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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: (y₂ – y₁)/(x₂ – x₁)
   2. point-slope of a line: y – y₀ = m(x – x₀); where (x₀, y₀) is any point on the line, m is slope of line
   3. order of operations when evaluating a mathematical expression: use either mnemonic device - PEMDAS or Please Excuse My Dear Aunt Sally
2. procedural knowledge
   1. how to calculate slope when given:
      • two points: (x₁, y₁) and (x₂, y₂)
      • a function, f(x), and a closed interval for the independent variable, [x₁, x₂]
      • a function, f(x), and two values for the independent variable: x₁, x₂
   2. how to evaluate trigonometric ratios of any multiple of the reference angles:
      • Use the unit circle: http://euclid.colorado.edu/~ernstd/Unit_Circle.pdf
      • OR
      • Step 1: Identify the reference angle, 0 ≤ θ ≤ π/2
      • Step 2: Evaluate trig(θ) using table located at http://calc.jjw3.com/math2431/trigRatios.doc
      • Step 3: Determine sign of answer by identifying the quadrant and the signs of x = cos(θ) and y = sin(θ) on the unit circle
   3. how to write the equation of the line given the slope, m, and a point, (x₁, y₁)
3. conditional knowledge
   1. know the difference between independent variable [usually x or t but NOT always; it will be the variable inside the parenthesis in function notation, f(var)] and dependent variable
   2. know the difference between closed intervals, [x₁, x₂]; open intervals, (x₁, x₂); and points, (x, y)
   3. know the difference between evaluating an expression (e.g., know x-value) and solving an equation (x-value is unknown)

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, given:
      1. a table of ordered pairs;
      2. a function and a closed interval;
      3. an initial point, P₀(x₀, y₀), a function and a list of x-values.
   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
   2. the tangent line to a curve may intersect the curve at more than one point (can you think of an example?)
   3. know the difference between speed and velocity

Section 2.2: Limit of a Function and Limit Laws

Knowledge Prerequisites

1. declarative knowledge (definitions)
   1. factors of a polynomial, P(x) = (x – x₀)(x – x₁)··· each (x – x_i) is a factor
   2. rationalizing the denominator of a rational expression
   3. difference quotient: (f(x + h) – f(x))/h
   4. quadratic polynomial: P(x) = ax² + bx + c; a ≠ 0
2. procedural knowledge
   1. how to factor a polynomial:
      1. Move all terms to one side of equation;
      2. Use distributive propoerty to factor out common factors from all terms, if possible;
      3. If polynomial is binomial, i.e., only two terms:
         • The difference between two squares: a² – b² = (a – b)(a + b)
         • The difference between two cubes: a³ – b³ = (a – b)(a² + ab + b²)
         • The sum of two cubes: a³ + b³ = (a + b)(a² – ab + b²)
      4. Other cases, see http://www.germanna.edu/tutor/Handouts/Math/Factoring_Methods.pdf
   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. understand the sum of two squares is not factorable over the reals (i.e., not factorable using real numbers)
   2. in general, (a + b)ⁿ ≠ aⁿ + bⁿ
   3. understand that radicals in the denominator is valid; we rationalize in order to find the limit

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. equation for horizontal asymptote
   7. 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 what the limit means
   2. know when to use the appropriate methods for evaluating limits
   3. know why the appropriate methods for evaluating limits are needed
   4. know two-sided limits relate to the one-sided limits
   5. know when and why a limit does not exist
   6. know when and why a limit is represented as ±∞
   7. 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
   5. vertical line test
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 use the vertical line test to determine if the graph of an equation is a function
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
   3. the equation for a 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
   7. know the difference between lim x → a^– and lim x → –a
   8. know when and why lim x → a is needed in an expression
   9. know how to correctly write lim x → a for an expression

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): all x_i so that f(x_i) = 0; if x_i ∈ ℜ, then x_i is an x-intercept
   6. domain of a function
   7. range of a function
   8. argument of a function
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
   5. how to write domain and range of a function in interval notation
   6. how to find domain of a function
      • polynomials; radicals with odd indices; exponentials, f(x) = a^x, a > 0, a ≠ 0; sin(x); cos(x) have domain of (–∞, ∞)
      • to find domain of radical funcions with even indices: set argument of radical ≥ 0 and solve polynomial inequality
      • to find domain of rational funcions: set denominator ≠ 0, solve for x and remove these x-values from (–∞, ∞)
      • to find domain of logarithm funcions: set argument of logarithm > 0 and solve polynomial inequality
      • to find domain of piecewise-defined funcions: use conditions
      • to find domain from the graph of a funcion
3. conditional knowledge
   1. know when to use the appropriate methods for evaluating limits, the methods include:
      • Direct Substitution
      • factor and simplify
      • rationalize the numerator
      • rationalize the denominator
      • divide all terms by xⁿ, where xⁿ is in the denominator and n is the largest exponent of the xs in the denominator, simplify each term, then evaluate the limit
      • know how factors of polynomials are related to roots of a function
      • know when to use ( or [ and ] or ) in the domain and range of a function

Learning Goals

1. declarative knowledge (definitions)
   1. continuity at a point
   2. continuity at a point from the left
   3. continuity at a point 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
      1. identify conditions
      2. evaluate function at endpoints
      3. make conclusion
3. conditional knowledge
   1. relate the definition of continuity at a point with the different types of discontinuties [removable, infinite, or jump]
   2. know how to identify the type of discontinuity [removable, infinite, or jump] from the function without a graph and use the definition of continuity to explain
   3. know how to identify the type of discontinuity [removable, infinite, or jump] from the graph of a function and use the definition of continuity to explain

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
   2. how to identify extraneous solutions

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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