MATA35 - Calculus II for Biological Sciences - Summer 2021

Week 1: Friday, May 7 and Wednesday, May 12

Techniques of integration, including partial fractions decomposition, u-substitution, and integration by parts.

Recommended reading: Bittinger, Chapter 5

Lecture 1a Video (YouTube/UofTMyMedia) - Notes

Recommended problems (Friday): Bittinger, Exercise Set 5.1, pg 345.

Lecture 1b/c pre-class notes

Lecture 1b Video (YouTube/UofTMyMedia) - Notes

Lecture 1c Video (YouTube/UofTMyMedia) - Notes

Recommended problems (Wednesday): Bittinger, Exercise Set 5.2, pg 359.

Recommended problems (Wednesday): Bittinger, Exercise Set 5.5, pg 385.

Recommended problems (Wednesday): Bittinger, Exercise Set 5.6, pg 393.

Recommended problems (partial fractions): https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracdirectory/PartialFrac.html#PROBLEM%208

Week 2: Friday, May 14 and Wednesday, May 19

Repeat of u-substitution and integration by parts. Numerical integration, improper integrals, volumes, and areas.

Lecture 2a (review of u-substitution and integration by parts; numerical integration): pre-class notes; post-class notes; YouTube/UofTMyMedia

• Recommended problems: https://www.integral-calculator.com/
• Recommended problems (u-substitution): Bittinger, Exercise Set 5.5, pg 385 (again)
• Recommended problems (integration by parts): Bittinger, Exercise Set 5.6, pg 393 (again)
• Recommended problems (numerical integration): Bittinger, Exercise Set 5.7, problems 43-53, pg. 405

Lecture 2b (volume and area integrals): pre-class notes; post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 5.8, pg 411

Lecture 2c (improper integrals): pre-class notes; post-class notes; YouTUbe/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 5.9, pg 420

If time permits: Lecture 3a (introduction to matrices): pre-class notes; post-class notes; YouTube/UofTMyMedia (incomplete, as we didn't finish these slides, and we'll finish them next week)

• Recommended problems: Bittinger, Exercise Set 6.1, pg 440

Week 3: Wednesday, May 26 and Friday, May 28

Matrix operations. Systems of linear equations. Matrix inverses and cofactor expansions.
Applications: Leslie diagrams and population models.

Lecture 3a, part 2 (matrix algebra): pre-class notes; post-class notes; YouTube/UofTMyMedia (slides extended from last week's 3a slides)

• Recommended problems: Bittinger, Exercise Set 6.1, pg 440

Lecture 3b (Solving systems of linear equations, reduced row echelon form, Gaussian elimination): pre-class notes; post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 6.2, pg 455

Lecture 3c (Matrix inverses and determinants): pre-class notes; post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 6.3, pg 469

Week 4: Wednesday, June 2 and Friday, June 4

Eigenvalues and eigenvectors.
Applications: population growth and long-term growth rate.

Lecture 4a: pre-class notes; post-class notes; YouTube/UofTMyMedia

Lecture 4b: pre-class notes; post-class notes; YouTube/UofTMyMedia

Lecture 4c: pre-class notes; post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 6.4, pg. 484

Week 5: Wednesday, June 9 and Friday, June 11

Functions of several variables. Multiple integration. Partial derivatives. Minimum-Maximum problems.
Applications: body surface area, wind speed of tornado, optimization problems.

Lecture 5a (multivariable functions): pre-class notes;post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 7.1, pg. 507

Lecture 5b (partial derivatives): pre-class notes;post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 7.2, pg. 517

Lecture 5c (multiple integration): pre-class notes;post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 7.5, pg. 539

Lecture 5d (maximums and minimums): pre-class notes;post-class notes; YouTube/UoftMyMedia

• Recommended problems: Bittinger, Exercise Set 7.3, pg. 525

Week 6: Wednesday, June 16 and Friday, June 18

Elements of regression analysis. Method of least squares. Best fit line. Quadratic, exponential, and power dependencies.
Applications: Experimental Data Fitting.

Lecture 6a (introduction to regression analysis): pre-class notes; post-class notes; YouTube/UofTMyMedia

Lecture 6b (Python regression practicum): pre-class notes; Google Colab; YouTube/UofTMyMedia

Lecture 6c (Multilinear and nonlinear regression): pre-class notes; Google Colab; post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 7.4, pg. 533

Week 7.1: Wednesday, June 30

Integrals as general and particular solutions. Pure time and separable first order differential equations. Linear first order differential equations exact differential equations.

Lecture 7a (introduction to ODEs): pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 8.1, pg. 550, problems 1-34, 39-46

Lecture 7b (pure-time and separable): pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 8.4, pg. 581

Lecture 7c (exact differential equations): pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: None, because we didn't finish the lecture

Week 7.2: Wednesday, July 7

Integrating factors and linear 1st-order ODEs. Autonomous differential equations. Direction fields and solution curves. Phase line.
Applications: general population models, logistic models, carrying capacity of populations.

Lecture 7d (integrating factors and linear 1st-order): pre-class notes/post-class notes; YouTube/UoFTMyMedia

• Recommended problems: Bittinger, Exercise Set 8.1, pg. 551
• Supplemental text and problems: Trench, Section 2.5-2.6

Lecture 7e (Autonomous ODEs, direction fields, phase line): pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 8.3, pg. 571

Week 8: Friday, July 9 and Wednesday, July 14

Differential equations and mathematical modelling. Mixture problems. One and two-compartment models. Substitution methods. Numerical solutions.

Applications: pollution of Great Lakes, crop yield, mixing chemicals

Lecture 8a: pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 8.2, pg. 561, problems 35-52

Lecture 8b: pre-class notes/post-class notes; YouTube/UofTMyMedia

• Note that the recommended problems come from Tenenbaum and Pollard, and are sometimes more advanced than what you're expected to know for this class.
• Recommended problems: Tenenbaum and Pollard, Exercise 7, pg. 61.
• Recommended problems: Tenenbaum and Pollard, Exercise 8, pg. 69.
• Recommended problems: Tenenbaum and Pollard, Exercise 11, pg. 97, problems 3, 7, 8, 18.

Lecture 8c: pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 8.5, pg. 590

Week 9: Friday, July 16 and Wednesday, July 21

Higher-order differential equations with constant coefficients. Homogeneous and non-homogeneous equations.
Applications: mechanical and electrical vibrations, parallel reactions.

Lecture 9a: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 9b: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 9c: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 9d: pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 9.1, pg. 605
• Recommended problems: Bittinger, Exercise Set 9.2, pg. 611
• Review of complex arithmetic, including problems

Week 10: Friday, July 23 and Wednesday, July 28

Systems of autonomous differential equations and stability of equilibria using qualitative analysis.

Lecture 10a: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 10b: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 10c: pre-class notes/post-class notes; YouTUbe/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 9.3, pg. 619
• Recommended problems: Bittinger, Exercise Set 9.4, pg. 633

Week 11: Friday, July 30 and Wednesday, August 4

Systems of nonlinear differential equations and using linearization to analyze them.
Power series expansions, including Taylor Series and other series representation of functions. Linear and quadratic approximations.

Lecture 11a: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 11b: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 11c: pre-class notes/post-class notes; YouTube/UofTMyMedia

• Recommended problems: Bittinger, Exercise Set 9.5, pg. 643
• Extra resource for Taylor Series: https://people.math.sc.edu/girardi/m142/handouts/10sTaylorPolySeries.pdf

Week 12: Friday, August 6 and Wednesday, August 11

Applications: Pandemic modelling

If extra time, we will review the semester on Wednesday, Aug 11

Lecture 12a: pre-class notes/post-class notes; YouTube/UofTMyMedia

Lecture 12b: pre-class notes/post-class notes; YouTube/UofTMyMedia

Review/Office hours: YouTube/UofTMyMedia

"Lecture 13": extra review pre-class notes/post-class notes; YouTube/UofTMyMedia

• Taylor Series practice problems: Paul's Online Notes