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It's common for majors in biology to take calculus courses, and many calculus textbooks (and calculus professors) try to cater to these students by including applications to biology.

My question is, in what specific ways is a calculus course actually helpful for biology majors?

For example, are there any courses typically taken by biology majors that involve ideas from calculus? If so, what ideas come up? Do any biology courses actually require students to take derivatives, compute integrals, or solve differential equations?

I'm also curious in what ways a two-semester calculus course could be made more helpful for biology majors. For example, would it help to cover basic multivariable functions and partial derivatives? Applications of calculus to probability and statistics? Systems of differential equations? Fourier series?

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It seems to me to be a question for biologists. But here are links describing a calc/stats course for bio majors: ncbi.nlm.nih.gov/pmc/articles/PMC2931686, mathcs.emory.edu/math4bio –  Michael E2 Apr 29 at 19:50
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@MichaelE2: There's also Lior Pachter's math.berkeley.edu/~lpachter/courses/Math10a and math.berkeley.edu/~lpachter/courses/Math10b. –  J W Apr 30 at 5:48
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May I suggest taking a look at amazon.com/Dynamic-Models-Biology-Stephen-Ellner/dp/0691125899. Mostly the use of calculus helps students with dynamic modeling (what the book is about) and statistical modeling. You really need to know the basic concepts of calculus to understand statistics at the level of truly thinking about your data critically and not just applying tests haphazardly (you don't need to know how to do the calculations, but you need to know enough calculus to tell the stats software what to calculate for you). –  MHH May 6 at 0:46

14 Answers 14

I'm an old-school biologist (animal physiology) who works with mostly cell biologists. I sent out an email to a bunch of grad students and postdocs I work with. Here is the data so far:

  1. Senior undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her.
  2. Grad student: Undergrad biophysics course used modeling with differential equations. Graduate class in systems cellular biology used modeling with differential equations.
  3. Grad student: Undergrad physical chemistry used calculus, no biology
  4. Grad student: none, other than watching some derivatives and integrals in an engineering-level physics. Suggests perhaps a course on bioinformatics might use calculus.
  5. Grad student: none. Suggests systems biology might have some.
  6. Grad student: none. Some algebra for bacterial growth curves.
  7. Postdoc: no actual calculus used, but calculus helpful for understanding diffusion of molecules in space

I will add to the list (open-source data!) as emails come in, but it seems safe to say that calculus is rarely used by biology students outside of calculus class.

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Thanks for reaching out. As Matt F. mentioned, there are some things from calculus which might be of help when working with data, multivariate functions, log transforms, shape of normal distributions. These might not be apparent as things from calculus, but can be part of a calculus curriculum. –  Roland Apr 29 at 21:43
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What they do and what they should do are completely separate things. –  Carl Witthoft Apr 29 at 21:43
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To add to what Carl Witthoft writes, I think there's a difference between justifiably not using math because mathematical knowledge isn't appropriate/necessary to understand/solve the problem at hand and not using it out of ignorance, when it could in fact be beneficial. –  J W Apr 30 at 20:52
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I'm not surprised that the only positive response you found was differential equations modeling. Having taught that course a lot, the modeling examples fit nonlinear systems as perfectly as physics examples fit linear systems (and almost everything else in basic calculus). They felt real, not contrived. –  Ryan Reich May 1 at 2:50

Most bio majors won't need calculus in their bio classes. They will take chemistry classes in which understanding rates of change is useful, so:

  • partial derivatives will help them.

More importantly, many bio majors will work in quantitative areas in the life sciences, where data science is key. Think of developing drugs from chemical compounds, or clinical tests of drugs, or genomics. A calculus class with this in mind will definitely include:

  • The normal curve -- since the expression $$\frac{1}{\sigma\sqrt{2\pi}}\Large e^{\Large-(x-\mu)^2/2\sigma^2}$$ and its integrals, which are ubiquitous in statistical thinking, will not become natural to them in any other way.

  • Transforming data with log and exp, e.g. reading log-log plots.

  • Different ways to visualize functions, e.g. contour graphs.

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Absolutely. Every field of science (and even the pseudo-ones like economics) should require not only Calc. but Statistics as well. –  Carl Witthoft Apr 29 at 21:42
    
-1, I find this answer very alarming. The fact that biology students will work with data does not mean they need to use the equation for the normal curve or attempt to integrate it! Are you a biologist / do you have any experience in this area? I guess it's possible that biologists are using these equations all the time, but I find this an extraordinary claim! –  Chris Cunningham Apr 30 at 0:48
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@ChrisCunningham, you're attacking a straw man. 1) Neither the question nor my answer is about biologists. My relevant experience is talking with friends and colleagues in professional roles that biology majors often pursue. 2) I am not making the extraordinary claim you suggest. I am saying that a calculus class could help a biology major by helping them understand cumulative normals and the p-values or z-tests which depend on them. Is it so much to ask for including $\exp(-x^2)/\sqrt{2\pi}$ as an example of a way to use exponentials? –  Matt F. Apr 30 at 2:48
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An observation: The latter three points are all subjects that would be at home in one form of calculus or another, but (former) students using these afterwards would probably not think of themselves as "using calculus." –  Charles Staats May 2 at 17:23
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I'd like to highlight "p-values" here. You can teach students "What p-values really mean" using the concepts of integration. This will be super useful to biolgists! I work with them a lot and the ones who truly understand what a p-value is tend not to abuse statistics as much as the ones who don't. –  MHH May 6 at 0:39

I happen to have revised our calculus syllabus for first year biology majors about one year ago (in a French university, for that matter). I benefited a lot from my wife's experience as a math-friendly biologist.

The main point of the course is to get students able to deal with quantitative models. For example, my wife studied the movement of cells under various circumstances.

A common model postulates that the average distance $d$ between two positions of a cell at times $t_0$ and $t_0+T$ is given by $$d = \alpha T^\beta$$ where $\alpha>0$ is a speed parameter and $\beta\in[\frac12,1]$ is a parameter that measures how the movement fits between a Brownian motion ($\beta=\frac12$) and a purely ballistic motion ($\beta=1$).

This simple model is a great example to show how calculus can be relevant to biology.

My first point might be specific to recent French students: first-year students are often not even proficient enough with basic algebraic manipulations to be able to do anything relevant with such a model. For example, even asking to compute how $d$ changes when $T$ is multiplied by a constant needs to now how to deal with exponents. In fact, we even had serious issues with the mere use of percentages.

One of the main point of our new calculus course is to be able to estimate uncertainties: in particular, given that $T=T_0\pm \delta T$, $\alpha=\alpha_0\pm\delta\alpha$ and $\beta=\beta_0\pm\delta\beta$, we ask them to estimate $d$ up to order one (i.e. using first-order Taylor series). This already involves derivatives of multivariable functions, and is an important computation when you want to draw conclusions from experiments.

Another important point of the course is the use of logarithms and exponentials, in particular to interpret log or log-log graphs. For example, in the above model, it takes a (very) little habit to see that taking logs is a good thing to do: $\log d = \beta\log T+\log \alpha$ so that plotting your data in log-log chart should give you a line (if the models accurately represent your experiments).

This then interacts with statistics: one can find the linear regression in log-log charts to find estimates for $\alpha$ and $\beta$. But then one really gets an estimate of $\beta$ and... $\log\alpha$, so one should have a sense of how badly this uncertainty propagates to $\alpha$ (one variable first-order Taylor series: easy peasy).

The other main goal of the course is to get them able to deal with some (ordinary) differential equations. The motivating example I chose was offered to me by the chemist of our syllabus meeting.

A common model for the kinetics of a chemical reaction $$A + B \to C$$ is the second-order model: one assumes that the speed of the reaction is proportional to the product of the concentrations of the species A and B. This leads to a not-so-easy differential equation of the form $$ y'(t) = (a-y(t))(b-y(t)).$$ This is a first-order ODE with separable variables. One can solve it explicitly (a luxury!) by dividing by the second member, integrate in $t$, do a change of variable $u=y(t)$ in the left-hand-side, resolve into partial fractions the rational fraction that comes out, and remember how log is an antiderivative of the inverse function (and how to adjust for the various constants the appeared in the process). Then, you need some algebraic manipulations to transform the resulting equation into the form $y(t) = \dots$. Unfortunately and of course, we are far from being able to properly cover all this material, but we try to get the student able to follow this road later on, with their chemistry teachers.

In fact, I would love to be able to do more quantitative analysis of differential equations, but it is difficult to teach since it quickly goes beyond a few recipes. For example, I would like them to become able to tell in a glimpse the variations of solutions to $$y'(t)=a\cdot y(t)-b \sqrt{y(t)}$$ (a model of population growth for colonies of small living entities organized in circles, where death occur mostly on the edge - note how basic geometry makes an appearance here to explain the model) in terms of the initial value. Or to be able to realize that solutions to $$y'(t)=\sqrt{y(t)}$$ must be sub-exponential (and what that even means...). For this kind of goals, one must first aim to basic proficiency in calculus.

To sum up, dealing with any quantitative model needs a fair bit of calculus, in order to have a sense of what the model says, to use it with actual data, to analyze experimental data, to interpret it, etc.

To finish with a controversial point, it seems to me that, at least in my environment, biologists tend to underestimate the usefulness of calculus (and statistics, and more generally mathematics) and that improving the basic understanding of mathematics among biologists-to-be can only be beneficial.

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In the model of cell motion, is $d$ the average of the magnitude of the displacement? The root-mean-square distance? I would be interested in seeing this worked out in more detail. It's not obvious to me how to apply calculus to this example, since the derivative $dd/dT$ can't be interpreted as a velocity except perhaps in the case $\beta=1$. –  Ben Crowell May 5 at 3:14
    
It looks like a great course, though ambitious for first-year students. (The U.S. has plenty of students unable to deal with exponentials too.) A student who understands even the half of your syllabus before differential equations may be more mathematically sophisticated than most academic biologists. –  Matt F. May 6 at 1:24
    
@BenCrowell: in the model of cell motion, $d$ is indeed the root-mean-square distance. Any model involving a reasonably simple relation between variables would work here: calculus is used mostly to deal with uncertainties, and to discuss change of variables and log-log plots. –  Benoît Kloeckner May 7 at 15:17
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@MattF.: this course, especially the idealized version I presented here, is indeed ambitious. However the current calculus level of academic biologists should not be taken as the target for students, but as something that needs to be improved in the future. –  Benoît Kloeckner May 7 at 15:20

I am not a biologist, and this question asks for the contribution of a biologist, nevertheless I might be contribute about the practice in our university in Budapest.

We have a special two-semester calculus-type mathematics course for biologists developed together with biology departments. The curriculum is:

  • First semester:

    • complex numbers, matrices, eigenvalues, Leslie model
    • elements of one-and higher dimensional calculus (very quickly, mostly through examples)
    • discrete dynamical systems
  • Second semester:

    • differential equations (mostly geometric theory with phase diagrams on computer), Lotka-Volterra model
    • elements of probability theory

This looks extremely quick for a mathematician but we have to solve somehow the problem that some parts of biology need deep mathematical results but there is no time to develop the theory.

Later and in the master/PhD program they can choose specialized courses held by biologists about game theory in ecology and population models (based on Lotka-Volterra type models), disease transition or tumor growth models use heavy ODE theory.

Added: Here are some links to Hungarian course materials (at least the literature is in English).

population biology

evolutionary game theory

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Could you publish a link to the department, or syllabi of the courses, or some other details? I'm sure OP would appreciate them. –  vonbrand Apr 29 at 20:22
    
It is a bit awkward for me, but I do not find the English files only the Hungarian ones on the homepage... –  András Bátkai Apr 29 at 20:25
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Could you add a link to it anyhow? A link to a page in Hungarian is more useful than no link at all. –  Joe Apr 29 at 21:58

An all-inclusive neurobiology class, which is normally appropriate for upper-division undergraduates, will present the physiology of excitable membranes.

Modeling at this level can be as simple as the Nernst equation for the equilibrium potential of a particular ionic species:

enter image description here

http://en.wikipedia.org/wiki/Nernst_equation

By taking into account ion permeability, the Goldman–Hodgkin–Katz equation can be used to illustrate the reversal potential for a given membrane:

enter image description here http://en.wikipedia.org/wiki/Goldman_equation

Neither of these models uses calculus explicitly, but more advanced students (especially those who are interested in computational modeling) can be introduced to the Hodgkin-Huxley model:

enter image description here http://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model

As mentioned in some of the other answers, a thorough knowledge of statistics is incredibly useful to students pursuing undergraduate research or those with plans to continue their education, but the aforementioned example is an opportunity for students to directly employ differential equations-based models in the undergraduate biology curriculum.

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One division of Biology that can be quite mathematical is Ecology and Evolutionary Biology. There are definitely courses that require calculus and differential equations quite similar to what you would teach an engineer for example. From what I understand this can come as quite a surprise to the biology students who go into Ecology because they like the outdoors and the plants/animals. But if you want to understand something like how it is possible that different animals can occupy what seems like the same evolutionary niche, then mathematical models really are the best way to do it.

From the University of Arizona course catalog (that link will require some clicking around, sorry):

ECOL 447 - Introduction to Theoretical Ecology Population growth and density dependence; predation; competition and apparent competition; coexistence mechanisms: niches, spatial and temporal variation; food web concepts and properties; applications. Emphasis on understanding through models and examples. Prerequisite: Calculus I

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Differential equations are used to model e.g. predator/prey interactions in ecology, spread of diseases in epidemiology.

Much of (molecular) biology is chemical reaction kinetics, again calculus/differential equations.

[The above just as somebody with an interest in biology in general, no formal relation to the subject.]

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Purely anecdotal, but I knew biology undergrads studying epidemiology were using some models that I never looked into but I presume were differential equations, discrete dynamic systems, or both. However, they were mostly using software to study the models, so I suppose you could argue over how much calculus they actually needed to know. It's entirely possible I (a maths undergrad) would have been unable to solve them other than by numerical methods. However, this was in the UK, US biology syllabuses might be completely different for all I know. –  Steve Jessop May 1 at 10:13
  1. Mathematics courses encourage analytical thinking in a way that may be helpful for biology majors.
  2. There is some argument that calculus ought to be more widely known within the biology community. For example, see the following infamous paper, which has gained over 200 citations according to Google scholar:

    M.M. Tai, A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves. Diabetes Care, Vol 17, Issue 2, 152–154.

    The "mathematical model" discussed in the trapezoid rule, which is often covered in second-semester calculus courses.

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I find this offensive towards biology majors. –  Mark Fantini Apr 30 at 13:00
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It might be worth mentioning that Tai's paper have been quite widely discussed on internet, for example here is related question in SE network: academia.stackexchange.com/questions/9602/… –  Martin Sleziak Apr 30 at 14:03
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@Fantini I have edited this answer to improve politeness while preserving the content as much as possible. –  Jim Belk Apr 30 at 14:43
    
@JimBelk I have removed my downvote and turned into an upvote. –  Mark Fantini May 1 at 17:58

Some years ago I taught a one-semester course on mathematics for pharmacy students. (They also got a semester of statistics in another course.) I looked at some of the second and third-year prescribed books for the pharmacy degree and they had quite a lot of calculus in them. Physical pharmacy: rates of diffusion of various things. Interpreting the elimination of a drug given orally from the body by looking at measurements in the blood at different times: the drug goes first into the stomach and then into the bloodstream, so you end up with two coupled DEs (or even three, if some organ or tissue is acting as a reservoir). Chemistry: in pharmacy you are generally dealing with weak acids and weak alkalis, so the situation is considerably more complicated than in the usual beginning chemistry.

Certainly things like semi-log plots occurred quite a lot - not exactly calculus, but often taught with it. And we did teach the trapezoidal rule!

There wasn't any other maths/stats as such except the two one-semester courses in the Pharmacy program. They did a lot of chemistry and biology, and specialised courses on Pharmacy topics. This course was in Australia.

I'm a bit surprised at the Pharmacology major mentioned above.

And I would say that anyone who is good at both maths and biology has got some fantastic opportunities.

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When you want to discuss the rate something happens, you will find that differential equations of calculus are helpful.

Some examples in biology:

  1. population growths: dx/dt = Rx, describes unlimited/exponential growth of a population that could be rabbits, cells, etc.

  2. kinetics of a chemical reaction: reversible [A][B] <-> [AB]. d[AB]/dt = k1*[A][B]-k2[AB] formation rate of d[AB]/dt slows as you use up [A] and [B]

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Calculus is seldom helpful for biology majors, if "helpful" means useful in a utilitarian, professional sense. The vast majority of biology majors are going into allied health fields: they intend to be doctors, pharmacists, physical therapists, vets, optometrists, and dentists. These professions are not like engineering, in which calculus is used from day to day. Here in California, the UC system decided ca. 1997 to start requiring biology majors to take calculus-based physics. The motivation was pretty transparent: they had too many biology majors (the major was "impacted"), and they wanted to get rid of some. This is similar to the fact that in 19th century Britain, if you wanted to be a military officer, you had to pass a test on Greek and Latin.

Does this mean that future military officers have nothing to gain from learning ancient Greek, or that future dentists have nothing to gain from taking calculus? Absolutely not. It simply means that for the future dentist, learning calculus is one possible ingredient in that quaint notion of a general education. It's a way of gaining broad knowledge about the world and getting experience in varied intellectual pursuits and ways of thinking.

For comparison, it may be helpful to ask the similar question of whether biology coursework is helpful to biology majors. A lot of it clearly is not, if helpful is used in the sense of day-to-day professional utility. For example, biology majors learn about the reproduction of ferns and club mosses, which is likely to be of very little practical utility to an optometrist.

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This is only true of professional biologists not academic ones. Most academic biologists do in fact use some concepts from calculus, even if they aren't doing calculus explicitly. –  MHH May 6 at 19:15

One important application of calculus in biology is called the predator-prey model, which determines the equilibrium numbers of predator and prey animals in an ecosystem.

It's actually an application of "differential equations" but you will need calculus to "get there."

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There is at least one very good reason to know calculus as a Biologist. There was a certain paper published, I don't know the details, but could probably look it up, by a biologist to a bio journal detailing how to calculate the area under a curve by using this amazing approximation using rectangles and trapezoids. This of course was peer reviewed and hailed as a major advance for some part of bio that constantly needed to do this. The story goes on to say that the biologist knew that this came for math somewhere, but so many other biologists wanted to use the technique and needed something to cite, so he published the paper. However, the issue remains: Biologists, didn't know basic integration. I'm sure you could find this story online. I am not sure if it is valid, but I find it likely to be in part true at the very least. So being a respectable scientist is a good enough reason to learn something like calculus.

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The question on Academia SE has some more discussion about this story. –  scaaahu Jul 15 at 11:30
    
Thank you for the link. That provides sourcing and credibility. –  Thoth19 Jul 15 at 11:55

At the end of the day, all science is 'applied mathematics'...without the math supporting your observations, you greatly limit yourself in your chosen field. Can you get through life in a science career without math? Sure...if all you care about is qualitative observations. With post-trig math knowledge though (e.g.- Calculus, Differential Equations, Linear Algebra, etc)...you're provided a deeper, quantitative understanding of your chosen field.

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Could you make your answer more focused and provide evidence on these claims? We all agree in our heart with you, but some data is always better... –  András Bátkai Apr 30 at 18:58
    
Niels Bohr was most of the most influential physicists of the 20th century, with essentially no math: he relied instead on his brother Harald. So Craig, I would say yes, and @Andras, I disagree. –  Matt F. May 1 at 3:35
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@MattF. What I meant was that as teachers of mathematics, we dream of a world where these claims are true, but it would be great to support them. As your example shows, it is just a dream and we should know our place. –  András Bátkai May 1 at 7:41
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The question was not "Is math helpful?" but "How are the specific topics classified as 'calculus' helpful?" You didn't address "how" in the slightest. –  Ben Voigt May 3 at 23:35

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