PUMP Level 2 focuses on reading and understanding definitions and theorems, and writing basic mathematical proofs. Within each topic, students will see definitions, theorems, ways to use each, and basic proofs. There will be plenty of opportunities to write proofs in the style of each topic, explore the topic at a higher level of rigour than before, and receive feedback.
Topics include:
- Symbolic Logic and Understanding Theorems: Basics of first-order logic, well formed formulas, and quantifiers; Translating between mathematical theorems as logical statements; Implications, the contrapositive, converse, and inverse; Basic proof techniques including direct proof, proof by contradiction, and proving the contrapositive.
- Polynomials and Other Functions: Factoring polynomials of degree two, three, and higher; Factor Theorem; Simplifying rational expressions; Domain/Range; Injective/Surjective properties; Graphing and basic transformations.
- Planar and 3-D Geometry: Equations of lines and planes; Normal and tangent lines; Triangles, circles, rectangles; Pythagorean Theorem, distance formula between points, a point and a line, planes; Adding/Subtracting vectors algebraically and geometrically.
- Trigonometry: Definition of radians; Converting between degrees and radians - how to do it and motivation; Trigonometric rations; inverse trigonometric rations and right-angle triangle geometry; The unit circle; Sine and Cosine laws; Techniques for proving trigonometric identities; Proofs of the double/half angle formulas and adding/subtracting formulas; Graphs of trigonometirc functions and their tranformations; Applications.
- Exponential and Logarithmic Functions: Definitions of "e", "In", and "log"; Exponential growth, decay, and modelling real-world problems; Inverse functions and solving exponential/logarithmic equations; Properities of logarithmic and exponential functions and their graphs.