Calculus: Single Variable Part 4 - Applications

About this course: Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this fourth part--part four of five--we cover computing areas and volumes, other geometric applications, physical applications, and averages and mass. We also introduce probability.

Created by:  University of Pennsylvania

• Taught by:  Robert Ghrist, Professor

  Mathematics and Electrical & Systems Engineering

Commitment 6-8 hours/week Language English How To Pass Pass all graded assignments to complete the course. Coursework

Each course is like an interactive textbook, featuring pre-recorded videos, quizzes and projects.

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University of Pennsylvania The University of Pennsylvania (commonly referred to as Penn) is a private university, located in Philadelphia, Pennsylvania, United States. A member of the Ivy League, Penn is the fourth-oldest institution of higher education in the United States, and considers itself to be the first university in the United States with both undergraduate and graduate studies.

Syllabus


WEEK 1


Computing Areas and Volumes



Having seen some calculus before, you may recall some of the motivations for integrals arising from area computations. We will review those classical applications, while introducing the core idea of this module -- a differential element. By computing area and volume elements, we will see how to tackle tough geometry problems in a principled manner.


6 videos, 2 readings, 4 practice quizzes expand

1. Reading: How to Get Started in this Course
2. Reading: How Grading Works
3. Video: Simple Areas
4. Practice Quiz: Challenge Homework: Simple Areas
5. Video: Complex Areas
6. Video: BONUS!
7. Practice Quiz: Challenge Homework: Complex Areas
8. Video: Volumes
9. Video: BONUS!
10. Practice Quiz: Challenge Homework: Simple Volumes
11. Video: Complex Volumes
12. Practice Quiz: Challenge Homework: Complex Volumes

Graded: Core Homework: Simple Areas
Graded: Core Homework: Complex Areas
Graded: Core Homework: Simple Volumes
Graded: Core Homework: Complex Volumes

WEEK 2


Other Geometric Applications



There's more to geometry than just area and volume! In this module, we will take things "to the next level", ascending to higher dimensions. Coming back to the 3-d world, we will return to problems of length and area, but this time in the context of curves and surfaces. As always, the emphasis will be on how to construct the appropriate differential element for integrating.


4 videos, 3 practice quizzes expand

1. Video: Volume & Dimension
2. Video: BONUS
3. Practice Quiz: Challenge Homework: Volume and Dimension
4. Video: Arclength
5. Practice Quiz: Challenge Homework: Arclength
6. Video: Surface Area
7. Practice Quiz: Challenge Homework: Surface Area

Graded: Core Homework: Volume and Dimension
Graded: Core Homework: Arclength
Graded: Core Homework: Surface Area

WEEK 3


Physical Applications



There is so much more to applications of integrals than geometry! So many subjects, from physics to finance, have, at heart, the need for setting up and computing definite integrals. In this short but intense module, we will cover applications including work, force, torque, mass, and present & future value.


3 videos, 2 practice quizzes expand

1. Video: Work
2. Practice Quiz: Challenge Homework: Work
3. Video: Elements
4. Video: BONUS!
5. Practice Quiz: Challenge Homework: Elements

Graded: Core Homework: Work
Graded: Core Homework: Elements

WEEK 4


Averages and Mass



There is a statistical aspect to integrals that has not yet been brought up in this course: integrals are ideal for computing averages. Motivated by physical problems of mass, centroid, and moments of inertia, we will cover applications of integrals to averages.


5 videos, 3 practice quizzes expand

1. Video: Averages
2. Practice Quiz: Challenge Homework: Averages
3. Video: Centroids
4. Video: BONUS!
5. Practice Quiz: Challenge Homework: Centroids
6. Video: Moments and Gyrations
7. Video: BONUS!
8. Practice Quiz: Challenge Homework: Moments and Gyrations

Graded: Core Homework: Averages
Graded: Core Homework: Centroids
Graded: Core Homework: Moments and Gyrations

WEEK 5


An Introduction to Probability



This capstone module gives a very brief introduction to probability, using what we know about integrals and differential elements. Beginning with common-sense uniform probabilities, we move on to define probability density functions and the corresponding probability element. Building on the physical intuition obtained from centers of mass and moments of inertia, we offer a unique perspective on expectation, variance, and standard deviation.


3 videos, 1 reading, 3 practice quizzes expand

1. Video: Fair Probability
2. Practice Quiz: Challenge Homework: Fair Probability
3. Video: Probability Densities
4. Practice Quiz: Challenge Homework: Probability Densities
5. Video: Expectation and Variance
6. Practice Quiz: Challenge Homework: Expectation and Variance
7. Reading: About the Chapter 4 Exam

Graded: Core Homework: Fair Probability
Graded: Core Homework: Probability Densities
Graded: Core Homework: Expectation and Variance
Graded: Chapter 4: Applications - Exam