1. Logic: Logic is the study of reasoning and argument structure. It involves understanding propositions, truth values, logical connectives (and, or, not), and methods of proof such as direct proof, contradiction, and contrapositive. 2. Sets and Counting: Sets are collections of distinct objects. Counting principles include the rule of sum, rule of product, permutations, combinations, and the pigeonhole principle. 3. Functions and Linear Models: A function relates inputs to outputs. Linear models describe relationships with equations of the form $y=mx+b$, where $m$ is the slope and $b$ the intercept. 4. Nonlinear Functions and Models: These include quadratic, exponential, logarithmic, and other functions that do not form a straight line. They model more complex relationships. 5. Introduction to the Derivative: The derivative measures the rate of change of a function. It is defined as $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$. 6. Techniques of Differentiation with Applications: Includes rules like product, quotient, and chain rule to find derivatives of complex functions. Applications include finding slopes, rates, and optimization. 7. Further Applications of the Derivative: Use derivatives to analyze increasing/decreasing behavior, find local maxima/minima, and solve real-world problems. Second Semester: 1. Systems of Linear Equations and Matrices: Solve multiple linear equations simultaneously using substitution, elimination, and matrix methods. 2. Matrix Algebra and Applications: Operations on matrices such as addition, multiplication, inverses, and determinants with applications in solving systems and transformations. 3. The Integral: The integral represents area under a curve and accumulation. Defined as $$\int_a^b f(x) dx$$. 4. Further Integration Techniques and Applications: Methods like substitution, integration by parts, partial fractions, and applications in area, volume, and physics. 5. Functions of Several Variables: Study functions with multiple inputs, partial derivatives, and gradients. 6. Trigonometric Models: Use sine, cosine, and other trig functions to model periodic phenomena like waves and oscillations.