Double final exam sample: this set of sample questions is formed from combining together two previous real final exams (Winter 2016 and Summer 2021), and is slightly less than twice as long as the real final will be. However, the real final will cover exactly the same material. If you can successfully do all the questions on the sample in a reasonable amount of time, you will ace the final. Our job this semester is to get you to that point.
Review of basic integration. Techniques of integration, including partial fractions decomposition, u-substitution, and integration by parts.
Recommended reading: Bittinger, Chapter 5; OpenStax Strang (Vol 2) 1.1-1.5, 3.1
Lecture 1a: pre-class notes; post-class notes.
Lecture 1b: pre-class notes; post-class notes.
Lecture 1c: pre-class notes; post-class notes.
More detail on u-substitution and integration by parts. Numerical integration, improper integrals, volumes, and areas.
Recommended reading: Bittinger 5.5-5.9, 6.1; OpenStax Strang (Vol 2) 3.1, 2.1-2.4
Lecture 2a: pre-class notes; post-class notes.
Lecture 2c: pre-class notes; post-class notes. (note swapped order of b/c)
Lecture 2b: pre-class notes; post-class notes. (this material not on quiz 1, though it may show up later)
Review of Week 1 and Practice Quiz 1 (and Solutions)
Matrix operations. Systems of linear equations.
Applications: Leslie diagrams and population models.
Recommended reading: Bittinger 6.1-6.3; OpenStax Abrahamson 7.1-7.2, 7.5-7.7
Lecture 3a: pre-class notes; post-class notes.
Lecture 3b: pre-class notes; post-class notes.
Lecture 3c: pre-class notes; post-class notes. (barely got started, so we wrapped most of this lecture into next week's.
Covers material from Weeks 1 and 2.
Matrix inverses and determinants. Bases, eigenvalues and eigenvectors.
Applications: population growth and long-term growth rate.
Recommended reading: Bittinger 6.1-6.3; OpenStax Abrahamson 7.5-7.7
Lecture 4a: pre-class notes; post-class notes.
Lecture 4b: pre-class notes; post-class notes.
Lecture 4b: pre-class notes; post-class notes.
Review of linear algebra material. Some practice problems.
Functions of several variables. Multiple integration. Partial derivatives. Minimum-Maximum problems.
Applications: body surface area, wind speed of tornado, optimization problems.
Recommended reading: Bittinger 7.1, 7.2, 7.3, 7.5; OpenStax Strang (Vol 3) 4.1, 4.3, 4.7, 5.1.
Lecture 5a: pre-class notes; post-class notes.
Lecture 5b: pre-class notes; post-class notes.
Lecture 5c: pre-class notes; post-class notes.
Lecture 5d: pre-class notes; post-class notes.
Review of Week 4. Also, Practice Quiz 2 (and Solutions (corrected 2023-02-13))
Elements of regression analysis. Method of least squares. Best fit line. Quadratic, exponential, and power dependencies.
Applications: Experimental Data Fitting.
Lecture 6a: pre-class notes; post-class notes.
Lecture 6b: pre-class notes; post-class notes. Google Colab
Lecture 6c: pre-class notes; post-class notes. Google Colab
Covers material from Lecture sets 3 and 4.
Integrals as general and particular solutions. Pure time and separable first order differential equations. Exact differential equations.
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 7a: pre-class notes; post-class notes.
Lecture 7b: pre-class notes; post-class notes.
Lecture 7c: pre-class notes; post-class notes.
Lecture 7d: pre-class notes; post-class notes.
Review of Weeks 5 and 6. Also, Practice Quiz 3 / Solutions / Solutions as IPython notebook
Covers material from Weeks 5 and 6
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.
Lecture 8b: pre-class notes; post-class notes.
Lecture 8c: pre-class notes; post-class notes.
Review of Weeks 7-8. Practice Quiz 4 / Solutions
Higher-order differential equations with constant coefficients. Homogeneous and non-homogeneous equations.
Applications: mechanical and electrical vibrations, parallel reactions.
Note: a few slides near the end of 9b and 9c were added after lecture
Lecture 9a: pre-class notes; post-class notes.
Lecture 9b: pre-class notes; post-class notes.
Lecture 9c: pre-class notes; post-class notes.
Covers material from Weeks 7 and 8
Systems of autonomous differential equations and stability of equilibria using qualitative analysis.
Lecture 10a: pre-class notes; post-class notes.
Lecture 10b: pre-class notes; post-class notes.
Lecture 10c: pre-class notes; post-class notes.
Review of Weeks 9 and 10. Practice Quiz 5 / Solutions
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.
Lecture 11b: pre-class notes; post-class notes.
Lecture 11c: pre-class notes; post-class notes.
Covers material from Weeks 9 and 10
Applications: Pandemic modelling
Lecture 12a: pre-class notes; post-class notes.
Lecture 12b: pre-class notes; post-class notes.
Lecture "13" (Review): pre-class notes; post-class notes.
Review of Weeks 11 and 12.
The final examination will be cumulative and cover all the material from the semester, at a faster pace than in the quizzes.