Karl Castleton

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Karl Castleton

• Research Scientist
• Pacific Northwest National Laboratory

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What is this good for?

• Intent To produce a set of practical calculus
  problems that can be used at certain points in a
  typical series of calculus courses
• Assumptions
• Not all students in a calculus class are Math
  majors
• Many students (even Math majors) benefit from
  practical hands on experiments in calculus

3
Inspiration

• Car talk hosts Tom and Ray Magliozzi say ooooh
  this requires calculus
• A practical question leads to calculus.
• What other practical questions lead to calculus?
• Wouldnt it be fun to have a list of such
  questions?

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Any examples from the audience?

• I know many of these might be characterized as
  engineering questions but a well placed concrete
  example helps many understand the math concept
  better.
• What problems seem to satisfy students when they
  ask you What is it good for?
• I could not seem to think of a concrete example
  for series.
• Please bring up any examples that strike you as I
  speak.

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A diesel truck driver needs to know how much fuel
is in the tank
The area above the 1/4 mark should be equal to
the area below. Only 1/4 of the tank need be
considered.The Area of an angular segment will be
useful. Area AB C -B Lets name the angle t
t
A
B
C
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Diesel Tank continued
C-BBA tr2 - ½ r2cos(t)sin(t) ½
r2cos(t)sin(t)(pi/2-t)r2 assume r1 (for
simplicity) tcos(t)sin(t)(pi/2-t) arrange so t
is on one side pi/22t-cos(t)sin(t) Not very
satisfying! But where is the calculus? Were Tom
and Ray wrong as they so often are? The answer
from above is roughly 30 of r for ¼ and 70
for ¾ found by experimentation.
Well the assumption that tr2 is the area of C A
is essentially a calculus result. But the hand
check clearly is. Put the 1/4 circle on a grid.
Count the total in the quarter. Now count until
you reach half that number. Split the remaining
amount. Want a better answer use a finer grid.
(Clearly the mark of Calculus)
1/4 approx.
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So what other questions?

• How many sprinkler heads?
• Getting the most inside the fence.
• Measure totals with sampled rates?
• Whats going to happen in the future?

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A wacky gardener wants to know how many heads to
put in his garden.
Assume the gardener has coordinates of the
corners. Clearly this is just integration (for
those who know what it is) in hiding. But the
strange shape might initially make it seem
difficult. Trapezoid summation of the areas
under the line segments. Area of a trapezoid
A1/2h(b1b2)
h65-10, b110,b210 h will be negative for the
lines that go towards the Y axis. So the line
(120,100) - (75,25) will have h-45
Our gardener likes non rectangular shapes. Each
sprinkler head can covers 2000ft2. How many does
he need.
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Wacky Gardener Continued
The math skill required can be kicked up a notch
by not giving the students the coordinates of the
vertices and have them devise a technique for
measuring them. I would suggest the you give
them the picture of the plot on weird shape
paper. The technique is simply to draw a line r
you do know the length of, then measure the
distance between the ends of the line segment r
and any corner. From these two distances the X
and Y can be computed relative to ruler r. It is
just some algebra.
r
A11/2(65-10)(10100) A21/2(75-65)(10070) A31/2
(120-75)(70100) A41/2(75-120)(10025) A51/2(10-
75)(2510) A1A2A3A4A53750ft2
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Getting the most inside a piece of fence.
Students should get used to the idea that if you
are maximizing or minimizing something you are
going to be taking the derivative and setting it
equal to 0. The most important part of this
question is setting it up properly. Assume the
small side length is x then the large has to be
100-2x. Students may try to call this distance y
and be stuck. Ax (100-2x)100x-2x2 dA/dx100-2
2x0 1004x or x25
This one does appear in many calculus texts. A
farmer has 100 feet of fence and he wants to
enclose the largest rectangular area along side
his barn. What should the dimensions of the area
be?
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Measure the total with sampled rates.
Your company produces pop/soda. Estimate the
total number of bottles leaving the plant without
adding equipment to count every bottle.(cheap
boss) You know that the plants production rate
in bottles/day does not change instaneously but
slowly increases or decreases.
We should assume that we can every once in a
while measure the number of bottles that left
over a short period of time or monitor how long
it takes for a certain amount of bottles to leave
the plant. Either would give you b/day estimates
(slopes) at given points in time. Lets assume we
got the following measures. Day 1 25 b/day, Day 2
40 b/day Day 3 12 b/day, Day 4 45 b/day
b/day
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Bottle Counting continued
Bt1/2(1)(2540) 1/2(1)(4012)
1/2(1)(1245)87 Check your calculus intuition
and see if you can see what is wrong with the two
pictures to the right? Remember ymxb would
represent the line between the points on the rate
graph. Integrate with respect to x.
115
b
This should start to look like the wacky gardener
again. You could integrate and find the total
number of bottles.
77
65
25
1 2 3 4
day
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Whats going to happen in the future
This is a Constantly Stirred Tank Reactor (CSTR)
model and is the bread and butter of civil
engineering. VdC/dtQCi-QC-lCV dC/dtQ/VCi-Q/VC-lC
V/QT dC/dt(Ci-C-lCT)/T dC/dt
(Ci-C(1lCT))/T dC/C(1lCT)1/Tdt uCi-C(1lT) du
-(1lT)dC -(1lT)dC/C(1lCT)-(1lT)dt/T du/uutuo
ln(ut/uo)-(1lT)t/T
How long do you need to put clean water into your
swimming pool if you accidentally put 10 times
the chlorine you should have. The pool is
already full.
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Whats going to happen continued
Now this equation can be rearranged for t and
assuming .1CoC and Ci0 With this result you
could even account for the fact that the water
you are putting into the pool has chlorine as
well. The students need to realize that this
problem really does make a prediction of the
future based on how the system works.
Environmental issues are described and decided
upon using such equations.
ln(ut/uo)-(1lT)t/T ln((Ci-C(1lT))/(Ci-Co(1lT))
)(-t/T)(1lT) Ci-C(1lT)exp((-t/T)(1lT))(Ci-Co
(1lT)) C(1lT) exp((-t/T)(1lT))(Ci-Co(1lT))C
i C exp((-t/T)(1lT))(Ci/(1lT)-Co)Ci/(1lT) C
exp((-tQ/V)(1lV/Q))(Ci/(1lV/Q)-Co)Ci/(1lV/Q)
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Conclusions?

• Thanks for the time.
• I hope this gives you some ideas that you can use
  to inspire students.
• More examples will be added to this set and
  available at http//home.mesastate.edu/kcastlet/c
  alculus