Math 1025: Elementary Calculus I _Spring 2022
Week 1 Day 01 : Module 1.1
Mathematical Applications : Mathematics is applied in various fields and disciplines.
Mathematical concepts and procedures are used to solve problems for example in engineering
and physics. What are some other disciplines?
Application of Mathematics in Life Sciences
Calculus is an important tool to study change.
How do we measure the rate at which something is changing? What do we examine? Can we
predict how fast an organism will grow months from now simply by examining its current rate
of growth?
Lesson 1.1.1 : Rate of Change of Volume Compared to Surface Area
In Biology, diffusion is a process in which cells obtain nutrients. Molecules move through a
cell membrane through the process of diffusion. This process depends on
1. The size of the cell – the volume the cell takes up
2. The size of the membrane – the surface area of the cell
3. The relationship between the surface area and volume of a cell.
To get an idea of growth rate of a cell, we can look at the change in the volume, the changes
in the surface area, and the ratio between the surface area of the cell and the volume of the
cell over time.
The coccus bacterial cell is nearly spherical in shape. The following table shows the volume
and surface area of a growing coccus cell over a period of 10 minutes.
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time (min)
0
1
2
3
4
5
6
7
8
9
10
volume (𝜇m3)
0.521
0.59
0.661
0.737
0.817
0.908
1.003
1.099
1.207
1.322
1.449
area (𝜇m2)
3.132 3.401
3.67
3.947
4.225
4.535
4.846
5.151
5.482
5.826
6.193
ratio (area/volume)
6.012 5.764
5.552
5.355
5.171
4.994
4.832
4.687
4.542
4.407
4.274
What are some of the things we can tell about the cell’s changes from this chart?
If we want a rate of change for volume, we are looking to find:
How did the volume change in the first minute?
How did the volume change over the third interval of time?
If we want a rate of change for surface area, we are looking to find:
What is the change in SA between 5 and 6 seconds?
If we want to find out what is happening to their ratio’s rate of change:
The ratio _____________
over the first interval of time.
The ratio _____________________________________ over the 4th interval of time.
Explanation: When the cell increases in size, the volume increases faster than the
surface area. When there is more volume and less surface area, the ratio of SA/V is
decreasing and diffusion takes longer and is less effective.
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Example 1: You have been asked to analyze the following cell data:
Find the first time interval when the change in volume is at least 115.72
1) 1-2 minutes
(2) 2-3 minutes (3) 3-4 minutes (4) 4-5 minutes (5) None of these
Example 2: You have been asked to analyze the following cell data:
Which of the following are true? (Calculations rounded to two decimal places.)
I. The ratio A/ B decreases over the interval 2-3 minutes
II. The ratio A/ B decreases over the interval 3-4 minutes
III. The ratio A/ B increases over the interval 2-3 minutes
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1.1.2 Graphs of Volume Compared to Surface Area
Let us examine how the Volume, Surface area, and ratio of SA/V change visually.
As time increases from left to right, the Volume_____________________
As time increases from left to right, the SA _________________________
As time increases from left to right, what can we say about SA/V? How would the graph
look like?
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Example 1: Which of the following graphs best represents the ratio SA/V?
Ans :
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Example 2: Which of the following graphs best represents the ratio SA/V?
Ans:
Increasing and Decreasing Functions:
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Example: Determine the intervals where f(x) is increasing or decreasing:
1.1.3 Logarithmic Functions and Log Graphs
Some populations, such as bacteria, are better modelled using exponential functions. That
means that their population growth more closely follows a curve of the form:
𝑦 = 𝑦0 𝑒 𝑘𝑡
In 𝑦 = 𝑦0 𝑒 𝑘𝑡 , “k” is called relative growth rate or specific growth rate
𝑦0 is the amount of bacteria at the start
Recall: Log properties
ln(𝑎𝑏) = ln(𝑎) + ln(𝑏)
𝑎
ln ( ) = ln(𝑎) − ln(𝑏)
𝑏
ln(𝑒) = 1
ln(1) = 0
ln(𝑎𝑏 ) = 𝑏 𝑙𝑛(𝑎)
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Suppose the mass a bacteria culture grows exponentially and the amount of bacteria over a
period of time is given by the following table. Determine how fast the bacteria grows.
Determine the function that best estimates the data.
Hour
2
6
Bacteria Mass
583
2888
Rule: Checking for Exponential Relationship
To check if data is exponential, you can take the natural log of the outputs, graph them and
see if the graph is a line.
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Example: The data below is a log plot for a bacterial population. How can we find the
exponential function that best describes the data?
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