| Topic |
Pre-requisite |
Objective |
Examples
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| The Limit of a Function |
- Evaluation of a function including the absolute value, rational, and piecewise functions
- Domain and Range of a Function
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- Describe the limit of a function using correct notation.
- Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
- Use a graph to estimate the limit of a function or to identify when the limit does not exist.
- Define one-sided limits and provide examples.
- Explain the relationship between one-sided and two-sided limits.
- Describe an infinite limit using correct notation.
- Define a vertical asymptote.
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| The Limit Laws
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- Simplifying algebraic expressions.
- Factoring polynomials
- Identifying conjugate radical expressions.
- Evaluating expressions at a value.
- Simplifying complex rational expressions by obtaining common denominators.
- Evaluating piecewise functions.
- The trigonometric functions and right triangle trigonometry.
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- Recognize the basic limit laws.
- Use the limit laws to evaluate the limit of a function.
- Evaluate the limit of a function by factoring.
- Use the limit laws to evaluate the limit of a polynomial or rational function.
- Evaluate the limit of a function by factoring or by using conjugates.
- Evaluate the limit of a function by using the squeeze theorem.
- Evaluate left, right, and two sided limits of piecewise defined functions.
- Evaluate limits of the form K/0, K≠0.
- Establish and use this to evaluate other limits involving trigonometric functions.
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| Continuity
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- Domain of function.
- Interval notation.
- Evaluate limits.
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- Continuity at a point.
- Describe three kinds of discontinuities.
- Define continuity on an interval.
- State the theorem for limits of composite functions and use the theorem to evaluate limits.
- Provide an example of the intermediate value theorem.
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| Limits at infinity and asymptotes
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- Horizontal asymptote for the graph of a function
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- Calculate the limit of a function that is unbounded.
- Identify a horizontal asymptote for the graph of a function.
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| Defining the Derivative
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- Evaluation of a function at a value or variable expression.
- Find equation of a line given a point on the line and its slope.
- Evaluate limits.
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- Recognize the meaning of the tangent to a curve at a point.
- Calculate the slope of a secant line (average rate of change of a function over an interval).
- Calculate the slope of a tangent line.
- Find the equation of the line tangent to a curve at a point.
- Identify the derivative as the limit of a difference quotient.
- Calculate the derivative of a given function at a point.
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| The Derivative as a Function
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- Graphing functions.
- The definition of continuity of a function at a point.
- Understanding that derivative of a function at a point represents the slope of the curve at a point.
- Understanding when a limit fails to exist.
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- Define the derivative function of a given function.
- Graph a derivative function from the graph of a given function.
- State the connection between derivatives and continuity.
- Describe three conditions for when a function does not have a derivative.
- Explain the meaning of and compute a higher-order derivative.
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| Differentiation Rules
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- Radical and exponential notation.
- Convert between radical and rational exponents.
- Use properties of exponents to re-write with or without negative exponents.
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- State the constant, constant multiple, and power rules.
- Apply the sum and difference rules to combine derivatives.
- Use the product rule for finding the derivative of a product of functions.
- Use the quotient rule for finding the derivative of a quotient of functions.
- Extend the power rule to functions with negative exponents.
- Combine the differentiation rules to find the derivative of a polynomial or rational function.
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| Derivatives as Rates of Change
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- Function evaluation at a value or variable expression.
- Solving an algebraic equation.
- Find derivatives of functions using the derivative rules.
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- Determine a new value of a quantity from the old value and the amount of change.
- Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
- Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
- Predict the future population from the present value and the population growth rate.
- Use derivatives to calculate marginal cost and revenue in a business situation.
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| Derivatives of the Trigonometric Functions
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- State and use trigonometric identities.
- Graphs of the six trigonometric functions.
- Power, Product, and Quotient Rules for finding derivatives.
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- Find the derivatives of the sine and cosine function.
- Find the derivatives of the standard trigonometric functions.
- Calculate the higher-order derivatives of the sine and cosine.
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| The Chain Rule
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- Composition of functions.
- Solve trigonometric equations.
- Power, Product, and Quotient Rules for finding derivatives.
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- State the chain rule for the composition of two functions.
- Apply the chain rule together with the power rule.
- Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
- Recognize and apply the chain rule for a composition of three or more functions.
- Use interchangeably the Newton and Leibniz Notation for the Chain Rule.
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| Derivatives of Inverse Functions
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- Determine if a function is 1-1.
- The relationship between a 1-1 function and its inverse.
- Knowing customary domain restrictions for trigonometric functions to define their inverses.
- Rules for differentiating functions.
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- State the Inverse Function Theorem for Derivatives.
- Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
- Derivatives of the inverse trigonometric functions.
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| Implicit Differentiation
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