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

1. Technology
2. Student Resources and Support - links to hints/solutions
3. Universal Design for Learning - multiple representations, student choice, multiple attempts
4. Diversity and Inclusion - Content is appealing to instructors in a wide variety of educational settings
5. Images and Representations
6. Human Bias
7. Content Meaning - students connect course content to their sociocultural backgrounds
8. 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

Braess Paradox

A

Z

https://www.maa.org/press/periodicals/convergence/braess-paradox-in-city-planning-a-mini-primary-source-project-for-multivariable-calculus-students

11

Braess Paradox

A

Z

https://www.maa.org/press/periodicals/convergence/braess-paradox-in-city-planning-a-mini-primary-source-project-for-multivariable-calculus-students

11

Braess Paradox

A

Z

Adding a new bridge/connector can actually cause more congestion and longer travel times.

https://www.maa.org/press/periodicals/convergence/braess-paradox-in-city-planning-a-mini-primary-source-project-for-multivariable-calculus-students

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

https://www.maa.org/press/periodicals/convergence/braess-paradox-in-city-planning-a-mini-primary-source-project-for-multivariable-calculus-students

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

https://openwebwork.org/

https://c3d.libretexts.org/CalcPlot3D/index.html