May 18

This is a problem I normally give at the end of semester in calculus 1 as part of a worksheet composed entirely of application problems from various sections of the book (Some of the other problems from this worksheet appear else where in the blog.  Links to them are at the bottom of this post).  This kind of mixed review can be very powerful because the students have to figure out which of the many techniques they have learned apply to the situation.  This is a skill that employers recruiting at our university’s career fairs have told me the are seeking.  I include the references on the worksheet that I hand out, not just the questions.

A variation on this problem (using function composition and similar triangles instead of related rates) is suitable for use in a precalculus course.  The precalculus variation will be appearing as [4].


Here’s the problem:

(Hint: In this problem you need to be very careful with units). The Maldives is an example of a poor low-lying country which will be hit soon and hard by the effects of global climate change including sea level rise [3], [2].  The Maldives is located south-southwest of India.  It consists of approximately 1,190 islands and is the lwest country in the world with a maximum elevation of 2.4 m.  The Maldives has a total area of 298 km squared and a coastline of 644km [2].  Since the actual shape of the Maldives is complicated, we will use a simpler shape to compute with and assume this is a good stand-in for the real islands.  We will pretend the nation is one triangular prism as pictured below.  The width of the base is 1 km and the length of the base is 320 km.  The height of the prism is 2.4m.

a) Convert all units to kilometers and label the picture with the measurements in kilometers.  What is the map area of the prism in kilometers squared?   By map area, in this problem, we mean the area of the rectangle you would see by looking down on the prism from above.


b) At what rate is the area changing when the elevation is 2meters if the sea level is rising at a rate of 4.3325 mm per year?

c) The Intergovernmental Panel on Climate Change (IPCC) predicts between 0.28 and 0.98 m of sea level rise by 2100 [1].  Some reputable sources predict up to 2 m of sea level rise  by 2100 [3].  What is the annual rate of sea level rise in mm per year for each of these estimates?  What is the rate of change of the area for each of these estimates?



[1] J. A. Church, et. al. “Sea Level Change“. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. 2013.
Full text available at\_Chapter13\_FINAL.pdf

[2] The CIA World Factbook,, accessed on May 6, 2015.

[3] K. Dow and T. E. Downing, The Atlas of Climate Change: Mapping the world’s greatest challenge, 3rd ed.
University of California Press, Berkeley,  2011.

[4] Archey, Dawn. “Sea Level Change and Function Composition.” In Mathematics and Social Justice Modules for the Classroom.  Eds. Karaali, Gizem and Khadjavi, Lily. to appear.


Other problems in the Calculus 1 mixed review worksheet included:



Sep 22

Here are two problems I used in my calculus 1 class, just after teaching them the definition of the derivative and discussing average and instantaneous rates of change.


  1. My car has a digital read out with the average fuel economy.  I used to have a car which also had the instantaneous fuel economy.
    What is the difference between these two fuel economies?  Under what circumstances is each more useful?  How can I use the average fuel economy feature to approximate my instantaneous fuel economy?  (Your answer should be  2-5 complete sentences).
  2. Read the article at .
    You can use this QR code to access it

    1. What is the average rate of change of the mass of Antarctica’s ice?
    2. What is the average rate of change of the mass of Greenland’s ice?
    3. Looking at the graphs, do you think it is reasonable to talk about the instantaneous rate of change for these quantities?  Why or why not?
      Then answer the following questions. Give all answers in complete sentences with units.


Jul 27

I was so excited about New Horizon’s fly by of Pluto, that I decided to create some math problems about it.  Here they are.  The first one uses the concepts of percent change and volume of a sphere.  The second two use right triangle trigonometry.


1) Answer the question posed in @MathInTheNews ‘s tweet from July 14, 2015:
New Horizons says Pluto’s diameter is 1473mi, 50mi larger than believed. What % does that change its volume estimate?


(Yes, the picture was part of the tweet)

2) Pick two mountains in the large picture of Pluto.  The angle of elevation from the tip of the mountain’s shadow to the tip of the mountain is approximately y/736.5 radians where y is the distance in miles from the base of the mountain to the edge of Pluto in the picture.

a)  What is the height of each of your two mountains?

b) How precise do you think our method is?

c)  Do you think NASA’s claim that there are 11,000 ft high mountains on Pluto is credible?

(Note to other instructors, the estimation of the angle given above is very imprecise, so the students ended up finding mountain heights no more than 7,000 feet high.  In order to improve precision, I printed the following picture out as a 11 by 17 picture).




3) In a July 13, 2015 article for Reuter’s Irene Klotz wrote, “Mysterious Pluto looms large and turns out to be larger than expected as NASA’s New Horizons spacecraft wraps up a nearly decade-long journey, with a close flyby on track for Tuesday, scientists said on Monday.  The nuclear-powered probe was in position to pass dead center of a 60-by-90-mile (97-by-145 km) target zone between the orbits of Pluto and its primary moon, Charon, at 7:49 a.m. EDT (1149 GMT) on Tuesday, said managers at New Horizons mission control center, located at the Johns Hopkins University Applied Physics Laboratory outside of Baltimore. After a journey of 3 billion miles (4.88 billion km), threading that needle is like golfer in New York hitting a hole-in-one in Los Angeles, project manager Glen Fountain told reporters.”

Is this analogy correct?  (You will need to look up some additional numbers and make additional assumptions to answer this question).

Jul 27

Here’s a warm up activity to get students thinking mathematically and thinking about what math is:

One of my friends posted this picture on Facebook.  Use it and whatever other resources you need to answer the following questions.


a) Assuming the first statement is correct, is the second statement correct?

b)  What (if anything) is wrong with this “solution” to homelessness?

c) To what extent was this problem “doing math”?  Were both parts a and b math problems?

Jul 07

This is a problem I work through in my pre-calculus class when we are discussing the meaning of the slope and y-intercept.  It also gives me the opportunity to discuss the perils of extrapolating far outside of the data set.

Consider the following data and the linear model that fits it.


The equation for the linear model is y=0.019x+4.76.  Answer the following questions in complete sentences with units.

  1. What are the units for the slope?
  2. What does the slope mean in this particular situation?
  3. What are the units for the y-intercept?
  4. What does the y-intercept mean in this particular situation?

(data from and )

Apr 13

The FoxTrot Strip from November 6, 2005 contained a “numerical word search”

You can find the strip here.

There are seven clues which are arithmetic problems, but there is also the derivative of 43981 x , the definite integral of x^3 from 0 to 16, and the sum from k equals 0 to 47 of k squared.    I asked my calculus students to perform all these calculations.

Apr 13

I asked my calculus students to read this xkcd strip from

Fairy Tales

Then I asked them:

a) What is $\lim_{x\rightarrow \infty} (x )$?
b) Explain what might have been weird about the $\lim_{x\rightarrow \infty} (x )$ little pigs.

Mar 23

Randall Munroe drew a lovely comic strip graphing the effects of proximity to a cat on the human intellect:


Cat Proximity

(You can view it here:

In differential calculus I asked students to describe the graphs using their knowledge of calculus.  Statements about both the first and second derivative are expected.

In Intermediate Algebra here at UDM we study qualitative graphs, so I also asked my Algebra students to describe the graph.  These students have to say things like the intelligence of the human is decreasing, but at an ever faster rate.

One caution with this comic: some students do not know what “inanity” means, so I use this problem on the review sheet rather than the exam.

Mar 18

In this FoxTrot strip Paige has been cramming for her math final and thus is answering all questions in mathematical notation.

(Its the May 22, 2005 strip, if the link doesn’t work.)

In precalculus I asked the students:

1) What temperature will it be tomorrow?  Is it going to be hot tomorrow?

(The answer is sin^{-1}(1) =90 degrees, so it will be hot).

2) What does the girl (Paige) want for snack?

(The answer is 3.14159265359…, so she wants pie).

In calculus of sequences and series, you could also ask what the discount was in the fourth panel, which requires computing the sum of an infinite geometric series.

In an algebra class one could ask what time Paige went to bed (\sqrt{121}= 11:00) or What is on TV? (4!=24).

Mar 16

In this FoxTrot strip Jason is “helping” Paige do her homework without even looking in the book.  He claims the answer to question 36 is lim_{x \rightarrow \infty} \frac{x+2}{x-2}.

I asked my students to evaluate this limit, but think this wasn’t a very good use of the comic strip.  If you have a better idea, please comment or email me.

(If the link doesn’t work try going to , entering the keyword “question” and checking the FoxTrot box.  It was the first hit when I used this search.  The fact that it is from September 13, 2001    may also help you find it.)

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