Calculus Reform in LB 118
Robert Bell & Aklilu Zeleke
STEM Alliance Presentation
May 05, 2016
HHMI Calculus Reform Project
This project is supported by a grant from the Howard Hughes
Medical Institute:
LEVERS: Leveraging Engagement and Vision
to Encourage Retention in STEM.
Project Overview
This is an e↵ort to develop a first semester calculus course with
the following features:
1. There is an emphasis on examples from the life sciences.
2. Developing mathematical models motivates the study of
functions, derivatives, and integrals.
3. Calculus concepts are explored using numerical data.
4. Conceptual understanding is emphasized.
Comparison with a Traditional Course
The traditional courses are
I
LB 118 Calculus I
I
MTH 132 Calculus I
These courses are interchangeable for the purpose of prerequisites
and requirements for minors and majors. The course could be
roughly described as “calculus for scientists and engineers”.
Comparison with a Traditional Course
The traditional courses are
I
LB 118 Calculus I
I
MTH 132 Calculus I
These courses are interchangeable for the purpose of prerequisites
and requirements for minors and majors. The course could be
roughly described as “calculus for scientists and engineers”.
LB 118 serves 1st and 2nd year students considering majors in all
STEM fields. However, most students intend to major in the life
sciences (especially human biology or physiology).
What is first semester calculus (LB 118 / MTH 132)?
Topic: Application of the derivative
The top of a 13 foot ladder slides down a vertical wall at a rate of
2 feet per second. How quickly is the base of the ladder moving
away from the wall when it is 5 feet from the wall?
Traditional Final Exam Question
Topic: Fundamental Theorem of Calculus
Compute
d
dx
Z
p
⇡
x
2 sin (t 2 ) 1
p
dt
t4 + 1
A Better Traditional Final Exam Question
Topic: Geometric and conceptual meaning of the derivative
The student is given the graph of the velocity of a particle moving
along the real line and is asked to answer thought provoking
questions about the relationship between derivatives and concepts
from physics (e.g. displacement, distance, speed, acceleration,
speeding up/ slowing down).
A Modeling Based Calculus Exam Question
Topic: Mathematical models and data
First, fill in the missing entries in the table below, plot Yt+1 Yt
versus Yt , and use this plot to describe the relationship that is
expected to hold between Yt and t.
t
Yt
Yt+1
0
0.5
Yt
1
1.0
2
2.5
3
4.5
4
8.0
Another Modeling Based Calculus Exam Question
Topics: Mathematical models, chain rule for derivatives
Suppose that a mold colony has a circular shape. The following
observations are made:
I
the colony has a radius of r = 1 centimeter on the zeroth day,
t = 0.
I
the radius, r , of the colony increases at a rate of 0.5
centimeters per day.
1. Express the radius of the colony as a function of the day. Use
r for the radius and t for the day, where t = 0 refers to the
zeroth day.
2. Express the area of the colony as a function of the day.
3. At what rate is the area of the colony changing on the fourth
day? Express your answer using the correct units.
Challenges
I
Modeling is difficult and is unfamiliar to students.
I
Realistic applications can obscure and overcomplicate
fundamental mathematical concepts.
I
Students may enroll in LB 119 or MTH 133.
I
Textbook options are insufficient for our objectives.
I
Online homework resources are insufficient.
I
If you add features (e.g. modeling, R & RStudio, numerical
data), then others must be omitted (e.g. sliding ladder
problems, the calculus that results from the theory).
Depth Layer
Id
Id
0
wM / cm
0.400
1
2
3
4
5
6
7
8
9
0.330
0.270
0.216
0.170
0.140
0.124
0.098
0.082
0.065
2
1
Id
wM / cm 2
Some Notation:
I d is the light intensity at the bottom of the d th layer or depth.
f is the fraction of light absorbed by each layer. Assume for now f to be the same for all
layers, though in reality it should depend on the thickness of the layer and the distribution
of suspended particles in the water and atomic interactions with light.
I d 1 I d is the change in light intensity between layer d and d 1
1) What is the change in light intensity? i.e. fill in the third column?
2) Plot the following data using your graphing calculator and/or R describe (guess) the type of function.
Give as much detail as you can about the function.
a) Light intensity ( I d ) vs Depth Layer ( d )
b) Change in Light intensity I d
1
I d vs Light intensity ( I d )
2) Derive the Dynamic Equation for I d
3)
Depth
Layer
Id
wM / cm
0
0.400
1
2
3
4
5
6
7
8
9
0.330
0.270
0.216
0.170
0.140
0.124
0.098
0.082
0.065
2
Light
Intensity
computed
from the
model
Absolute
Error
Id
1
Id
wM / cm 2
4) Comment on how good the model is to predict the light intensity
Class Activity
1. Use the graph below to rank the value of each expression from smallest (1) to largest (5).
8
6
4
2
0
-2
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
-4
-6
x
_____ f (4)
_____ slope of f ( x) at x
_____ lim
h
_____
0
f (28 h)
h
22
f (28)
f (20) f (10)
20 10
_____ slope of the tangent line at x 14
2. Sketch the graph of the derivative of the above function