Course Information

Lectures: MWF at 9am in Rock Hall 194. Attendance not mandatory. Lectures are recorded: recordings are available on Canvas in the "My Media" section.

Textbook: Thomas' Calculus Early Transcendentals (15th Edition) by G. B. Thomas, , and J. Hass, Christopher E. Heil, P. Bogacki and M. Weir. We will cover Chapters 10, 12, 13 and 14.

Syllabus: The course syllabus for course guidelines. Please do read the syllabus carefully, many questions you might have are answered in the syllabus.

Calculus Room: The Calculus Room is open every weekday in 11am-6pm. This is a valuable resource at the Department of Mathematics. I strongly encourage MAT-21C students to use the Calculus Room.

Office Hours and Contact Information: The office hours of Professor Casals (casals--at--ucdavis.edu) are Mondays 3:30-4:30pm and Wednesday 3:30-4:30pm at MSB 3214. Always feel free to ask after class. The office hours for the instructors are:

• Kelli Loritsch (keloritsch--at--ucdavis.edu): Tuesdays 12-2pm at MSB 2141.
• Marco Ruiz (mru--at--ucdavis.edu): Mondays 4-5pm at MSB 2232.
• Ana Sammel (apsammel--at--ucdavis.edu): Mondays 12-1pm and Wednesdays 5-6pm at MSB 2117. (Zoom link option.)
• Haotian Sun (hatsun--at--math.ucdavis.edu): Thursdays 2-4pm at MSB 3137.
• Oliver Yau (opyau--at--ucdavis.edu): Thursdays 4-5pm at MSB 2131

Recitations: The hours for the recitations according to your section are:

• Section B01: taught by Ana Sammel on Tuesdays 6-7pm at Young 192.
• Section B02: taught by Ana Sammel on Tuesdays 7-8pm at Young 192.
• Section B03: taught by Marco Ruiz on Tuesdays 8-9pm at Young 192.
• Section B04: taught by Oliver Yau on Tuesdays 5-6pm at Young 192.
• Section B05: taught by Kelli Loritsch on Tuesdays 4-5pm at Young 192.
• Section B06: taught by Kelli Loritsch on Tuesdays 3-4pm at Young 192.
• Section B07: taught by Haotian Sun on Tuesdays 5-6pm at ART 217.
• Section B08: taught by Haotian Sun on Tuesdays 6-7pm at ART 217.

Piazza: The Piazza for this course is "MAT 021C B01-B08 SQ2023". You can access the Piazza forum through the UC Davis Canvas MAT-21C course. Piazza is daily monitored by the recitation instructors as follows: Sun (Mon), Sammel (Tue), Loritsch (Wed), Yau (Thu) and Ruiz (Fri).

Important Dates: First Day (Apr 3), First Midterm Test (Apr 28), Second Midterm Test (May 26), Final Exam (June 12).

Additional dates: Add Date (Apr 18), Drop Date (Apr 28), Last Day of Instruction (June 7).

Lecture Notes: Taken by K. Folkers, who kindly agreed to share them with the class.

Problem Sets: Weekly assignments due on Fridays at 9am, a combination of Gradescope and Webwork.

Problem Sets and Solutions

1. Problem Set 1: Due Friday Apr 14, available Apr 6.
   
   
2. Solutions Problem Set 1: Available Wed Apr 19.
   
   
3. Problem Set 2: Due Friday Apr 21, available Apr 13.
   
   
4. Solutions Problem Set 2: Available Tue Apr 25.
   
   
5. Practice Midterm: practice only, no due date, available Thu Apr 20.
   
   
6. Solutions Practice Midterm: Available Thu Apr 20.
   
   
7. Practice Midterm II: practice only, no due date, available Mon Apr 24.
   
   
8. Solutions Practice Midterm II: Available Mon Apr 24.
   
   
9. First Midterm.
   
   
10. First Midterm Solutions.
    
    
11. Practice Problems with solutions for Lecture 1 (Part II).
    
    
12. Practice Problems with solutions for Lecture 2 (Part II).
    
    
13. Practice Problems with solutions for Lecture 3 (Part II).
    
    
14. Practice Problems for Lecture 4 (Part II).
    
    
15. Solutions Practice Problems Lecture 4.
    
    
16. Practice Problems for Lecture 5 (Part II).
    
    
17. Solutions Practice Problems Lecture 5.
    
    
18. Practice Problems for Lecture 6 (Part II).
    
    
19. Solutions Practice Problems Lecture 6.
    
    
20. Practice Problems for Lecture 7 (Part II).
    
    
21. Solutions Practice Problems Lecture 7.
    
    
22. Practice Problems for Lecture 8 (Part II).
    
    
23. Solutions Practice Problems Lecture 8.
    
    
24. Practice Problems for Lecture 9 (Part II).
    
    
25. Solutions Practice Problems Lecture 9.
    
    
26. Practice Problems for Lecture 10 (Part II).
    
    
27. Practice Midterm Part 2: practice only, no due date, available Mon May 22.
    
    
28. Solutions Practice Midterm Part 2.
    
    
29. Practice Midterm II Part 2: practice only, no due date, available Mon May 22.
    
    
30. Solutions Practice Midterm II Part 2.
    
    
31. Midterm II.
    
    
32. Practice Problems for Lecture 11 (Part II).
    
    
33. Solutions Practice Problems Lecture 11.
    
    
34. Practice Problems for Lecture 12 (Part II).
    
    
35. Solutions Practice Problems Lecture 12.
    
    
36. Practice Final I: practice only, no due date, available Tue Jun 6.
    
    
37. Practice Final II: practice only, no due date, available Wed Jun 7.
    
    
38. Solutions Practice Final I.
    
    
39. Solutions Practice Final II.
    
    

Math Diaries

1. Monday Apr 3: Introduction to MAT-21C (Part I).
   The use of sequences and series in calculus. Examples.
   Approximating functions. Convergence and divergence.
   
   
   Lecture 1 Notes: Part 1 and Part 2.
   
   
   
2. Wednesday Apr 5: Sequences I.
   
   Definition of a sequence. Examples of convergent and divergent sequences.
   Decision table given a sequence and reasons for convergence and divergence.
   Definition of a limit of a sequence. Hierarchy of growth and example of use.
   
   Textbook Reading (Apr 5): Section 10.1 (up to Theorem 1, not included).
   Lecture 2 Notes: Part 1 and Part 2.
   
   
   
3. Friday Apr 7: Sequences II.
   
   Theorems to prove convergence: Theorems 1,2,3,4,5. Examples for Theorems 2 & 3.
   A reason for divergence: unboundedness. Monotone Convergence Theorem (Theorem 6).
   Recursive Sequences through example: how to prove convergence and find their limit.
   
   Textbook Reading (Apr 7): Section 10.1 (rest).
   Lecture 3 Notes: Part 1 and Part 2.
   
   
   
4. Monday Apr 10: Infinite Series.
   
   Recall main goal: approximating functions with polynomials. Exponential example.
   Meaning of an infinite series. Definition of partial sums associated to a sequence.
   Three Examples. Geometric Series and criteria for convergence. Test for divergence (Theorem 7).
   
   Textbook Reading (Apr 10): Section 10.2.
   Lecture 4 Notes: Part 1 and Part 2.
   
   
   
5. Wednesday Apr 12: Criteria for convergence of Infinite Series I.
   
   Consequence of Monotone Convergence Theorem. Example of reciprocals of squares.
   Integral Test. Application to convergence and divergence of p-harmonic series.
   Limit comparison test and two examples.
   
   Textbook Reading (Apr 12): Sections 10.3 and 10.4.
   Lecture 5 Notes: Notes.
   
   
   
6. Friday Apr 14: Criteria for convergence of Infinite Series II.
   
   Absolute convergence. Example of convergent but not absolutely convergent.
   Alternating Series Test and applications. Examples of alternating.
   Ratio test. Example (e.g. factorials). Root test. Example (e.g. powers of n).
   
   Textbook Reading (Apr 14): Sections 10.5 and 10.6.
   Lecture 6 Notes: Notes.
   
   
   
7. Monday Apr 17: Power Series.
   
   Definition of a power series. Example of geometric series.
   Convergence Theorem (Theorem 18). Radius of convergence and Corollary of Theorem 18.
   Differentiation and Integration of Power Series. Examples and applications.
   
   Textbook Reading (Apr 17): Sections 10.7.
   Lecture 7 Notes: Part 1 and Part 2.
   
   
   
8. Wednesday Apr 19: Taylor Series.
   
   Motivation for Taylor Series. Ideal algorithm.
   Taylor's Theorem (Theorem 23). Taylor's formula. Example with exponential.
   Example with finite radius of convergence. Comments of validity of algorithm.
   
   Textbook Reading (Apr 19): Sections 10.8 and 10.9.
   Lecture 8 Notes: Notes.
   
   
   
9. Wednesday Apr 21: Error bounds for Taylor approximations.
   
   More examples of Taylor series. Approximating a value via Taylor truncation.
   Error Formula in Taylor truncation of order n: Taylor's Theorem. Remainder Estimation (Theorem 24)
   Emphasis on unknown value c and upper boudns. Example of error bounds for cos(x).
   
   Textbook Reading (Apr 21): Section 10.9.
   Lecture 9 Notes: Part 1 and Part 2.
   
   
   
10. Monday Apr 24: Error bounds for Taylor approximations II.
    
    Recall of overall strategy to estimate errors. Theorem 24 again.
    Estimating sin(0.2) with Taylor series and bounds on errors (error decays factorially).
    Estimating ln(2) with Taylor series and error bounds (error decays as slow as 1/n).
    
    Textbook Reading (Apr 24): Section 10.9.
    Lecture 10 Notes: Notes.
    
    
    
11. Wednesday Apr 26: Review of material in Part I (Chapter 10).
    
    Textbook Reading (Apr 26): Sections 10.1 through 10.9.
    Review Lecture Notes: Notes.
    
    
    
12. Friday Apr 28: First Midterm Exam.
    
    Material covered by Midterm 1 (Apr 28): Sections 10.1 through 10.9.
    
    

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13. Monday May 1: Introduction to MAT-21 C (Part II).
    
    The real plane. Points in the plane. Verifying that a point belong to a set.
    Example of a line. General equation for a line. Steps to draw solutions to an equation.
    Example of a circle. General equation for a circle.
    
    Lecture 1 Notes: Notes for Lecture 1.
    Practice Problems (with solutions) for Lecture 1.
    
    
    
14. Wednesday May 3: Three-dimensional coordinate systems.
    
    Real three-dimensional space. Points in space. Drawing points in space.
    General equation for a plane. Steps to draw a plane and checking if points belong.
    Three different ways to describe a plane and how to transition between them. Examples.
    Distance between points. Comments on the distance function. Example of distances.
    
    Textbook Reading (May 3): Section 12.1.
    Lecture 2 Notes: Notes I and Notes II.
    Practice Problems (with solutions) for Lecture 2.
    
    
    
15. Friday May 5: Spheres, inequalities and intersections.
    
    Definition of a sphere in terms of radius and center: formula and geometric meaning.
    Example of sphere and deciding when a point is on a sphere. Interior and exterior of a sphere.
    Inequalities: geometric meaning. Analyzing equality and sampling points. Examples: half-spaces and balls.
    Intersections: geometric meaning of multiple equations. How to analyze intersections. Examples.
    
    Textbook Reading (May 5): Section 12.1.
    Lecture 3 Notes: Notes I and Notes II.
    Practice Problems for Lecture 3 (Part II).
    
    
    
16. Monday May 8: Vectors in three-dimensional space.
    
    Vectors between two points. Definition and notations for vectors. Examples.
    Length of a vector. Relation to distances. Practice examples.
    Two operations: addition of vectors and multiplication by a constant. Examples.
    Applications: finding the unit length vector in a direction and finding midpoints.
    Practice problems. General formula for midpoint.
    
    Textbook Reading (May 8): Section 12.2.
    Lecture 4 Notes: Notes I and Notes II.
    Practice Problems for Lecture 4.
    Solutions Problems for Lecture 4.
    
    
    
17. Wednesday May 10: Dot product.
    
    Definition of dot product. Example of computation.
    Projections and dot products: scalar components and projections of vectors.
    Exercise on how to compute the projection of a vector onto another vector.
    Dot product and angle formula. Example on how to compute the angle between two vectors.
    
    Textbook Reading (May 10): Section 12.3.
    Lecture 5 Notes: Notes I and Notes II.
    Practice Problems for Lecture 5.
    Solutions Problems for Lecture 5.
    
    
    
18. Friday May 12: Cross product.
    
    Shorthand notation: determinants. Definition cross-product.
    Example of computation. Cross product in terms of determinants.
    Properties: cross product is perpendicular and it measures area of parallelogram.
    Exercises computing areas of parallelograms. Cross-product angle formula.
    Real life applications: Lorentz force and Maxwell's Laws.
    
    Textbook Reading (May 12): Section 12.4.
    Lecture 6 Notes: Notes I and Notes II.
    Practice Problems for Lecture 6.
    Solutions Problems for Lecture 6.
    
    
    
19. Monday May 15: Lines and Planes in Space.
    
    Four descriptions of planes in space. Exercises on how to translate between them.
    Theorem: using the cross and dot products to obtain equation of a plane.
    Possible intersections of planes in space: parallel planes or intersection at lines.
    Using cross product to decide intersection and finding direciton of intersection line.
    
    Textbook Reading (May 15): Section 12.5.
    Lecture 7 Notes: Notes I and Notes II.
    Practice Problems for Lecture 7.
    
    
    
20. Wednesday May 17: Distances to Lines and Planes.
    
    Definition of distance from a point to a line. Formula for the distance from a point to a line.
    Detailed exercise computing the distance from a point to a line.
    Distance from a point to a plane. Formula for the distance from a point to a plane.
    Exercise computing the distance from a point to a plane.
    
    Textbook Reading (May 17): Section 12.5 (continued).
    Lecture 8 Notes: Notes I and Notes II.
    Practice Problems for Lecture 8.
    
    
    
21. Friday May 19: Trajectories in space.
    
    Definition of a trajectory: vectors with functions of t as components.
    Example of a particle moving in a helix trajectory in space.
    Theorem on parametrizing a trajectory along a line. Exercise finding such trajectories.
    Definition of velocity vector and speed. Example of velocity for helix trajectory.
    Definition of acceleration vector. Problem finding velocity and acceleration for a trajectory.
    
    Textbook Reading (May 19): Section 13.1.
    Lecture 9 Notes: Notes I and Notes II.
    Practice Problems for Lecture 9.
    
    
    
22. Monday May 22: Integrals and trajectories in space.
    
    How to solve trajectories from velocities or accelerations: differential equations.
    Problem recovering the trajectories from knowing velocities and just initial position.
    Problem recovering trajectories from knowing accelerations and just initial position and velocity.
    
    Textbook Reading (May 22): Section 13.1.
    Lecture 10 Notes: Notes I and Notes II.
    Practice Problems for Lecture 10.
    
    
    
23. Wednesday May 24: Review Midterm II.
    Lecture Notes: Notes.
    
    
24. Friday May 26: Midterm II.
    
    
25. Wednesday May 31: Partial Derivatives.
    
    Functions of two variables. Evaluating a function and the surface graph.
    Examples. Partial derivatives of a function of two variables. Conceptual meaning.
    How to compute partial derivatives. Several examples. Steps for finding critical points.
    
    Textbook Reading (May 31): Section 14.3.
    Lecture 11 Notes: Notes I and Notes II.
    Practice Problems for Lecture 11.
    
    
    
26. Friday Jun 2: Minima, saddles and maxima.
    
    Three types of critical points: minima, saddles and maxima. Running example.
    The computation of second derivatives: 4 types of 2nd derivatives. Examples.
    Evaluating 2nd derivatives. The characteristic polynomial of a critical point.
    Criteria for min, saddle or max in terms or roots of characteristic polynomial.
    
    Textbook Reading (Jun 2): Section 14.3 and 14.7.
    Lecture 12 Notes: Notes.
    Practice Problems for Lecture 12.
    
    
    
27. Monday Jun 5: Review Session I.
    Lecture 13 Notes: Notes I and Notes II.
    
    
    
28. Wednesday Jun 7: Review Session II.
    Lecture 14 Notes: Notes.
    
    
    
29. Monday Jun 12: Final Exam.
    
    

Further Resources

Books: For Part I of MAT-21C, Chapters 2 and 6 in "Understanding Analysis" by S. Abbott are a conceptual approach. Similarly, Chapters 2, 3 & 8 of "Real Analysis via Sequences and Series" by C.H.C. Little, K.L. Teo and B. van Brunt, Chapter 4 of "Calculus for Engineering Students" by J.M. Vaquero et al., Chapters 3 & 4 of "Real Analysis: A Long-Form Mathematics Textbook" by J. Cummings, and "Calculus" by M. Spivak can be alternative sources for Part I of MAT-21C. See "this thread" for a discussion on further sources.

Online resources: The calculus series in "3Blue1Brown" can serve as refresher for MAT-21A and MAT-21B. Our MAT-21C class starts at "Lecture on Taylor Series". The preceeding lectures is material from MAT-21A and MAT-21B. At the Kahn Academy, "Unit 10: Infinite sequences and series" is particularly relevant for Part I of our MAT-21C class. For Part II of MAT-21C, we cover material in line with "Units 1,2 & 3 in Multivariable calculus". The "Open Course Ware (MIT)" contains courses that align with MAT-21C. In particular, "Part 1C of Single Variable Calculus" is relevant for Part I of our MAT-21C, and "Parts I & II" of 18.02 Multivariable Calculus is relevant for Part II of our MAT-21C. There are many additional resources online, let me know if you are interested in more references for particular topic.