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- declarative knowledge (definitions)
- slope of a line: (y
_{2}– y_{1})/(x_{2}– x_{1}) - point-slope of a line: y – y
_{0}= m(x – x_{0}); where (x_{0}, y_{0}) is any point on the line, m is slope of line - order of operations when evaluating a mathematical expression: use either mnemonic device - PEMDAS or Please Excuse My Dear Aunt Sally

- slope of a line: (y
- procedural knowledge
- how to calculate slope when given:
- two points: (x
_{1}, y_{1}) and (x_{2}, y_{2}) - a function, f(x), and a closed interval for the independent variable, [x
_{1}, x_{2}] - a function, f(x), and two values for the independent variable: x
_{1}, x_{2}

- two points: (x
- 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

- how to write the equation of the line given the slope, m, and a point, (x
_{1}, y_{1})

- how to calculate slope when given:
- conditional knowledge
- 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
- know the difference between closed intervals, [x
_{1}, x_{2}]; open intervals, (x_{1}, x_{2}); and points, (x, y) - know the difference between evaluating an expression (e.g., know x-value) and solving an equation (x-value is unknown)

- declarative knowledge (definitions)
- average speed
- average rate of change
- tangent line to a curve
- secant line to a curve

- procedural knowledge
- how to estimate the slope of a tangent line to a curve, given:
- a table of ordered pairs;
- a function and a closed interval;
- an initial point, P
_{0}(x_{0}, y_{0}), a function and a list of x-values.

- how to write the equation of the line tangent to the curve at a point

- how to estimate the slope of a tangent line to a curve, given:
- conditional knowledge
- distinguish between calculating the slope between two points and estimating the slope of a curve at a point
- the tangent line to a curve may intersect the curve at more than one point (can you think of an example?)
- know the difference between speed and velocity

- declarative knowledge (definitions)
- factors of a polynomial, P(x) = (x – x
_{0})(x – x_{1})··· each (x – x_{i}) is a factor - rationalizing the denominator of a rational expression
- difference quotient: (f(x + h) – f(x))/h
- quadratic polynomial: P(x) = ax
^{2}+ bx + c; a ≠ 0

- factors of a polynomial, P(x) = (x – x
- procedural knowledge
- how to factor a polynomial:
- Move all terms to one side of equation;
- Use distributive propoerty to factor out common factors from all terms, if possible;
- If polynomial is binomial, i.e., only two terms:
- The difference between two squares: a
^{2}– b^{2}= (a – b)(a + b) - The difference between two cubes: a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - The sum of two cubes: a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2})

- The difference between two squares: a
- Other cases, see http://www.germanna.edu/tutor/Handouts/Math/Factoring_Methods.pdf

- how to rationalize the denominator of a rational expression
- how to rationalize the numerator of a rational expression
- how to evaluate trigonometric ratios of any multiple of the reference angles
- how to evaluate difference quotient

- how to factor a polynomial:
- conditional knowledge
- understand the sum of two squares is not factorable over the reals (i.e., not factorable using real numbers)
- in general, (a + b)
^{n}≠ a^{n}+ b^{n} - understand that radicals in the denominator is valid; we rationalize in order to find the limit

- declarative knowledge (definitions)
- limit
- Limit Laws
- The Sandwich Theorem (a.k.a., The Squeeze Theorem)

- procedural knowledge
- how to use Direct Substitution to evaluate the limit of a function
- how to graph to evaluate the limit of a function
- how to reduce a rational expression to evaluate the limit of a function
- how to rationalize the denominator of a rational expression to evaluate the limit of a function
- how to rationalize the numerator of a rational expression to evaluate the limit of a function
- how to use the Limit Laws to evaluate the limit of a function

- conditional knowledge
- know when to use the appropriate methods for evaluating limits
- know why the appropriate methods for evaluating limits are needed
- know when and why a limit does not exist

- declarative knowledge (definitions)
- Section 2.2
- factors of a polynomial
- rationalizing the denominator of a rational expression
- difference quotient
- piecewise-defined functions

- procedural knowledge
- how to factor a polynomial
- how to rationalize the denominator of a rational expression
- how to rationalize the numerator of a rational expression
- how to evaluate trigonometric ratios of any multiple of the reference angles
- how to evaluate difference quotient
- how to sketch a graph of a piecewise-defined function

- conditional knowledge
- none

- declarative knowledge (definitions)
- two-sided limit
- right-hand limit
- left-hand limit
- limit at infinity
- horizontal asymptote
- equation for horizontal asymptote
- slant asymptote

- procedural knowledge
- how to use Direct Substitution to evaluate the limit of a function
- how to graph to evaluate the limit of a function
- how to reduce a rational expression to evaluate the limit of a function
- how to rationalize the denominator of a rational expression to evaluate the limit of a function
- how to rationalize the numerator of a rational expression to evaluate the limit of a function
- how to evaluate the limit of a function as x approaches ±∞
- how to use the Limit Laws to evaluate the limit of a function

- conditional knowledge
- know what the limit means
- know when to use the appropriate methods for evaluating limits
- know why the appropriate methods for evaluating limits are needed
- know two-sided limits relate to the one-sided limits
- know when and why a limit does not exist
- know when and why a limit is represented as ±∞
- identify where limits exist or do not exist for functions

- declarative knowledge (definitions)
- Section 2.2
- Section 2.4
- factors of a polynomial
- rationalizing the denominator of a rational expression
- vertical line test

- procedural knowledge
- how to factor a polynomial
- how to rationalize the denominator of a rational expression
- how to rationalize the numerator of a rational expression
- how to evaluate trigonometric ratios of any multiple of the reference angles
- how to use the vertical line test to determine if the graph of an equation is a function

- conditional knowledge
- know when to use the appropriate methods for evaluating limits

- declarative knowledge (definitions)
- infinite limits
- vertical asymptote
- the equation for a vertical asymptote

- procedural knowledge
- how to use Direct Substitution to evaluate the limit of a function
- how to graph to evaluate the limit of a function
- how to reduce a rational expression to evaluate the limit of a function
- how to rationalize the denominator of a rational expression to evaluate the limit of a function
- how to rationalize the numerator of a rational expression to evaluate the limit of a function
- how to identify the vertical asymptotes of a function
- how to sketch functions from a set of limits

- conditional knowledge
- know when to use the appropriate methods for evaluating limits
- know why the appropriate methods for evaluating limits are needed
- know two-sided limits relate to the one-sided limits
- know when and why a limit does not exist
- know when and why a limit is represented as ±∞
- identify where limits exist or do not exist for functions
- know the difference between lim x → a
^{–}and lim x → –a - know when and why lim x → a is needed in an expression
- know how to correctly write lim x → a for an expression

- declarative knowledge (definitions)
- Section 2.2
- Section 2.4
- Section 2.5
- factors of a polynomial
- 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 - domain of a function
- range of a function
- argument of a function

- procedural knowledge
- how to factor a polynomial
- how to evaluate trigonometric ratios of any multiple of the reference angles
- how to evaluate the limit of a function
- how to sketch a graph of a piecewise-defined function
- how to write domain and range of a function in interval notation
- 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

- polynomials; radicals with odd indices; exponentials,
f(x) = a

- conditional knowledge
- 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 x
^{n}, where x^{n}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

- know when to use the appropriate methods for evaluating limits, the methods include:

- declarative knowledge (definitions)
- continuity at a point
- continuity at a point from the left
- continuity at a point from the right
- continuous on an interval
- point of discontinuity
- removable discontinuity (a.k.a., hole)
- jump discontinuity
- infinite discontinuity
- oscillating discontinuity
- properties of continuous functions
- Intermediate Value Theorem [IVT]

- procedural knowledge
- use the definition of continuity at a point
- identify points of discontinuity from a graph of a function
- identify points of discontinuity of a function
- identify interval on which a function is continuous
- rewrite a function as a piecewise-defined function to remove a removable discontinuity
- use IVT to identify roots of functions
- identify conditions
- evaluate function at endpoints
- make conclusion

- conditional knowledge
- relate the definition of continuity at a point with the different types of discontinuties [removable, infinite, or jump]
- 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
- 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

- declarative knowledge (definitions)
- Section 2.6
- solve polynomial equations
- solve rational equations
- tangent line to a curve
- point-slope equation of a line

- procedural knowledge
- how to evaluate the limit of a function
- how to determine if a function is continuous at a point

- conditional knowledge
- know when to use the appropriate methods for evaluating limits
- how to identify extraneous solutions

- declarative knowledge (definitions)
- derivative of a function at a point

- procedural knowledge
- find the derivative of a function at a point
- find the equation of a tangent line to a curve at a point
- find when a function has a horizontal tangent line
- find when a function has a vertical tangent line

- conditional knowledge
- know how derivative of a function at a point relates to the rate of change of a function
- know how derivative of a function at a point relates to the slope of the tangent line at the point

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