AP Calculus
Course Outcome Summary
Information
Credits
1
Organization
Hartford Union High School
Mission/Description
Mission Statement: Our mathematics program is designed to enable all students to function successfully
in a number-related world, to give students a basis to increase their mathematical knowledge and improve
their quality of life, and to develop an understanding and appreciation of mathematics and technology.
Course Description:
This is an advanced placement course in calculus that covers the concepts of a first semester college
calculus course. Topics to be covered are functions and graph analysis, limits and their properties,
differentiation and applications of differentiation, transcendental and inverse functions, integration and
applications of integration, integration techniques, differential equations and slope fields. Students taking
this course should have an excellent mathematical background, be highly motivated to learn the material
and possess and interest in the sciences or engineering.
Units and Timelines
Preparation for Calculus --- 5 Days
Limits and Their Properties --- 11 Days
Differentiation --- 20 Days
Applications of Differentiation --- 30 Days
Integration --- 24 Days
Logarithmic, Exponential, and Other Transcendental Functions --- 22 Days
Applications of Integration --- 10 Days
Preparation for the AP Calculus Test --- 10 Days
Integration Techniques, L'Hopital's Rule, and Improper Integrals --- 8 Days
Textbooks
1. Calculus of a Single Variable
Competencies, Linked Standards, Objectives and
Performance Standards
1.
Analyze graphs of functions.
Properties
Domain: Cognitive
Level: Application
Difficulty: Medium
Importance: Essential
Linked External Standards
WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
in a variety of ways
WI.MA.F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of
linear equations and inequalities
Performance Standards
o
learner graphs functions on coordinate systems by hand.
o
learner manipulates hand-held graphers and computers to graph functions.
o
learner graphs systems of functions by hand.
o
learner graphs systems of functions with technology.
o
o
learner finds zeros algebraically.
learner solves systems with algebra.
o
learner finds zeros using calculus.
o
o
learner identifies maximum and minimum values of a function.
learner identifies inflection points of a function.
o
learner discriminates between local and global extremum.
o
learner chooses suitable methods to represent functions.
Learning objectives
a.
Produce the graph of a function on a coordinate system with the aid of technology.
2.
b.
Identify the zeros of a function.
c.
Explore local and global behavior of a function.
d.
Analyze functions graphically, numerically, and analytically.
Interpret the meaning of the limit of a function at a point.
Properties
Domain: Cognitive
Level: Evaluation
Difficulty: High
Importance: Essential
Linked External Standards
WI.MA.F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse
variation) and numerical sequences, and then represent them with algebraic expressions
and equations
WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
in a variety of ways
Performance Standards
o
learner calculates the value of a function at a certain point.
o
learner estimates the value of a function at a point using a set of data.
o
learner examines behavior of functions at points where functions become undefined.
o
learner examines behavior of functions at points where functions approach infinity.
o
learner discriminates between functions based on rates of change and magnitude.
o
learner identifies graphs of functions based on continuity.
o
learner describes properties of functions based on the Intermediate Value Theorem.
o
learner describes properties of functions based on the Extreme Value Theorem.
Learning objectives
a.
Calculate limits using algebra.
3.
b.
Estimate limits using graphs and tables of data.
c.
Describe asymptotic and unbounded behavior in terms of limits.
d.
Compare relative magnitudes of functions and their rates of change.
e.
Interpret continuity in terms of limits.
f.
Apply the Intermediate Value Theorem and Extreme Value Theorem to graphs of
continuous functions.
Identify the derivative as the limit of the rate of change of a function.
Properties
Domain: Cognitive
Level: Analysis
Difficulty: Medium
Importance: Essential
Linked External Standards
WI.MA.F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse
variation) and numerical sequences, and then represent them with algebraic expressions
and equations
WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
in a variety of ways
Performance Standards
o
learner defines the instantaneous rate of change as a limit.
o
learner identifies the relationship of limits and continuity in functions.
o
learner describes the intervals where the derivative exists.
o
learner calculates the limit of a function from the right and the left.
o
o
learner defines the derivative as the instantaneous rate of change.
learner calculates the slope of the tangent line at a point.
o
learner solves problems using the information gained from the first derivative.
o
o
learner solves problems using second and higher order derivatives.
learner establishes the connection between graphs of functions and the graphs of the
higher order derivatives.
o
learner applies the Mean Value Theorem to solve problems of average rate of change.
o
learner estimates local approximation of the tangent to the derivative.
o
learner draws graphs of functions and their first and second derivatives.
Learning objectives
a.
Define the derivative as the limit of the difference quotient.
4.
b.
Analyze derivatives graphically, numerically, and analytically.
c.
Explain the relationship between differentiability and continuity.
d.
Interpret the derivative as the instantaneous rate of change of a function.
e.
f.
Apply the Mean Value Theorem
Relate the tangent line to a curve and local linear approximation to the derivative.
g.
Approximate the rate of change from graphs and tables of values.
h.
Calculate first and second derivatives.
i.
Verify and generate the graphs of functions using the graphs of f' and f''.
Solve a variety of problems using differentiation.
Properties
Domain: Cognitive
Level: Application
Difficulty: High
Importance: Important
Linked External Standards
WI.MA.A.12.3 Analyze non-routine problems and arrive at solutions by various means
WI.MA.D.12.1 Identify, describe, and use derived attributes (e.g., density, speed,
acceleration, pressure) to represent and solve problem situations
WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
in a variety of ways
WI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by
using algebraic expressions, equations, and inequalities
Performance Standards
o
learner uses the First Derivative Test to determine monotonicity.
o
learner uses the Second Derivative Test to determine concavity.
o
learner sketches higher order functions using extremum, concavity and inflection points.
o
o
learner models real world problems as rates of change.
learner models real world problems as related rates.
o
learner describes rate of change of particle motion as the derivative of the position
function.
o
learner applies differentiation rules for basic equations including sum, difference,
product and quotient rules.
o
learner applies differentiation rules for more difficult equations including chain, implicit
differentiation and substitution rules.
o
learner differentiates exponential, logarithmic, trigonometric and inverse functions.
o
learner differentiates inverse trigonometric and hyperbolic functions.
o
learner sketches a slope field for a given differential equation.
o
learner sketches a solution curve through a given point on a slope field.
o
learner matches a slope field to a differential equation.
o
learner matches a slope field to a solution of a differential equation.
o
learner completes Free Response Questions AB5-1991, AB1-1992, AB3-1991 with 90%
accuracy.
Learning objectives
a.
Analyze curves using the notions of monotonicity and concavity.
5.
b.
Utilize differentiation rules for basic functions including exponential, logarithmic,
trigonometric and inverse functions.
c.
Apply basic differentiation rules for sums, products, quotients.
d.
Utilize the chain rule and implicit differentiation.
e.
f.
Model rates of change and related rates in real-word problems.
Solve optimization problems examining both absolute and relative extrema.
g.
Interpret the derivative as a rate of change in varied applied contexts, including velocity,
speed, and acceleration.
h.
Model rates of change, including related rates problems.
i.
Interpret differential equations geometrically via slope fields.
j.
Describe the relationship between slope fields and solution curves for differential
equations.
Identify the integral as the limit of Riemann Sums or as the net accumulator of the rate of
change.
Properties
Domain: Cognitive
Level: Analysis
Difficulty: Medium
Importance: Essential
Linked External Standards
WI.MA.F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse
variation) and numerical sequences, and then represent them with algebraic expressions
and equations
WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
in a variety of ways
Performance Standards
o
learner estimates the area under a curve using rectangles with right or left endpoints.
o
learner estimates the area under a curve using rectangles at their midpoints.
o
o
learner estimates the area under a curve using n-subintervals of equal length.
learner estimates the area under a curve using the trapezoidal rule.
o
learner applies the concept of Riemann Sums to approximate the area under a curve.
o
learner applies the limiting process to Riemann Summing.
o
o
learner applies the limit of Riemann Sums to develop the integral.
learner calculates the exact area under a curve using integration.
o
learner uses the integral as the accumulator function.
o
learner solves problems using basic integration rules including power and substitution
rules.
o
learner solves problems using integration of exponential, inverse, trigonometric,
logarithmic and hyperbolic functions.
o
learner calculates the area between curves using integration.
Learning objectives
a.
Estimate the area under a curve using geometric formulas.
b.
Compute Riemann sums using left, right, and midpoint evaluation points.
6.
c.
Define the definite integral as a limit of Riemann sums over equal subdivisions.
d.
Identify the integral as the accumulator of the quantity of the rate of change function.
e.
Use basic properties of integrals.
Solve a variety of problems using integration.
Properties
Domain: Cognitive
Level: Application
Difficulty: High
Importance: Important
Linked External Standards
WI.MA.A.12.3 Analyze non-routine problems and arrive at solutions by various means
WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
in a variety of ways
Performance Standards
o
learner calculates definite integrals using antidifferentiation techniques.
o
learner solves problems of growth and decay using integration.
o
o
learner calculates work, pressure and force using integration.
learner integrates to find fluid pressure in biological and environmental sciences.
o
o
learner calculates the volumes of geometric shapes of known cross sections using
integration.
learner solves separable differential equations.
o
learner determines volumes and areas of shapes of revolution.
o
learner calculates surface area and volume of irregular objects.
o
learner approximates integrals using the Average Value Theorem.
o
o
learner predicts economic and demographic outcomes using techniques of integration.
learner completes Free Response Questions AB1-1987, AB1-1990, AB1-1991, AB61987, AB2-1988, AB3-1989, AB2-1992 with 90% accuracy.
Learning objectives
a.
Recognize that the techniques of antidifferentiation follow directly from derivatives of
basic functions.
7.
b.
Determine antiderivates by substitution of variables including change of limits for
definite integrals.
c.
d.
Evaluate definite and indefinite integrals using integration by parts.
Investigate physical, social, and economic situations with appropriate integrals.
e.
Determine the area and volumes of geometric figures using integrals.
f.
Solve separable differential equations.
g.
h.
Find specific antiderivatives using initial conditions.
Use separable differential equations to study the equation y' = ky.
i.
Find the volume of a solid with known cross sections.
j.
Use the Average Value Theorem to approximate integrals.
k.
Approximate definite integrals of functions represented algebraically, graphically, and
by tables of values using the Trapezoid Rule, Simpson's Rule, and Riemann Sums.
Identify the parts of the Fundamental Theorem of Calculus as expressed by the derivative
and the integral.
Properties
Domain: Cognitive
Level: Analysis
Difficulty: Medium
Importance: Important
Performance Standards
o
learner relates the parts of the Fundamental Theorem of Calculus to differentiation and
integration.
o
learner applies the Fundamental Theorem of Calculus to real world problems.
o
o
learner defines the Fundamental Theorem of Calculus using antiderivatives.
learner uses the Fundamental Theorem of Calculus to solve problems.
o
learner calculates volume using the disk method.
o
learner establishes predictions based on analysis of graphs using the Fundamental
Theorem of Calculus.
o
learner calculates definite integrals.
Learning objectives
a.
Identify the parts of the Fundamental Theorem of Calculus.
b.
Use the Fundamental Theorem of Calculus to calculate definite integrals.
c.
Represent the particular antiderivatives using the Fundamental Theorem of Calculus.
d.
8.
Analyze graphs of data using the Fundamental Theorem of Calculus.
Communicate calculus both orally and in well written sentences.
Properties
Domain: Cognitive
Level: Synthesis
Difficulty: High
Importance: Essential
Linked External Standards
WI.MA.A.12.4 Develop effective oral and written presentations employing correct
mathematical terminology, notation, symbols, and conventions for mathematical
arguments and display of data
Performance Standards
o
learner demonstrates confidence by speaking and writing calculus as evidenced by
observation.
o
learner applies properties of limits in written form.
o
learner describes the meaning of the derivative in oral statements.
o
learner calculates the first, second and nth derivative of a function.
o
learner displays understanding of integration through oral participation as evidenced by
observation.
o
o
learner integrates functions of varying forms to calculate area and volume.
learner presents a calculus lesson to the class complete with objectives, definitions,
theorems, an investigation using technology, free response and multiple choice
assessment questions.
Learning objectives
a.
Explain properties of limits orally and in writing.
b.
Describe calculations of derivatives orally and in writing.
c.
9.
Manipulate integrals orally and in writing.
Model written descriptions of a physical situation with a function, differential equation, or
integral.
Properties
Domain: Cognitive
Level: Synthesis
Difficulty: High
Importance: Essential
Linked External Standards
WI.MA.E.12.1 Work with data in the context of real-world situations
Performance Standards
o
learner describes all aspects of particle motion using technology.
o
learner investigates particle motion using first and second derivatives.
o
learner demonstrates the use of appropriate calculus to solve real world problems.
o
o
learner calculates rates of change.
learner derives solutions to specific problems using differentiation.
o
learner solves problems using algebraic, analytic and graphing tools.
Learning objectives
a.
Illustrate projectile motion using the position function, f' and f''.
b.
Model physical, social and economic problems in biological, natural and social sciences
using calculus.
10.
c.
Calculate rates of change in the natural sciences.
d.
Derive solutions to problems using differential equations.
Determine reasonableness of solutions including, size, accuracy and units of measure.
Properties
Domain: Cognitive
Level: Evaluation
Difficulty: Medium
Importance: Important
Linked External Standards
WI.MA.B.12.6 Routinely assess the acceptable limits of error
Performance Standards
o
learner checks work when solving problems.
o
learner examines solutions to real world situations for accuracy and reliability.
o
learner disregards unreasonable answers and reworks solutions.
o
learner uses correct units of measurement in solutions to real world problems.
o
learner estimates answers prior to solutions to check for reasonableness.
Learning objectives
a.
Recognize reasonable and unreasonable solutions.
11.
b.
Employ unitization of measurement to check correctness of solution.
c.
Evaluate approximation and estimation techniques for solving problems.
Foster an appreciation of calculus as a coherent body of knowledge and as a human
accomplishment.
Properties
Domain: Affective
Level: Valuing
Difficulty: Medium
Importance: Important
Performance Standards
o
learner demonstrates enthusiasm for learning and mastering problem solving strategies
as evidenced by observation.
o
learner uses technology to visualize theoretical concepts.
o
o
learner explores the development of calculus through Interactive Calculus.
learner speaks the language of calculus as evidenced by observation and written essays.
o
learner projects the power of calculus through classroom discussions as evidenced by
observation.
o
learner completes Coliseum Problem, Flower Vase Problem, and Favorite Problem with
90% accuracy.
Learning objectives
a.
Foster a belief that calculus is a "coherent body of knowledge".
b.
Respect calculus as a human accomplishment.
c.
Find pleasure in the beauty and wonder of calculus.