Appendix C Theory
1.1 What is a function?
Theorem 1.1.36 Vertical line test
1.2 Linear and quadratic functions
Theorem 1.2.18 Properties of linear functions
Procedure 1.2.21 Equations of lines
Theorem 1.2.24 Zero product
Theorem 1.2.27 Quadratic formula
Procedure 1.2.29 Solving quadratic equations
Theorem 1.2.31 Quadratic factoring identities
Theorem 1.2.33 Properties of quadratic functions
1.3 Modeling with Linear and Quadratic Functions
Theorem 1.3.4 Linear function equivalence
Theorem 1.3.16 Height of falling object
1.4 Algebraic functions
Theorem 1.4.5 Power identities
Corollary 1.4.8 Radical identities
1.5 Algebraic functions: continued
Theorem 1.5.10 Roots of polynomials
Procedure 1.5.11 Polynomial long division
Theorem 1.5.15 Integer root theorem
Procedure 1.5.18 Factoring polynomials and finding roots
Theorem 1.5.21 Factoring tools
Procedure 1.5.22 Solving polynomial equations
Corollary 1.5.26 Polynomial equality
Theorem 1.5.27 Quotients of functions
Corollary 1.5.28 Domain and zeros of rational functions
1.6 Transformations and symmetry
Theorem 1.6.14 Function transformations
Theorem 1.6.33 Vertex form
1.8 Solving inequalities
Theorem 1.8.1 Elementary inequality rules
Theorem 1.8.8 Sign of a product
Theorem 1.8.11 Constancy of sign: algebraic functions
Procedure 1.8.12 Sign diagram
Theorem 1.8.15 Inequalities and squares
1.10 Piecewise-defined functions and absolute value
Theorem 1.10.14 Absolute value function is algebraic
Theorem 1.10.16 Properties of absolute value
Procedure 1.10.21 Graphing \(f(x)=\abs{g(x)}\)
1.11 Introduction to limits
Theorem 1.11.15 One-sided limits test
1.12 Limit rules
Theorem 1.12.1 Constant and identity function formulas
Theorem 1.12.2 Limit rules
Theorem 1.12.6 Polynomial and rational function evaluation
Theorem 1.12.11 Absolute value evaluation
1.14 Limits: formal definition
Procedure 1.14.10 Epsilon-delta proof for limits
Theorem 1.14.15 Triangle inequality
1.15 Continuity
Theorem 1.15.10 Continuous functions
Theorem 1.15.11 Continuity rules
Theorem 1.15.12 Continuity composition rule
Theorem 1.15.13 Algebraic functions are continuous
Theorem 1.15.14 Algebraic function evaluation
Theorem 1.15.18 Intermediate value theorem (IVT)
Theorem 1.15.23 Constancy of sign: continuous functions
1.16 Limits at infinity
Theorem 1.16.7 Limit at infinity formulas
1.17 Infinite limits
Theorem 1.17.5 Power functions and their reciprocals
Theorem 1.17.6 Infinite limit rules
Procedure 1.17.9 Infinite limit computation
Theorem 1.17.12 Limit at infinity: rational functions
1.19 Derivative: function
Theorem 1.19.9 Differentiable implies continuous
1.20 Derivative: rules
Theorem 1.20.1 Derivative formulas: constant and power functions
Theorem 1.20.5 Derivative rules
Theorem 1.20.9 Derivative: polynomials
1.21 Derivative as rate of change
Procedure 1.21.1 Rate of change interpretations
1.22 Chain rule
Theorem 1.22.1 Chain rule
Procedure 1.22.2 Chain rule
1.23 Implicit differentiation
Procedure 1.23.4 Implicit differentiation
1.24 Related rates I
Procedure 1.24.4 Related rates
2.1 Extreme value theorem
Theorem 2.1.1 Extreme value theorem
Theorem 2.1.11 Critical points and extreme values
Procedure 2.1.13 Extreme value theorem
2.2 Mean value theorem
Theorem 2.2.1 Rolle’s theorem
Theorem 2.2.3 Mean value theorem (MVT)
Corollary 2.2.5 Constant function characterization
Corollary 2.2.6 Functions with identical derivatives
Corollary 2.2.10 Taylor’s theorem (\(k=1\))
2.3 Linearization
Procedure 2.3.8 Linear approximation
2.4 Monotonicity and first derivative test
Theorem 2.4.2 Derivative and monotonicity
Procedure 2.4.5 Intervals of monotonicity
Theorem 2.4.6 First derivative test
Procedure 2.4.7 Classify critical points: first derivative test
2.5 Concavity and inflection points
Procedure 2.5.7 Concavity and inflection points
Theorem 2.5.10 Second derivative test
2.6 Curve sketching
Procedure 2.6.1 Curve sketching
2.7 Applied optimization
Procedure 2.7.4 Applied optimization
Procedure 2.7.7 Generalized extreme value theorem
2.8 Exponential functions
Theorem 2.8.21 Properties of Exponential Functions
