Really Need Help I seriously do not get this

Question

When a new pain killing drug is administered, the effect (in efficacy units) is modelled by, 

𝐸(𝑥) = 640𝑥𝑒−2.8𝑥, where 𝑥 ≥ 0 is the time in hours after the drugs are given.

1. How effective is the drug after 30 minutes?
2. Find the rate at which the drug efficacy is changing after 30 minutes?
3. At what time (to the nearest minute) is the drug most effective?
4. Suppose we want to compare this drug with other available painkillers. The effect of three different drugs (where 𝐸 is measured in efficacy units 𝑥 hours after being administered) is given below. Find the rate of change of drug efficacy for each of the following drugs, one hour after being administered:

i. Aspirin: 𝐸(𝑥) = 20 ln (1 − 𝑥/6) + 40

ii. Codeine: 𝐸(𝑥) = √(0.5𝑥² + 0.3𝑥)

iii. Morphine: 𝐸(𝑥) = 4𝑥²−8𝑥

1−5𝑥

Answer 1

Hello!

For part 1, we would plug in x=0.5 to represent 30 minutes (0.5 hours) to the original equation, E(x)= 640𝑥𝑒−2.8𝑥

For part 2, you would find the derivative of the original equation, E(x)= 640𝑥𝑒−2.8𝑥, and then plug in x=0.5 to represent the rate at which it changes.

For part 3, you would use your derivative to find the max value for the original equation, which would be a point where x=0 (a zero, where the graph of the derivative changes from negative to positive). If you could use a graphing calculator, you would be able to plot the graph and find the "maximum" for efficiency.

For part 4, you would need to find the derivatives of all of those functions, and then plug in x=1 to find the rate of change of the efficiency.

A good rule of thumb with calculus is to always remember that "rate of change" means derivative!