STEM Program
Department of Math and Computer Science
Lansing Community College
Prof. Jing Wang, Ph. D.
MATH after 112
MATH 121
Calculus Sequence:
Computer Science
Precalculus I
MATH 151: Calculus I
CPSC 131: MATLAB
MATH 161: Honors Calculus I
CPSC 230: C++
Math 152: Calculus II
CPSC 231: Data
Structures
MATH 112
Intermediate
MATH 122
Algebra
Precalculus II
Math 162: Honors Calculus II
or
High School
Graduates
STEM Programs
MATH 126
Accelerated
Precalculus
Math 253: Calculus III
Math 254: Diff Equation
CPSC 260: Computer
Science Structures
Math 260: Linear Algebra
Degree/Curriculum
Mathematics
Engineering/Physics
Computer Science
Math 281: Honors Seminar
Calculus Projects
Problems adapted from Stewart’s
Calculus: Concepts and Contexts, 4e
Calculus I Project: Rates of Change
Purpose: Apply Differential Calculus to Authentic Problems
Theme: Blood Flow in Human Body
Example:
Using the Law of Laminar Flow by Poiseuille:
v( r ) =
P
R2 - r 2 )
(
4h l
www.nhlbi.nih.gov
Figure from Stewart’s Calculus: Concepts and Contexts, 4e
Where: v ( r ) = the velocity of the blood at the distance r from the
center.
We evaluate the rate of change of v with respect to r . (Also called the Velocity
Gradient)
Assignments
· Using the flux F = kR 4 (the volume of blood per unit time that flow past a given point),
Evaluate the relative change F with respect R :
DF
DR
Result: Realizing
»4
F
R
Interpret the real world meaning of this result.
Figure from Stewart’s Calculus: Concepts and Contexts, 4e
· Using the resistance of the blood,
L
R = C 4 (One of the Poiseuille’s Law)
r
Where L is the length of the blood vessel, r is the radius, C positive constant.
Figure from Stewart’s Calculus: Concepts and Contexts, 4e
Find the optimal branching angle (correct to nearest degree) when radius of the smaller blood vessel is two-thirds
the radius of the larger vessel.
Results from doing this project:
Students experience how differential calculus is applied to model and solve the blood flow problems and gain a better
understanding of the vascular system. They should also develop critical thinking and problem solving skills.
Calculus II Project: Applying Integrals
Calculus
Figure from Stewart’s Calculus: Concepts and Contexts, 4e
Calculus II Project: Applying Integral
Calculus
Calculus III Project: Modeling Tumors using Bumpy
and Wrinkled Spheres
Purpose: Using three dimensional calculus and graphing technique to model tumors
Theme: Bumpy and Wrinkled Spheres.
The members of the family of surfaces given in spherical coordinates by the equation
r = a + b sin(mq )sin(nf )
have been suggested as models for tumors.
www.valstarsolution.com/images/turb.jpg
In this project, students will use a math software program to analyze this family of surfaces.
Questions: What roles do the values m and n play in the shape of the surface?
Student Work
Zach Richardson
Math 253 Project
Fall 2012
Assignment 1
n = 10
n = 25
n=5
As n grows larger, more wedges protrude from the service of the sphere. The number of wedges appears to be equal to the value of n.
Assignment 2
m=3
m=7
m = 30
The value of m seems to shift horizontal sections of the sphere alternately so that they appear “off center”. As m grows larger, there are more such shifted sections.
Student Work
Zach Richardson
Math 253 Project
Fall 2012
Assignment 3
(2,3)
(8,4)
(6,5)
Rather than dividing the sphere vertically or horizontally, when both n and m vary the sphere becomes deformed by bumps which could be caused by the two types
of wedges intersecting. The number of bumps appears to be dependent on the product of n and m so if you know their values you can predict how many bumps there
will be.
Assignment 4
b = .4
b = .6
b = .8
b=1
As b grows larger, the space between the bumps, the valleys, becomes more pronounced and seems to cut deeper into the sphere.
Student Work
Zach Richardson
Math 253 Project
Fall 2012
Assignment 5
a = .5
a=5
a = 50
a = 500
As a grows larger, the valleys grow less noticeable and soon appear to disappear altogether. Also, as a increases so does the radius of the sphere. When a > 5b the valleys
are either gone or extremely shallow. When 5b > a the valleys become more noticeable as the difference between the two increases.
Result from doing this project:
Students should realize the importance of spherical coordinates. Gain experience
analyzing a family of functions. Appreciate the power of computer software
programs such as mathematica .