Download Calculus AB Educational Learning Objectives Science Academy

Calculus AB
Educational Learning Objectives
Science Academy
J. Schofield
2006-07
PreCalculus Unit
The learner will
• Solve and graph linear and quadratic inequalities
• Define and apply the concept of absolute value
• Relate absolute value to distance
• Apply the distance formula
• Write equations of circles
• Find x and y intercepts of functions
• Find points of intersection for various functions and evaluate the feasibility of
these intersections
• Test the symmetry of a graph with respect to an axis and the origin
• Write and graph linear equations using the general form
• Interpret slope as a rate in a real-life application
• Write equations of lines parallel and perpendicular to given lines
• Use function notation to represent and evaluate a function
• Find the domain and range of a function
• Sketch the graph of a function
• Identify different types of transformations of functions
• Find expressions for composite functions
• Test for even and odd functions
• Evaluate trigonometric functions using special triangles and the unit circle
• Graph the basic trigonometric functions
• Solve trigonometric equations
• Fit linear, quadratic and trigonometric models to real-life data sets
Limits and Their Properties Unit
The learner will
• Estimate limits using graphical and numerical approaches
• Learn different ways a limit can fail to exist
• Study and use the ε−δ definition of a limit
• Evaluate limits using properties of limits
• Develop and use a strategy for finding limits
• Evaluate a limit using division and rationalization techniques
• Evaluate a limit using the Squeeze Theorem
• Determine continuity at a point and on an open interval
• Determine one-sided limits and continuity on a closed interval
• Use properties of continuity
• Understand and apply the Intermediate Value Theorem
• Determine infinite limits from the left and right
• Find and sketch the vertical asymptotes of the graph of a function
• Use limits to justify vertical asymptotes
• Find limits as x approaches infinity
• Find and sketch the horizontal asymptotes of the graph of a function
Differentiation Unit
The learner will
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Find the slope of the tangent line to a curve at a point
Use the limit definition to find the derivative of a function
Understand the relationship between differentiability and continuity
Find the derivative of a function using the basic differentiation rules
Find the derivative of sine and cosine functions
Use derivatives to find rates of change
Find the derivative of a function using the Product Rule
Find the derivative of a function using the Quotient Rule
Find the derivative of the six basic trigonometric functions
Find higher order derivatives
Find the derivative of a composite function using the Chain Rule
Find the derivative of a function using the General Power Rule
Simplify the derivative of a function using algebra
Distinguish between functions written in implicit and explicit forms
Use implicit differentiation to find the derivative of a function
Find a related rate
Use related rates to solve real-life problems
Applications of Differentiation Unit
The learner will
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Understand the definition of extrema of a function on an interval
Understand the definition of relative extrema of a function on an open interval
Find extrema on a closed interval
Understand and apply the Mean Value Theorem including Rolle’s Theorem
Determine intervals on which a function is increasing or decreasing
Apply the First Derivative Test to find relative extrema
Determine intervals on which a function is concave upward or concave downward
Find and justify points of inflection
Apply the Second Derivative Test to find relative extrema
Determine limits at infinity
Determine horizontal asymptotes
Analyze and sketch the graph of a function
Solve applied minimum and maximum problems
Approximate a zero of a function using Newton’s Method
Understand the concept of a tangent line approximation
Compare the value of the differential with the actual change
Estimate a propagated error using a differential
Find the differential of a function
Integration Unit
The learner will
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Write the general solution of a differential equation
Use indefinite integral notation for antiderivatives
Use basic integration rules to find antiderivatives
Find a particular solution of a differential equation
Use sigma notation to write and evaluate a sum
Understand the concept of an area
Approximate the area of a plane region
Find the area of a plane region using limits
Understand the definition of a Riemann Sum
Evaluate a definite integral using limits
Evaluate a definite integral using the properties of definite integrals
Evaluate a definite integral using the Fundamental Theorem of Calculus
Understand and use the Mean Value Theorem for Integrals
Find the average value of a function over a closed interval
Understand and use the Second Fundamental Theorem of Calculus
Use a change of variable to evaluate definite and indefinite integrals
Evaluate definite integrals of even and odd functions
Approximate a definite integral using Trapezoidal and Simpson’s Rules
Transcendental Functions Unit
The learner will
• Develop and use properties of the natural logarithmic function
• Understand the definition of the number e
• Find derivatives of functions involving the natural logarithmic function
• Integrate a rational function using the log rule
• Integrate trigonometric functions
• Verify that one function is the inverse function of another
• Determine whether a function has an inverse
• Find the derivative of an inverse function
• Develop properties of the natural exponential function
• Differentiate natural exponential functions
• Integrate natural exponential functions
• Define exponential functions that have other bases than e
• Differentiate and integrate exponential functions that have other bases than e
• Use exponential functions to model compound interest and exponential growth
• Develop properties of inverse trigonometric functions
• Differentiate inverse trigonometric functions
• Integrate functions whose antiderivatives involve inverse trigonometric functions
• Use completing the square to integrate a function
Differential Equations Unit
The learner will
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Use initial conditions to find particular solutions of differential equations
Use slope fields to approximate solutions of differential equations
Use Euler’s Method to approximate solutions of differential equations
Use separation of variables to solve simple differential equations
Use exponential functions to model growth and decay in applied problems
Applications of Integration Unit
The learner will
• Find the area of a region between two curves using integration
• Describe integration as an accumulation process
• Find the volume of a solid with known cross sections
• Find the volume of a solid using the disk method
• Find the volume of a solid using the washer method
• Find the volume of a solid using the shell method
• Compare the use of washer and shell methods
• Find the arc length of a smooth curve
• Find the area of a surface of revolution