St. Pius X High School Ms. Hernandez AP Calculus I F06 Q1 Derivatives Unit

2. Some Differentiation Rules! • Yeah, we have some rules that make finding the derivative so much EASIER! • Constant • Power • Constant Multiple • Sum and Difference • Sine and Cosine

3. Constant k is a constant

4. Power Special case n=1  f(x)=x  f’(x)=1

5. Constant Multiple

6. Sum and Difference

7. Sine and Cosine

8. TS 2 Rates of  • PVA = Position, Velocity, & Acceleration • RATE OF CHANGE • Rate = distance/time • The function s gives the position of an object as a function of time • Average velocity = change in distance change in time Average velocity = s / t s = s(t +t) – s(t)

9. Find average velocity of a falling object • If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function s = -16t2 + 100 s(t) is the position function of the billiard ball measure in feet t = time measured in seconds 100 = “ORIGIN”AL HEIGHT aka Initial Height

10. Find the average velocity s(t) = -16t2 + 100 find average velocity over the time period [1,2] s(1) = 84 feet and s(2) = 36 feet So average velocity is –48 ft/s Why is it 36 – 84 ? Why is the velocity negative?

11. Velocity function LOOKS LIKE THE DERIVATIVE!!!!!  So the velocity function is the Derivative of the position function !!!!!  YEAH!!!

12. Average velocity vs instant velocity • Average velocity between t1 and t2 is the slope of the secant line • Instantaneous velocity at t1 is the slope of your tangent line

13. Position function of a FREE falling object Neglecting air resistance…. s0 = initial height of the object v0 = initial velocity of the object g~ -32 ft/s2 or –9.8 m/s2 (acceleration due to gravity on earth)

14. Example • At time t=0, a diver jumps from a platform diving board that is 32 feet above water. The position of the diver is given by the following position function: • Where s is measured in feet and t is measured in seconds. • When does the diver hit the water? • What is the diver’s velocity at impact?

15. Example cont’d • 32 is the initial height (height of board above water) • From the middle term, 16t, 16 is the initial velocity of the diver To find the time t when the diver hits the water, let s = 0 and solve for t. If s = 0 then the position is 0, right b/c the diver HITS the water…..

16. Example cont’d • So we let s = 0 and solve for t to find the time it takes for the diver to hit the water t can not be negative… no negative time… this is not back to the future, ok?  So t = 2 is the only logical answer At t = 2 seconds, the diver hits the water –that’s fast!

17. Example cont’d • Next, lets solve for the diver’s velocity at impact. • We use t=2, b/c we just found out that’s the time it takes for the diver to hit the water and we want velocity at impact (like you know when the diver hits the water, duh) • Remember, velocity is the derivate of position