BC Midterm Review Topics

Unit 1: Functions, Limits, and Continuity

  1. The Cartesian Plane and Functions
  • Absolute Value
  • Symmetry
  • Even and Odd Functions
  • Domain and Range
  1. Limits and Their Properties
  • One and two-sided limits
  • Squeeze Theorem (look at proof of and confirm graphically)
  • Calculate limits of polynomial and rational functions graphically, analytically, and by using a table of values
  • Infinite Limits and Limits at Infinity
  • Horizontal and Vertical Asymptotes of a Function
  1. Continuity
  • Develop definition of continuity
  • Continuous and Discontinuous Functions
  • Removable, Non-removable, Infinite, Jump Discontinuity (discuss y = int(x) )
  • Intermediate Value Theorem
  • Extreme Value Theorem (introduce and revisit in Unit 3)

Unit 2: Differentiation

  1. Rates of Change of a Function
  • Average Rate of Change
  • Tangent Line to a Curve
  • Instantaneous Rate of Change
  1. The Derivative
  • Definition of the Derivative (difference quotient)
  • Derivative at a Point
  • One-sided derivatives
  • Numerical Derivative of a Function (using nDeriv on the calculator)
  • Graphing f`(x) using the graph of f(x)
  • The Derivative as a Function
  • Graphing the Derivative (explore using Y2 = nDeriv(Y1,X,X) on the calculator)
  1. Differentiability
  • Define differentiability
  • Differentiability and Continuity
  • Local Linearity
  • Symmetric Difference Quotient
  • Intermediate Value Theorem for Derivatives
  1. Differentiation Rules
  • Sum and Difference Rules
  • Constant, Power, Product, and Quotient Rules
  • Chain Rule
  • Higher Order Derivatives
  1. Applications of the Derivative
  • Position, Velocity, Acceleration, and Jerk (show that vertical motion formulas from physics are related through differentiation)
  • Particle Motion
  1. Implicit Differentiation
  2. y` notation
  3. Expressing derivatives in terms of x and y.
  4. Related Rates

Unit 3: Applications of Differentiation

  1. Extema and Related Theorems
  • Absolute Extrema
  • Extreme Value Theorem
  • Relative Extrema
  • Critical Values
  • Rolle’s Theorem
  • Mean Value Theorem
  1. Determining Function Behavior
  2. Increasing and Decreasing Functions
  3. First Derivative Test to Locate Relative Extrema
  4. Concavity
  5. Using the Second Derivative to Locate Points of Inflection
  6. Second Derivative Test to Locate Relative Extrema
  7. L’Hôpital’s Rule
  8. The Relationship Between f(x), f`(x), and f``(x).
  9. Optimization
  10. Differentials
  11. Local Linearity
  12. Tangent Line Approximation

Unit 4: Integration

  1. Antiderivatives
  • Indefinite Integrals
  • Initial Conditions and Particular Solutions
  • Basic Integration Rules
  1. Area Under a Curve
  • RAM (Rectangle Approximation Method)
  • Riemann Sums
  • Left sums, right sums, midpoint sums
  • Definite Integrals
  1. The Fundamental Theorem of Calculus
  • FTC Part 1
  • Numerical Integral (using fnInt on the calculator)
  • FTC Part 2
  • Mean Value Theorem for Integrals
  • Average Value of a Function
  • Integration by Substitution
  • Integrating with Respect to the x and y axes
  1. Trapezoidal Rule

Unit 5: Transcendental Functions

A.Trigonometric Functions

  • Differentiation
  • Integration

B.Inverse Trigonometric Functions

  • Differentiation
  • Integration
  • General Rule for Derivative of an Inverse Function

C.Exponential and Logarithmic Functions

Unit 6: Advanced Integration

  1. Substitution with Complete Change of Variable
  2. Integration by Parts
  3. Partial Fractions (non-repeating linear factors only)
  4. Improper Integrals

Unit 7: Differential Equations

  1. Slope Fields
  2. Euler’s Method
  3. Separable Differentiable Equations
  4. Exponential Growth and Decay (including their use in modeling)
  5. Logistic Differential Equations (including carrying capacity and their use in modeling)