Unit 1: Review of Pre-Calculus
PI 1.1Complete operations on functions
1.1Coordinate Increments and the Equation of a Line
1.2aUsing the vertical line test and finding the domain and range of functions
1.2bWorking with even and odd functions
1.2cGraphing piecewise-defined functions
1.2dWorking with composite functions
PI 1.2Graph, simplify, and solve exponential equations
1.3aGraphing exponential functions and finding their domain and range
1.3bRewriting exponential functions with different bases and finding zeroes
1.5aFind the inverse of a function
1.5bSolve exponential and logarithmic equations
1.5cDetermine the domain and range of logarithmic functions
PI 1.3Use parameterized expressions to describe functions
1.4Graphing parameterized functions and writing parameterizations for curves
PI 1.4Graph and solve trigonometric equations
1.6aWorking with radian measure and trigonometric values of angles
1.6bDetermine whether trigonometric functions are even or odd
1.6cGraphing and interpreting graphs of trigonometric functions
1.6dEvaluate expressions involving trigonometric functions and solve trigonometric equations
Unit 2: Introduction to Limits
PI 2.1Interpret and evaluate simple limits
2.1aFinding average and instantaneous speeds
2.1bApply the properties of limits
2.1cDetermine limits by substitution, tables, or graphs
2.1dFinding one-sided limits
PI 2.2Deduce and interpret behavior of functions using limits
2.2aFinding limits involving infinity
2.2bFinding vertical and horizontal asymptotes of functions
2.2cFinding power function end behavior models for functions, and identify any horizontal asymptotes
PI 2.3Analyze and determine intervals of continuity and points of discontinuity
2.3Determine where functions are continuous and find/classify points of discontinuity
PI 2.4Solve problems involving the slope of a tangent line
2.4aFind average rates of change of functions over intervals
2.4bFind equations of tangent and normal lines
2.4cFind instantaneous rates of change
Unit 3: Differentiability
PI 3.1Identify the derivative of a function as the limit of a difference quotient and estimate derivatives
3.1aFind derivatives of functions at indicated points
3.1bMatch graphs of functions with graphs of derivatives
PI 3.2Recognize the connection between differentiability and continuity
3.2aDetermine where functions are continuous and/or differentiable
3.2bDetermine whether functions have corners, cusps, vertical tangents, or discontinuities
3.2cFind numerical derivatives of functions at indicated points
PI 3.3Determine higher order derivatives
3.3aFind derivatives using the product and quotient rules
3.3bFind horizontal tangents of curves with product and quotient rules
3.3cFind the first four derivatives of functions
3.4Find rates of change related to velocity, including estimating from a graph
3.5aFind derivatives of trigonometric functions
3.5bFind velocity, speed, acceleration, and jerk of a function
Unit 4: Advanced Differentiation
PI 4.1Evaluate derivatives using the chain rule
4.1aUse given substitutions and the Chain Rule to find derivatives
4.1bEvaluate derivatives of composite functions at given values
4.1cFind equations of lines tangent to curves using the chain rule
PI 4.2Evaluate derivatives using implicit differentiation
4.2aFind derivatives using implicit differentiation
4.2bFind slopes of curves using implicit differentiation
4.2cFind equations of lines that are tangent or are normal to curves at given points using implicit differentiation
PI 4.3Evaluate derivatives of the inverse trigonometric functions
4.3aFind derivatives of inverse trigonometric functions
4.3bFind equations for tangent lines at indicated points of inverse trigonometric functions
PI 4.4Evaluate derivatives of exponential and logarithmic functions
4.4aFind the derivative of exponential and logarithmic functions
4.4bFind equations for tangent lines at indicated points for exponential and logarithmic functions
Unit 5: Application of Derivatives
PI 5.1Use derivatives to analyze properties of a function
5.1Find the extreme values of functions and where they occur using critical points
5.3aDetermine local extrema, identify any absolute extrema, or find points of inflection of a function
5.3bDetermine the interval on which the graph of a function is concave up or concave down
5.3cUse the graph of f' to estimate the intervals on which a function is increasing or decreasing
PI 5.2Apply the Mean Value Theorem to describe the behavior of a function over an interval
5.2aFind the values that satisfy the hypotheses of the Mean Value Theorem
5.2bFind all possible functions with a given derivative
PI 5.3Calculate linearizations to estimate functions
5.5aFind linearizations of functions at given values
5.5bUse linearizations to approximate expressions
5.5cFind differentials
5.5dUse Newton's Method to estimate solutions
PI 5.4Interpret the meaning of a derivative within a problem
5.6Solve related rates application problems
Unit 6: Integration
PI 6.1Calculate a definite integral using areas and properties of definite integrals
6.1aFind the exact area under a curve using finite sums
6.1bUse the rectangular approximation method to approximate areas under a curve
6.5Use the Trapezoidal Rule to approximate values of integrals
PI 6.2Express the limit of a Riemann sum in integral notation
6.2aExpress limits of desired quantitites as definite integrals
6.2bEvaluate the integral of constant functions
6.2cUse graphs of integrands and areas to evaluate integrals
6.2dEvaluate the integral of a function using technology
PI 6.3Apply definite integrals to problems involving the average value of a function
6.3aEvaluate integrals
6.3bFind average values of functions
PI 6.4Recognize antiderivatives of basic functions
6.4aFind derivatives of integrals
6.4bConstruct functions that satisfy given conditions
6.4cEvaluate definite integrals using the Fundamental Theorem of Calculus
Unit 7: Advanced Integration
PI 7.1Estimate and verify solutions to differential equations
7.1aFind general solutions to exact differential equations
7.1bSolve initial value problems explicitly
7.1cMatch differential equations with the appropriate graphs
7.1dUse Euler's method to approximate the value of y
PI 7.2Calculate antiderivatives and evaluate definite integrals
7.2aFind indefinite integrals
7.2bUse substitution to evaluate definite and indefinite integrals
7.3aUse integration by parts to evaluate definite and indefinite integrals
7.3bUse tabular integration to evaluate indefinite integrals
PI 7.3Analyze, interpret, create, and solve differential equations from problems in context
7.4aSolve initial value problems related to exponential growth and decay
7.4bSolve problems where interest is compounded continuously
7.4cFind exponential functions whose graphs pass through two points
Unit 8: Applications of Integration
PI 8.1Apply definite integrals to problems involving motion
8.1aSolve particle motion and projectile motion problems
8.1bSolve application problems related to the integral as net change
PI 8.2Apply definite integrals to problems involving area and volume
8.2Find the areas of regions in the plane
8.3Find the volumes of solids