| 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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- Implicit and explicit equations.
- Point-slope and slope-intercept equation of a line.
- Function evaluation.
- Know all rules for differentiating functions.
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- Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x.
- Assuming, for example, y is implicitly a function of x, and given an equation relating y to x, find the derivative of y with respect to x.
- Find the equation of a line tangent to an implicitly defined curve at a point.
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| Derivatives of Exponential and Logarithmic Functions
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- Know the properties associated with logarithmic expressions.
- Rules for differentiating function (chain rule in particular).
- Implicit differentiation.
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- Find the derivative of functions that involve exponential functions.
- Find the derivative of functions that involve logarithmic functions.
- Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients.
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| Related Rates
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- Formulas from classical geometry for area, volume, etc.
- Similar triangles to form proportions.
- Right triangle trigonometry.
- Use trigonometric identities to re-write expressions.
- Rules for finding derivatives of functions.
- Implicit differentiation.
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- Express changing quantities in terms of derivatives.
- Find relationships among the derivatives in a given problem.
- Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
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| Linear Approximations and Differentials
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- Find the equation of the tangent line to a curve y = f(x) at a certain given x-value
- Understand the Leibnitz notation of the derivative
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- Approximate the function value close to the center of the linear approximation using the linearization.
- Given an expression to be evaluated/approximated, come up with the function and its linearization
- Understand the formula for the differential; how it can be used to estimate the change in the dependent variable quantity, given the small change in the independent variable quantity.
- Use the information above to estimate potential relative (and percentage) error
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| Maxima and Minima
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- Understand the definition of an increasing and a decreasing function.
- Solve an algebraic equation.
- Understand interval notation.
- Solve trigonometric equations.
- Use all rules to differentiate algebraic and transcendental functions.
- Understand definition of continuity of a function at a point and over an interval.
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- Know the definitions of absolute and local extrema.
- Know what a critical point is and locate it (them).
- Use the Extreme Value Theorem to find the absolute extrema of a continuous function on a closed interval.
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| Mean Value Theorem
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- Function evaluation.
- Solve equations.
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- Determine if the MVT applies given a function on an interval.
- Find c in the conclusion of the MVT (if algebraically feasible)
- Know the first 3 Corollaries of MVT (especially the 3rd)
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| Derivatives and the Shape of a Graph
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- Function evaluation.
- Solve equations.
- Know how to find the derivative and critical point(s) of a function.
- Know how to find the second derivative
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- Use the First Derivative Test to find intervals on which the function is increasing and decreasing and the local extrema and their type
- Use the Concavity Test (aka the Second Derivative Test for Concavity) to find the intervals on which the function is concave up and down, and point(s) of inflection
- Understand the shape of the graph, given the signs of the first and second derivatives
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| Applied Optimization Problems
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- Translate the information given into mathematical statements/formulas.
- Know frequently used formulas pertaining to area and volume.
- Solve Algebraic and trigonometric equations.
- Absolute extrema of a function
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Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.
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| L’Hôpital’s Rule
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- Simplifying algebraic and trigonometric expressions.
- Evaluating limits.
- Finding derivatives.
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- Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
- Recognize when to apply L’Hôpital’s rule.
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| Antiderivatives
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- Inverse Functions
- Finding derivatives
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- Find the general antiderivative of a given function.
- Explain the terms and notation used for an indefinite integral.
- State the power rule for integrals.
- Use antidifferentiation to solve simple initial-value problems.
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| Approximating Areas
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- Sigma notation
- Area of a rectangle
- Graphs of continuous functions
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- Calculate sums and powers of integers.
- Use the sum of rectangular areas to approximate the area under a curve.
- Use Riemann sums to approximate area.
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| The Definite Integral
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- Antiderivatives
- Limits of Riemann Sums
- Continuous functions over bounded intervals
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- State the definition of the definite integral.
- Explain the terms integrand, limits of integration, and variable of integration.
- Explain when a function is integrable.
- Describe the relationship between the definite integral and net area.
- Use geometry and the properties of definite integrals to evaluate them.
- Calculate the average value of a function.
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| The Fundamental Theorem of Calculus
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- Derivatives
- Antiderivatives
- Mean Value Theorem
- Inverse functions
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- Describe the meaning of the Mean Value Theorem for Integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 1.
- Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 2.
- Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
- Explain the relationship between differentiation and integration.
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| Integration Formulas and the Net Change Theorem
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- Indefinite integrals
- Collections of functions
- The Fundamental Theorem (part 2)
- Displacment vs. distance traveled
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- Apply the basic integration formulas.
- Explain the significance of the net change theorem.
- Use the net change theorem to solve applied problems.
- Apply the integrals of odd and even functions.
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| Substitution Method for Integrals
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- Solving basic integrals.
- Derivatives
- Change of Variables
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- Use substitution to evaluate indefinite integrals.
- Use substitution to evaluate definite integrals.
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| Integrals Involving Exponential and Logarithmic Functions
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- Exponential and logarithmic functions
- Derivatives and integrals of these two functions
- Rules for derivatives and integration
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- Integrate functions involving exponential functions.
- Integrate functions involving logarithmic functions.
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| Integrals Resulting in Inverse Trigonometric Functions
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- Trigonometric functions and their inverses
- Injective functions and the domain of inverse trigonometric functions
- Rules for integration
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- Integrate functions resulting in inverse trigonometric functions.
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