Dr Z selfmedicates himself with caffeine every morning While

Dr. Z self-medicates himself with caffeine every morning. While the coffee kicks in, he decides to make an equation describing his coffee. The equation he decides upon is C(t) = -t^2+100t where t is measured in minutes and caffeine is measured in mg.

a) Find the average rate of change of caffeine in Dr. Z\'s body between one minute after he started drinking his coffee and 30 minutes after he started drinking his coffee. What does the average rate of change tell you?

b) Find the instantaneous rate of change of coffee in Dr. Z\'s body at t=1 and t=30. What does the instantaneous rate of change tel us?

c) when does Dr. Z have the maximum amount of caffeine in his body?

d) How much caffeine is in Dr. Z when he hits his maximum amount?

Solution

a)

Average rate of change 1 minute after he starts drinking coffee is

(C(1)-C(0))/1=C(1)=-1+100=99 mg/minute

Average rate of change 30 minutes after he starts drinking coffee is

(C30)-C(0))/30=C(30)/30=(-30^2+100*30)/30=-30+100=70

Average rate of change gives us an idea of overall rate of change and is not a local but global concept

b)

Instantaneous rate of change is given by value of derivative

C\'(t)=-2t+100

C\'(1)=-2+100=99

C\'(30)=-2*30+100=40

Instantaneous rate of change gives us rate of change at any time and is a local concept

c)

C(t)=-t^2+100t=-(t^2-100t)=-(t^2-2*50t+50^2-50^2)=-(t-50)^2+50^2

-(t-50)^2 takes maximum value =0 at t=50

HEnce maximum amount of caffiene is: 50^2=2500 mg at t=50 minutes

d)

2500 mg