Calculus

1. Preliminaries
   1. Functions and Graphs
      1. Types of Functions
         1. Linear functions and their graphs
         2. Quadratic functions and parabolas
         3. Polynomial functions and their general behavior
         4. Rational functions and asymptotic behavior
         5. Exponential and logarithmic functions and transformations
         6. Trigonometric functions and their periodic nature
         7. Inverse functions and their derivations
      2. Understanding Limits and Continuity
         1. Concept of approaching a value
         2. Understanding and identifying discontinuities
            1. Removable discontinuities
            2. Jump discontinuities
            3. Infinite discontinuities
         3. Continuous functions and the Intermediate Value Theorem
      3. Piecewise Functions
         1. Definition and examples
         2. Graphing piecewise functions
         3. Evaluating piecewise functions at given points
         4. Continuity at the boundaries of pieces
   2. Basic Algebra and Trigonometry Review
      1. Polynomial Functions
         1. Operations with polynomials (addition, subtraction, multiplication, division)
         2. Factorization techniques for polynomials
         3. Roots and the Fundamental Theorem of Algebra
         4. The Remainder and Factor Theorems
      2. Exponential Functions
         1. Laws of exponents
         2. Graphical representation of exponential growth and decay
         3. Applications of exponential functions
         4. Solving exponential equations
      3. Trigonometric Identities
         1. Basic identities (reciprocal, quotient, Pythagorean)
         2. Verifying identities
         3. Solving trigonometric equations using identities
         4. Angle sum and difference identities
         5. Double-angle and half-angle identities
   3. Introduction to Limits
      1. Definition and Concept of a Limit
         1. Formal (epsilon-delta) definition of a limit
         2. Intuitive understanding and graphical interpretation
      2. One-sided Limits
         1. Left-hand limits
         2. Right-hand limits
         3. Interpreting and comparing one-sided limits at a given point
      3. Infinite Limits and Limits at Infinity
         1. Behavior of functions as they approach infinity
         2. Horizontal asymptotes
         3. Vertical asymptotes and unbounded behavior
      4. Limit Laws and Properties
         1. Basic limit laws (sum, difference, product, and quotient)
         2. Limits of polynomials and rational functions
         3. Squeeze theorem
         4. Limits involving radicals and trigonometric functions