Wize High School Grade 12 Calculus Textbook > Derivatives of Exponential (& Logarithmic) Functions

Optimization Problems
We want to find the maximum and minimum values in a real-world application problem.

Strategy
1. Determine what you are minimizing/maximizing and draw a picture if possible
2. Write an equation for the quantity you are trying to maximize or minimize 
     *If there is more than 1 independent variable, create another equation relating the variables
3. If there is more than 1 independent variable, create another equation relating the variables → combine into one function with 1 independent variable
4. Follow the 3 steps to find the maximum or minimum value required
Step 1: find the derivative of the function and set to 0
Step 2: evaluate the function values at the point(s) found in step 1 and the endpoints of the interval (sometimes the endpoints don't exist)
Step 3: compare the values you found in step 2 and make a conclusion

Practice: Optimization w/ Population
The population PPP of a certain species of fish is modelled by P=1000+5000200+300e−t100\displaystyle P=1000+\frac{5000}{200+300e^{-\frac{t}{100}}}P=1000+200+300e−100t​5000​, where ttt is measured in days.

a) Determine the initial population at time t=0t=0t=0

Enter the initial population without units

I don't know

The monthly profit, in hundreds of dollars, of a company is given by P(x)=100x2e−x−300e−xP(x)=100x^2e^{-x}-300e^{-x}P(x)=100x2e−x−300e−x, where xxx is the units of goods sold, measured in hundreds.

If the maximum production capacity of this company is 500 units per month, determine the number of units to produce in order to maximize profit

What is the maximum monthly profit, rounded to the nearest dollar?

How many units of goods should the company product to maximize proft?

I don't know

Practice: Optimization w/ Diseases
A certain disease is being spread across a population. The percent of people who are affected by the disease is given by P(t)=100(e−2t100−e−3t100)P\left(t\right)=100\left(e^{-\frac{2t}{100}}-e^{-\frac{3t}{100}}\right)P(t)=100(e−1002t​−e−1003t​), where ttt is measured in days.

What is the highest percent of people affected by this disease within the first 100 days?

Enter the highest percent rounded to 1 decimal place

I don't know

Practice: Optimization w/ Position and Velocity
A bird's position sss measured in metres from its nest is given by s(t)=5+10te2ts\left(t\right)=5+\frac{10t}{e^{2t}}s(t)=5+e2t10t​ for t≥0t\ge0t≥0, where ttt is measured in seconds

a) At what time is the bird furtherest away from its nest? (i.e. when is the position maximized?)

b) At what time is the bird's velocity at a minimum? (recall that velocity is the derivative of position -- v(t)=s′(t)v\left(t\right)=s'\left(t\right)v(t)=s′(t))

a) Enter the time in seconds, do not include units

b) Enter the time in seconds, do not include units

I don't know

Practice: Optimization w/ Probability Density
The radial probability density for the hydrogen atom can be written as P(r)=kr4e−raP\left(r\right)=kr^4e^{-\frac{r}{a}}P(r)=kr4e−ar​ where
P(r)P\left(r\right)P(r) is probability that an electron can be found at the ring with radius rrr,
rrr is the radius (r>0r>0r>0), 
kkk is a constant, and
a=0.529×10−10a=0.529\times10^{-10}a=0.529×10−10m
At what radius rrr is the probability P(r)P\left(r\right)P(r) maximized?

0.13225×10−100.13225\times10^{-10}0.13225×10−10m away from the center of the atom

0.2645×10−100.2645\times10^{-10}0.2645×10−10m away from the center of the atom

0.529×10−100.529\times10^{-10}0.529×10−10m away from the center of the atom

1.058×10−101.058\times10^{-10}1.058×10−10m away from the center of the atom

2.116×10−102.116\times10^{-10}2.116×10−10m away from the center of the atom

I don't know