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How to use this page:
- Click the chapter we are working on.
- Review notes/examples as you do homework in an organized handwritten notebook.
- Use review sheets and ask questions, then try test preps to check understanding.
- Use exam review archive before tests working through several old exams.
Reference
Chapter 12 - Vectors and Geometry in 3D
Extra Practice & Review
Test Preps & Solutions
Review Sheets & Study Guides
Chapter 12 Exam Review
12.1 - Coordinates and Spheres
12.2 - Vectors
Desmos & Visualizations
Supplemental Videos
Try the homework first, then use these if you want another explanation or example.
The Vector Concept (12 min)
Scalar Multiplication (7 min)
Vector Addition (12 min)
2D Vectors: Angles & Slope (10 min)
The Vector Concept (12 min)
Scalar Multiplication (7 min)
Vector Addition (12 min)
2D Vectors: Angles & Slope (10 min)
12.3 - Dot Products and Projections
Desmos & Visualizations
Supplemental Videos
Try the homework first, then use these if you want another explanation or example.
Dot Product Concept, Proof & Key Formula (15 min)
Angle of Intersection (10 min)
Dot Products & Projections (12 min)
Projections, Projections and More Projections (8 min)
Dot Product Concept, Proof & Key Formula (15 min)
Angle of Intersection (10 min)
Dot Products & Projections (12 min)
Projections, Projections and More Projections (8 min)
12.4 - Cross Product and Area
Desmos & Visualizations
Supplemental Videos
Try the homework first, then use these if you want another explanation or example.
Cross Product Concept, Formula & Computation (13 min)
Right-Hand Rule (7 min)
Right-Hand Rule — Goofy Version
Cross Product Length & Parallelogram Area (14 min)
Cross Product Concept, Formula & Computation (13 min)
Right-Hand Rule (7 min)
Right-Hand Rule — Goofy Version
Cross Product Length & Parallelogram Area (14 min)
Optional Extension
12.5 - Lines and Planes
Lecture Outline — Lecture 1
|
Word
Lecture Outline — Lecture 2 (Problem Session) | Word
12.5 Review Sheet — Key Ideas and Formulas
Lecture Outline — Lecture 2 (Problem Session) | Word
12.5 Review Sheet — Key Ideas and Formulas
Desmos & Visualizations
Supplemental Videos
What are you GIVEN, and what do you WANT?
These four short videos develop this essential problem-solving skill.
Lines — What Do We WANT? (6 min)
Planes — What Do We WANT? (8 min)
What Are We GIVEN? (8 min)
Line Notation — Vector and Parametric (5 min)
Lines — What Do We WANT? (6 min)
Planes — What Do We WANT? (8 min)
What Are We GIVEN? (8 min)
Line Notation — Vector and Parametric (5 min)
Extra Practice & Problem-Solving Guides
12.6 - Quadric Surfaces
Start Here — Watch Before Homework
Please watch these first two videos before starting the homework!
They will help you quickly recognize and remember quadric surfaces.
Introduction to 12.6 — Naming Shapes (12 min)
My "Thumb" Method for Naming Shapes (11 min)
Introduction to 12.6 — Naming Shapes (12 min)
My "Thumb" Method for Naming Shapes (11 min)
More Details — Paraboloids & Hyperboloids
Optional Visualizations & Drawing
Explore these if you want to develop your 3D visualization skills
or just have some fun with the mathematics.
Freehand Drawing of Quadric Surfaces
More Detailed Drawing Tips
Cool Animations & Interesting Facts
Dr. Loveless's Calculus Art Gallery
3D-printed models of many surfaces can also be found in the Math Study Center (CMU B-014).
Freehand Drawing of Quadric Surfaces
More Detailed Drawing Tips
Cool Animations & Interesting Facts
Dr. Loveless's Calculus Art Gallery
3D-printed models of many surfaces can also be found in the Math Study Center (CMU B-014).
Chapter 13 - Curves in 3D and Motion in Space
Extra Practice & Review
Test Preps & Solutions
Review Sheets
13.1 - Space Curves
Start Here — 3 Videos
Please watch these three introductory videos,
especially the examples involving intersections.
3D Curves: Notation & Visualization (10 min)
Curve–Surface Intersections (10 min)
Intersections of Two Curves (10 min)
3D Curves: Notation & Visualization (10 min)
Curve–Surface Intersections (10 min)
Intersections of Two Curves (10 min)
13.2 - Vector Calculus
Videos & More Help
Desmos & Visualizations
13.3 - Measurement Tools for Curves
Start Here — 4 Videos
Please watch these four videos introducing arc length,
the TNB frame, and curvature.
Three Curve Measurement Tools (3 min)
Tool 1 — Arc Length (8 min)
Tool 2 — TNB Frame (15 min)
Tool 3 — 3D Curvature (10 min)
Three Curve Measurement Tools (3 min)
Tool 1 — Arc Length (8 min)
Tool 2 — TNB Frame (15 min)
Tool 3 — 3D Curvature (10 min)
Desmos & Visualizations
13.4 - Acceleration and the TNB-Frame
Start Here — 2 Videos
Desmos & Visualizations
Optional Fun — Fireworks
Model fireworks in 3D by assuming gravity is
the only acceleration (ignoring air resistance).
Fireworks — 3D Projectile Motion (2 min)
Fireworks — 3D Projectile Motion (2 min)
Chapter 14 - Partial Derivatives and Optimization
Extra Practice & Review
Test Preps & Solutions
TP - 14.1 & 14.3/14.4 (Older Curriculum) — Solutions
TP - 14.7 — Solutions
Older materials may use different section numbering; check current course topics.
TP - 14.7 — Solutions
Older materials may use different section numbering; check current course topics.
Review Sheets & Study Guides
14.1 - Multivariable Functions and Level Curves
Start Here — 3 Videos
Watch these for domains, contour maps, and homework examples.
Functions of Two Variables & Domains (10 min)
Contour Maps & Level Curves (7 min)
14.1 Homework Walkthrough (5 min)
Functions of Two Variables & Domains (10 min)
Contour Maps & Level Curves (7 min)
14.1 Homework Walkthrough (5 min)
Desmos & Visualizations
14.3 - Partial Derivatives
Start Here — 4 Videos
Desmos & Visualizations
14.4 - Tangent Planes and Linear Approximations
Videos & Homework Help
Desmos & Visualizations
Older Notes & Practice
14.5 - The Multivariable Chain Rule
Review & Problem-Solving
Chain Rule & Gradient Review — Page 1 (Chain Rule)
Tree diagrams: multiply derivatives along each path and add across paths. The review's second page previews 14.6.
Tree diagrams: multiply derivatives along each path and add across paths. The review's second page previews 14.6.
14.6 - Directional Derivatives and Gradients
Updated lecture materials and visualizations coming later. | Older Gradient Review (Page 2)
14.7 - Critical Points and Extrema
14.8 - Lagrange Multipliers and Constrained Optimization
Updated lecture materials and practice coming later.
Chapter 15 - Double Integrals and Polar Coordinates
Lecture Outlines and Old Notes:
- 15.1 - Double Integrals over Rectangles: Lecture Outline | Notes
- 15.2 - Double Integrals over General Regions: Lecture Outline | Notes
- Polar Coordinates Review: Lecture Outline | Notes
- 15.3 - Double Integrals in Polar Coordinates: Lecture Outline | Notes
- 15.4 - Center of Mass: Lecture Outline | Notes
Taylor Polynomials and Series
Lecture Outlines and Old Notes:
- TN 1 - Tangent Lines and Intro to Error Bounds: Lecture Outline | Notes
- TN 2-3 - Higher Order Taylor Polynomials and Taylor's Inequality: Lecture Outline | Notes
- TN 4 - Taylor Series: Lecture Outline | Notes
- TN 5 - Manipulating Taylor Series: Lecture Outline | Notes
- TP - TN 1-3 - solns - video
- TP - TN 4-5 - solns