An Open Source Collection of Multivariable Calculus Problems
Monica VanDieren
Agenda
- PA GOAL
- Proposal
- Accessibility
- Equity
- Multivariable Calculus Problems
- Graphical
- Solar Energy
- Braess Paradox
- Computer Graphics
- Structural Engineering
2
PA Grants for Open and Affordable Learning
(PA GOAL)
Funded by the CARES Act, the program supports the use and development of zero-cost or open educational resources across Pennsylvania.
Image source: https://en.wikipedia.org/wiki/File:Congress_U.S_Capitol_Side_View.jpg
3
Multivariable Calculus Project
Proposal for a bank of homework problems
with an emphasis on
graphical problems and modern applications
3
Accessibility
Section 508 Revised Standards
- Alternative text and/or table of values included for all graphs.
- Pointers to tactile models of graphs.
-
Attributes other than color will be used for identification.
-
Still image alternatives provided for all animations.
4
Use the contour plot of \(f(x,y)\) to maximize \(f(x,y)\) subject to the graphed linear constraint \(g(x,y)=0\).
5
Peralta Online Equity Rubric
- Technology
- Student Resources and Support - links to hints/solutions
- Universal Design for Learning - multiple representations, student choice, multiple attempts
- Diversity and Inclusion - Content is appealing to instructors in a wide variety of educational settings
- Images and Representations
- Human Bias
- Content Meaning - students connect course content to their sociocultural backgrounds
- Connection and Belonging - activities deepen connections among class participants, and encourage students to connect to institution and the discipline more broadly.
6
Multivariable Calculus
Applications
- Computer Graphics
- Solar Energy
- Braess' Paradox
- Machine Learning
- Structural Engineering
7
Ray-tracing
8
Bezier Curves
9
Solar Energy Problems
- Vector Geometry
- Contour Plots
- Directional Derivatives
- Riemann Sums
Image sources: https://www.nrel.gov/gis/solar-resource-map.html
https://www.lrc.rpi.edu/programs/nlpip/lightinganswers/photovoltaic/14-photovoltaic-tilt-angle.asp
10
A
Z
11
A
Z
11
A
Z
Adding a new bridge/connector can actually cause more congestion and longer travel times.
11
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
B
C
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}=t_{BZ}=50+\phi\)
12
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}=t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
12
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}=t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
12
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}=t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
12
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}=t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
12
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
13
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
\(t_{AB} =10(2)\)
13
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
\(t_{AB} =10(2)\)
\(t_{BZ} =50+1\)
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
\(t_{AB} =10(2)\)
\(t_{BZ} =50+1\)
green travel time =72
13
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
\(t_{AB}=10*2\)
14
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
\(t_{AB}=10*2\)
\(t_{BC}=10+1\)
14
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
\(t_{AB}=10*2\)
\(t_{BC}=10+1\)
\(t_{CZ}=10*2\)
14
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
\(t_{AB}=10*2\)
\(t_{BC}=10+1\)
\(t_{CZ}=10*2\)
blue travel time =51
14
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
blue travel time =51
gray travel time =72
15
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
blue travel time =51
gray travel time =72
15
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72
blue travel time =51
gray travel time =72
Best case distribution of 3 cars, but unstable
Braess Example
Suppose that the travel times on each road depend on the number of cars \(\phi\) on the road. For the road connecting A to B, we'll denote the travel time \(t_{AB}(\phi)\).
A
Z
\(t_{AB}=t_{CZ}=10\phi\)
\(t_{BC}=10+\phi\)
\(t_{AC}= \) \(t_{BZ}=50+\phi\)
Assume we have 3 cars on the network
B
C
Travel times
green travel time =72 62
blue travel time =51 62
gray travel time =72 81
Best case distribution of 3 cars, but unstable
Braess Paradox works well as a
Group Project
- Individual "buy in" by each student computing one of several scenarios to determine optimal distributions
- Demonstrate how a problem might have different set-ups, solutions, and solution strategies
- Students reflect on how this paradox might occur in their neighborhoods or other non-vehicular contexts
- Discuss the assumptions and their relation to Google Maps and other real-time navigation programs
Problems due on the Contributed WeBWorK Directory
July 2022
vandieren@rmu.edu