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AP Bio Equations and Formulas Review Sheet, Cheat Sheet of Biology

AP biology formula sheet in rate, population, exponential and logistic growth.

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AP BIO EQUATIONS AND FORMULAS REVIEW SHEET #4 – Answer Key
Formulas:
Rate Population Growth Exponential Growth Logistic Growth
dY/dt dN/dt = B – D
Nr
dt
dN
max
=
=
K
NK
Nr
dt
dN
max
dY = amount of change B = birth rate D = death rate N = population size
K = carrying capacity r
max
= maximum per capita growth rate of population
Notes
dt
dN
=
t
N
=
timeinchange
sizepopulationinchange
= population growth rate
Example 1:
There are 300 falcons living in a certain forest at the beginning of 2013. Suppose that every year there are 50
falcons born and 30 falcons that die.
a. What is the population growth rate (include units)? Interpret the value.
dN/dt = B – D = 50 – 30 = 20 falcons/year. Each year the falcon population will increase by 20 falcons.
b. What is the per capita growth rate of the falcons over a year? Interpret the value.
Nr
dt
dN
max
=
so 20 = r
max
300 so r
max
= .067 The falcon population will increase by 6.7% each year.
c. Fill in the table and construct a graph.
Year Population
2013 300
2014 320
2015 341
2016 364
2017 388
2018 414
d. Find the average rate of change for the falcon population from 2013 to 2018 (include units). Interpret the
value.
We have two data points: (2013, 300) and (2018, 483). Average rate of change = slope
8.22
2013
2018
300414
=
=
=
x
y
slope
falcons/year.
Over the past five years, the falcon population has increased by 22.8 per year on average.
Example 2:
Kentwood, Michigan had a population of 49,000 in the year 2013. The infrastructure of the city allows for a
carrying capacity of 60,000 people. r
max
= .9 for Kentwood.
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AP BIO EQUATIONS AND FORMULAS REVIEW SHEET #4 – Answer Key

Formulas:

Rate Population Growth Exponential Growth Logistic Growth

dY/dt dN/dt = B – D r N dt

dN = (^) max  

K

K N

r N dt

dN max

dY = amount of change B = birth rate D = death rate N = population size

K = carrying capacity rmax = maximum per capita growth rate of population

Notes

dt

dN

t

N

changeintime

change in populationsize = population growth rate

Example 1:

There are 300 falcons living in a certain forest at the beginning of 2013. Suppose that every year there are 50

falcons born and 30 falcons that die.

a. What is the population growth rate (include units)? Interpret the value.

dN/dt = B – D = 50 – 30 = 20 falcons/year. Each year the falcon population will increase by 20 falcons.

b. What is the per capita growth rate of the falcons over a year? Interpret the value.

r N dt

dN = (^) max so 20 = rmax 300 so rmax = .067 The falcon population will increase by 6.7% each year.

c. Fill in the table and construct a graph.

Year Population

2013 300

2014 320

2015 341

2016 364

2017 388

2018 414

d. Find the average rate of change for the falcon population from 2013 to 2018 (include units). Interpret the

value.

We have two data points: (2013, 300) and (2018, 483). Average rate of change = slope

x

y slope falcons/year.

Over the past five years, the falcon population has increased by 22.8 per year on average.

Example 2:

Kentwood, Michigan had a population of 49,000 in the year 2013. The infrastructure of the city allows for a

carrying capacity of 60,000 people. rmax = .9 for Kentwood.

a. Is the current population above or below the carrying capacity? Will the population increase or decrease in

the next year?

Current population is below the carrying capacity so we would expect the population to increase.

b. What will be the population growth rate for 2013 (include units)?

K

K N

r N dt

dN max =^ = 

( 0. 9 )( 49 , 000 ) 8085 people/year

c. What will be the population size at the start of 2014.

49,000 + 8085 = 57,085 people

d. Fill in the following table:

Year Population size Population growth rate

2013 49,000 8085

2014 57,085 2496.

2015 59,581 374.

2016 59,955 40.

2017 59,996 4.

e. What happened to the population size over the years? What happened to the population growth rate over the

years?

Population size got closer and closer to the carrying capacity of 60,000.

Population growth rate got smaller and smaller.

f. Explain your answer from part (e) using what you know about carrying capacity.

As the population gets closer and closer to the carrying capacity, the resources of the environment start

to deplete and the growth rate slows. Limiting factors take over (especially density-dependent ones,

space, food, water, shelter, etc)

g. Explain your answer from part (e) using the formula: (^) 

K

K N

r N dt

dN max

Look at this part of the formula: (^) 

K

K N

As the size of the population (N) gets closer and closer to the

carrying capacity (K), then this part of the formula gets smaller and smaller, thus reducing population

growth rate.