Math7B – Integral Calculus for Bio
The full details of the logistics of this course,
as well as a list of resources and other nuances,
can be found in the syllabus,
but I’ll provide a brief summary here.
The lecture will be held in Skye 170 Monday through Thursday,
8:10–9:30am, starting on July 29.
The discussion section will be in Skye 171 on Wednesdays, 11:40am–1:30pm.
The final exam will be on Saturday (sorry) August 31 from 8–10am in Skye 170.
I’ll host office hours in Skye 284 on Mondays 10am–3pm.
If you have questions or comments, email me at
.
This course will follow the book
Calculus for Biology and Medicine, Third Edition, by Claudia Neuhauser,
but you do not need to buy the book for this course.
I will give you homework problems each week to think about,
but your grade in the course will be determined by assessments you take in discussion
which are based on the homework, and by the final exam.
As mentioned in the syllabus,
YouTube, and
WolframAlpha, and
Desmos
are great technological resources to help you learn the material
for this course.
Also, Paul’s Online Notes are generally great, especially the
end of Paul’s Calculus I
and the
beginning of Paul’s Calculus II.
Additionally, here is a thread online that is relevant to this course:
How is Calculus Helpful for Biology Majors?
Course Outline, Notes, Homework, & Assessments
Here is an outline of the content of the course,
along with my personal lecture notes (which I really just write for myself
to help me lecture, but might be helpful to students),
and links to the homework and the assessment for that week.
I’ll post my solutions for the weekly assessments
and for the final exam after they are given.
Week Zero – Review of Differential Calculus, Trigonometry, Etc

Homework

This homework provides a review of material that will
be useful in the course, and also informs students what I think
they’ve learned in previous math courses,
so they can tell me that I’m wrong.
Week One – Definite and Indefinite Integrals, and the FTC

Lecture Notes
·
Homework
·
Assessment Solutions

We’ll define integrals as a limit that gives us
the area under the curve of the graph of a function,
and talk about evaluating definite integrals
both numerically and geometrically.
Then relate these integrals to (anti)derivatives
via the Fundamental Theorem of Calculus (FTC)
and we’ll talk about definite versus indefinite integrals,
Lastly we’ll use the FTC to calculate the area bounded between two curves
that are the each the graph of a continuous function.
Here are a few links related to this week’s material.
Week Two – Interpretations & Applications of Integration

Lecture Notes
·
Homework
·
Assessment Solutions

We’ll talk about a few more interpretations and applications of integration,
like how an integral counts the cumulative change in the value of a function,
and how, for example, that can correspond to the
bioavailability
of a drug in the bloodstream.
We’ll talk about how we can find the average value of a function over a length of time.
Then we’ll return to geometry a bit to learn how we can calculate
the length of a curve and the volume of a solid of revolution
using integration.
You can see an example of this worked out here:
Week Three – Techniques of Integration

Lecture Notes
·
Homework
·
Assessment Solutions

This week we’ll start practicing the various techniques of integration,
including substitution (also called $u$sub, or change of variables),
integration by parts, integrating rational functions by using
polynomial long division and partial fraction decomposition.
And we’ll talk about the integrals
\begin{equation*}
\int\tan(x)\,\mathrm{d}x
\qquad\qquad
\int\ln(x)\,\mathrm{d}x
\qquad\qquad
\int\sec(x)\,\mathrm{d}x
\,.
\end{equation*}
And here are a few links related to this week’s material.
Week Four – Further Techniques of Integration

Lecture Notes
·
Homework
·
Assessment Solutions

This week we’ll talk about evaluating integrals
of functions that are products and quotients of trig functions.
Then we’re going to talk about trigonometric substitution,
and then evaluating improper integrals.
Here are a few links related to this week’s material.
Week Five – Differential Equations and Applications

Lecture Notes
·
Homework

This last week we’ll start looking at ordinary differential equations (ODEs),
and see how integration helps us solve separable differential equations.
Then we’ll talk about some modelling with differential equations,
and equilibrium solutions to differential equations.
Basically we’ll start a crash course of
UCR’s
introduction to differential equations, Math46.
Final Exam
Notes for When I Teach This Course Again
I Need to Better Explain the Implications of the FTC
I asked this question on the final,
Write down a calculation to verify that
\begin{equation*}
\int \frac{1}{\sin(x)\cos(x)} \,\mathrm{d}x
\;\;=\;\;
\ln\left(\tan(x)\right) +C\,.
\end{equation*}
and was shocked that only three students responded
by taking the derivative of the function $\ln(\tan(x))+C$.
Most students tried to evaluate the integral.
I didn’t ask them a question exactly like this before on homework or an assessment,
but I figured that they had internalized at least the part of the
FTC that says taking integrals and taking derivatives are “inverse” operations.
But I was evidently wrong.
I need to be mindful of this if I teach this class again.
Take an Active Role Advising Students how to Take Notes
Students struggle at learning math.
I think this is, in part, because they try to employ
the same techniques they use to learn in their other classes
to math as well, which doesn’t always work.
In particular, I noticed this in their notetaking habits.
They’ll meticulously write down
everything,
even things I explicitly said wasn’t that useful
but that they should see worked out once.
Even when I put them in groups and have them work on
an exercise on the board together,
they’ll all take the time to write the exercise carefully
in their notes before working.
Next class I teach, I need to tell students
how to take notes in a math lecture.
Like, writing stuff down is important, but
thinking
about what’s being said, making sense of the lecture,
is the most important thing.
Here’s some relevant reading:
And this opens the door to the more general questions:
In what ways is learning math
different from learning other university topics?
To what extent do we need to take the time to coach our students
in
how to learn mathematics?
Stress the Importance of Definitions and Explanations
I thought I did this quite a bit,
but it was probably new to the students to be expected to
explain, like in writing, what things are and how things work.
I asked them questions on the assessments like,
What is the definition of the definite integral
of a function on the interval $(a,b)$?
What is the definition of the indefinite integral
of a function?
Explain, as if explaining to a friend,
why $\int_a^b f(x)\,\mathrm{d}x = \int_a^0 f(x)\,\mathrm{d}x +
\int_0^b f(x)\,\mathrm{d}x$.
There aren't too many of these sorts of questions I could
have asked in the course; we only had a few definitions,
and there were only a few concepts to internalize.
Maybe it would've been a good idea to give the students
a list of all these sorts of questions at the start of the term
and tell them at least one would be on the final.
This would put some pressure on them to practice
their mathematical writing and their skills at explaining concepts.
Write More Examples when Working Through Techniques of Integration
When I introduce a technique of integration,
I should write down a whole bunch of integrals that
immediately require that trick,
to help students recognize the pattern.
On the final, there were way more instances of students
trying the wrong technique on very easy integrals than I expected.
Be Sure to Assess for Common Misconceptions
There is a common trend in this course where students don’t
fully understand the difference between taking an integral
and “simplifying” the integrand.
I should be aware of this when designing the final.
Explore Definite Integrals that are Exactly 1
There are few neat areas under a graph that turn out to be exactly $1$.
At least the two
\begin{equation*}
\int_0^1 \ln(x) \;\mathrm{d}x
\quad\text{and}\quad
\int_0^{\pi/2} \cos(x) \;\mathrm{d}x
\,,
\end{equation*}
and I could use these examples as tools
when asking questions that require the student
to think of definite integrals as areas.
Learn gnuplot
As an educator, I should learn how to use
gnuplot.
Personal Technical Notes
Lastly, just a few technical notes to myself.
(1) I should title all of my lectures. The students appreciate this
so they can add the titles to their notes for reference to specific lectures.
(2) I need to literally send the students a link to the iEvaluation website.
I only mentioned that they could go evaluate me at the beginning of a lecture,
and only two student responded to evaluations.
(3) I should download the CSV file from
iGrade.ucr.edu
and work with that from the very beginning. That way I can just upload my grades
at the end of the term, and reduce the risk of errors/typos.
(4) A couple students requested that I put solutions to
all
the exercises. I should do this for the computational exercises
if I teach this course again.