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Chapter 2: Limits and Continuity

Section 2.1: Rates of Change and Tangents to Curves

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. slope of a line
   2. point-slope of a line
2. procedural knowledge
   1. how to calculate slope when given:
      • two points
      • a function and an interval for the independent variable
      • a function and two values for the independent variable
   2. how to evaluate trigonometric ratios of any multiple of the reference angles
   3. how to write the equation of the line tangent given the slope and a point
3. conditional knowledge
   1. none

Learning Goals

1. declarative knowledge (definitions)
   1. average speed
   2. average rate of change
   3. tangent line to a curve
   4. secant line to a curve
2. procedural knowledge
   1. how to estimate the slope of a tangent line to a curve
   2. how to write the equation of the line tangent to the curve at a point
3. conditional knowledge
   1. distinguish between calculating the slope between two points and estimating the slope of a curve at a point

Section 2.2: Limit of a Function and Limit Laws

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. factors of a polynomial
   2. rationalizing the denominator of a rational expression
   3. difference quotient
2. procedural knowledge
   1. how to factor a polynomial
   2. how to rationalize the denominator of a rational expression
   3. how to rationalize the numerator of a rational expression
   4. how to evaluate trigonometric ratios of any multiple of the reference angles
   5. how to evaluate difference quotient
3. conditional knowledge
   1. none

Learning Goals

1. declarative knowledge (definitions)
   1. limit
   2. Limit Laws
   3. The Sandwich Theorem (a.k.a., The Squeeze Theorem)
2. procedural knowledge
   1. how to use Direct Substitution to evaluate the limit of a function
   2. how to graph to evaluate the limit of a function
   3. how to reduce a rational expression to evaluate the limit of a function
   4. how to rationalize the denominator of a rational expression to evaluate the limit of a function
   5. how to rationalize the numerator of a rational expression to evaluate the limit of a function
   6. how to use the Limit Laws to evaluate the limit of a function
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits
   2. know why the appropriate methods for evaluating limits are needed
   3. know when and why a limit does not exist

Section 2.4: One-Sided Limits and Limits at Infinity

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. Section 2.2
   2. factors of a polynomial
   3. rationalizing the denominator of a rational expression
   4. difference quotient
   5. piecewise-defined functions
2. procedural knowledge
   1. how to factor a polynomial
   2. how to rationalize the denominator of a rational expression
   3. how to rationalize the numerator of a rational expression
   4. how to evaluate trigonometric ratios of any multiple of the reference angles
   5. how to evaluate difference quotient
   6. how to sketch a graph of a piecewise-defined function
3. conditional knowledge
   1. none

Learning Goals

1. declarative knowledge (definitions)
   1. two-sided limit
   2. right-hand limit
   3. left-hand limit
   4. limit at infinity
   5. horizontal asymptote
   6. slant asymptote
2. procedural knowledge
   1. how to use Direct Substitution to evaluate the limit of a function
   2. how to graph to evaluate the limit of a function
   3. how to reduce a rational expression to evaluate the limit of a function
   4. how to rationalize the denominator of a rational expression to evaluate the limit of a function
   5. how to rationalize the numerator of a rational expression to evaluate the limit of a function
   6. how to evaluate the limit of a function as x approaches ±∞
   7. how to use the Limit Laws to evaluate the limit of a function
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits
   2. know why the appropriate methods for evaluating limits are needed
   3. know two-sided limits relate to the one-sided limits
   4. know when and why a limit does not exist
   5. know when and why a limit is represented as ±∞
   6. identify where limits exist or do not exist for functions

Section 2.5: Infinite Limits and Vertical Asymptotes

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. Section 2.2
   2. Section 2.4
   3. factors of a polynomial
   4. rationalizing the denominator of a rational expression
2. procedural knowledge
   1. how to factor a polynomial
   2. how to rationalize the denominator of a rational expression
   3. how to rationalize the numerator of a rational expression
   4. how to evaluate trigonometric ratios of any multiple of the reference angles
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits

Learning Goals

1. declarative knowledge (definitions)
   1. infinite limits
   2. vertical asymptote
2. procedural knowledge
   1. how to use Direct Substitution to evaluate the limit of a function
   2. how to graph to evaluate the limit of a function
   3. how to reduce a rational expression to evaluate the limit of a function
   4. how to rationalize the denominator of a rational expression to evaluate the limit of a function
   5. how to rationalize the numerator of a rational expression to evaluate the limit of a function
   6. how to identify the vertical asymptotes of a function
   7. how to sketch functions from a set of limits
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits
   2. know why the appropriate methods for evaluating limits are needed
   3. know two-sided limits relate to the one-sided limits
   4. know when and why a limit does not exist
   5. know when and why a limit is represented as ±∞
   6. identify where limits exist or do not exist for functions

Section 2.6: Continuity

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. Section 2.2
   2. Section 2.4
   3. Section 2.5
   4. factors of a polynomial
   5. roots of a function (a.k.a., zeros)
2. procedural knowledge
   1. how to factor a polynomial
   2. how to evaluate trigonometric ratios of any multiple of the reference angles
   3. how to evaluate the limit of a function
   4. how to sketch a graph of a piecewise-defined function
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits

Learning Goals

1. declarative knowledge (definitions)
   1. continuity at a point
   2. continuous from the left
   3. continuous from the right
   4. continuous on an interval
   5. point of discontinuity
   6. removable discontinuity (a.k.a., hole)
   7. jump discontinuity
   8. infinite discontinuity
   9. oscillating discontinuity
   10. properties of continuous functions
   11. Intermediate Value Theorem (IVT)
2. procedural knowledge
   1. use the definition of continuity at a point
   2. identify points of discontinuity from a graph of a function
   3. identify points of discontinuity of a function
   4. identify interval on which a function is continuous
   5. rewrite a function as a piecewise-defined function to remove a removable discontinuity
   6. use IVT to identify roots of functions
3. conditional knowledge
   1. know how to identify the type of discontinuity

Section 2.7: Tangents and Derivatives at a Point

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. Section 2.6
   2. solve polynomial equations
   3. solve rational equations
   4. tangent line to a curve
   5. point-slope equation of a line
2. procedural knowledge
   1. how to evaluate the limit of a function
   2. how to determine if a function is continuous at a point
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits

Learning Goals

1. declarative knowledge (definitions)
   1. derivative of a function at a point
2. procedural knowledge
   1. find the derivative of a function at a point
   2. find the equation of a tangent line to a curve at a point
   3. find when a function has a horizontal tangent line
   4. find when a function has a vertical tangent line
3. conditional knowledge
   1. know how derivative of a function at a point relates to the rate of change of a function
   2. know how derivative of a function at a point relates to the slope of the tangent line at the point

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