Study 3.4, p motion, population; # , 165 a, b,c. 3.4 Derivative as a Rate of Change. Derivative as a Rate of Change.
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1 GOALS: 1. Recognize the derivative as a rate of change of one variable with respect to another. change of position with respect to time: v(t) = s '(t) 3. Recognize acceleration as the instantaneous rate of change of velocity with respect to time: a(t) = v '(t) = s ''(t) 4. Consider rates of change of populations. Study 3.4, p motion, population; # , 165 a, b,c 3.2 Derivative of a Function (x+h, f(x+h)) f(x+h) f(x) (x, f(x)) h Home Page Class Notes G. Battaly, WCC, Class Notes G. Battaly
2 Average Rate of Change on Interval Slope of Secant Line eg: total distance total time average population over last 10 years vs Instantaneous Rate of Change Slope of Tangent Line eg: velocity at specific time 1. Recognize the derivative as a rate of change of one variable with respect to another. rate of population growth at a particular time: These are examples. There are many more. Part 1 change of position with respect to time: v(t) = s '(t) 3. Recognize acceleration as the instantaneous rate of change of velocity with respect to time: a(t) = v '(t) = s ''(t) G. Battaly
3 change of position with respect to time: v(t) = s '(t) change of position with respect to time: v(t) = s '(t) b) a(t)=v '(t)=s ''(t)= 18t a(1)= 18 m/s/s s(t)= 3t 3 7t a) v(t)=s '(t)= 9t 2 7 v(1)= 9 7 = 2 m/s c) v(t)=s '(t)= 9t 2 7 = 0 9t 2 = 7, t 2 = 7/9 t = ± 7/9 = + 7/3 sec a( 7/3) = 18 7/3 = 6 7 m/s/s G. Battaly
4 Motion: Speeding Up (SU) and Slowing Down () I am taking a trip to Montauk. What happens to my motion as I back our of my driveway to begin the trip? Car parked Start backing up Beginning to stop on road Stop on road Drive to Highway v(t) a(t) motion Getting ready to turn Motion: Speeding Up (SU) and Slowing Down () I am taking a trip to Montauk. What happens to my motion as I back our of my driveway to begin the trip? v(t) a(t) motion Car parked Start backing up Beginning to stop on road Stop on road Drive to Highway Getting ready to turn none SU none SU G. Battaly
5 Motion: Speeding Up (SU) and Slowing Down () I am taking a trip to Montauk. What happens to my motion as I back our of my driveway to begin the trip? Car parked Start backing up Beginning to stop on road Stop on road Drive to Highway Getting ready to turn v(t) a(t) motion none SU none SU Speeding Up (SU): v(t) and a(t) same sign Slowing Down (): v(t) and a(t) opposite sign change of position with respect to time: v(t) = s '(t) G. Battaly
6 b) slowing down when v and a have opposite signs speeding up wheen v and a have the same signs need to know the intervals where each is + and where each is b) slowing down when v and a have opposite signs speeding up wheen v and a have the same signs need to know the intervals where each is + and where each is G. Battaly
7 b) slowing down when v and a have opposite signs speeding up wheen v and a have the same signs need to know the intervals where each is + and where each is G. Battaly
8 c) a(t)=v '(t)=s ''(t)= 2t a(1)= 2(1) = 2 a(2)= 2(2) = 4 a(3)= 2(3) = 6 a(4)= 2(4) = 8 a) b) s(t)= 1t t v(t)=s '(t)= t v(1)= = 63 v(2)= = 60 v(3)= = 55 v(4)= = 48 Population is declining Population is declining at the rate of 2000/yr G. Battaly
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