Honors Calculus

College Board AP Calculus Standards

• CHA-1.A Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.
• LIM-1.A Represent limits analytically using correct notation.
• LIM-1.A Interpret limits expressed in analytic notation.
• LIM-1.C Estimate limits of functions.
• LIM-1.D Determine the limits of functions using limit theorems.
• LIM-1.E Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.
• LIM-2.A Justify conclusions about continuity at a point using the definition.
• LIM-2.B Determine intervals over which a function is continuous.
• LIM-2.C Determine values of x or solve for parameters that make discontinuous functions continuous, if possible.
• FUN-1.A Explain the behavior of a function on an interval using the Intermediate Value Theorem.
   

• Rates of change
• Limits of functions
• Continuity versus discontinuity of functions

• Calculus allows us to generalize knowledge about motion to diverse problems involving change.
• Reasoning with definitions, theorems, and properties can be used to justify claims about limits.
• Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.

• Calculate average and instantaneous rates of change.
• Define and calculate limits for function values and apply the properties of limits.
• Find and verify end behavior models for various functions.
• Define and calculate limits for functions.
• Calculate infinite limits and identify vertical and horizontal asymptotes.
• Find equations of tangent and normal lines to a curve.
• Apply the Intermediate Value Theorem and the properties of algebraic combinations and composites of continuous functions. 

• How are limits used to describe continuity, the derivative, and the integral: the ideas giving the foundation of Calculus?
• How do limits and continuity provide the foundation for understanding change and the behavior of functions in calculus?
   

• What is the difference between average rate of change and instantaneous rate of change?
• How do we calculate the limits of functions and use the properties of limits?
• How do we create an end behavior model for a function?
• How do we find vertical and horizontal asymptotes and relate those to limits?
• What is the difference between tangent and normal lines to a curve?
• How do we analyze functions to find intervals of continuity and discontinuity?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Average rate of change
• Vertical asymptote
• Squeeze Theorem
• Instantaneous rate of change
• End behavior mode
• Intermediate Value 
• Theorem
• Limit of a function
• Continuity
• Extended function
• Horizontal asymptote
• Discontinuity
• Sandwich Theorem

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments
   

College Board AP Calculus Standards

• CHA-2.A Determine average rates of change using difference quotients.
• CHA-2.B Represent the derivative of a function as the limit of a difference quotient.
• CHA-2.C Determine the equation of a line tangent to a curve at a given point.
• CHA-2.D Estimate derivatives.
• FUN-2.A Explain the relationship between differentiability and continuity.
• FUN-3.A Calculate derivatives of familiar functions.
• FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
• FUN-3.F Determine higher order derivatives of a function.
• LIM-3.A Interpret a limit as a definition of a derivative.
   

• Definition of a derivative.
• Basic derivative rules.
• Equations of tangent lines.
   

• Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.
• Recognizing that a function’s derivative may also be a function allows us to develop knowledge about the related behaviors of both.
• Recognizing opportunities to apply derivative rules can simplify differentiation.
• Reasoning with definitions, theorems, and properties can be used to determine a limit.

• Calculate slopes and derivatives using the definition of the derivative.
• Graph f from the graph of f', graph f' from the graph of f, and graph the derivative of a function given numerically with data.
• Find where a function is non-differentiable.
• Graph the derivative given the graph of a function.
• Derive functions using rules for differentiation. 
• Find derivatives using powers, multiples, sums, and differences.
• Differentiate composite functions using the Chain Rule.
• Find derivatives using the Power Rule for Rational Powers of x.
• Find derivatives of trigonometric functions.

• How does differentiation help us understand and describe the rates of change and behavior of functions, and how can it be applied to solve real-world problems?
• Why are there points where the derivative does not exist? 
• How can we find derivatives of functions analytically more efficiently?
• How do derivatives play a key role in describing periodic change? 

• What is the difference between the average rate of change and instantaneous rate of change?
• Why do we use the definition of a derivative to find the instantaneous rate of change of a function?
• How do we find the slope of a tangent line to a curve?
• How do we write the equation of the tangent line and the normal line to a curve?
• How can you tell if a function is not differentiable?
• How can we apply the rules of differentiation to find the derivative of a function?
• How do we find the derivative of a trig function?
• How do we relate composition and decomposition of functions to the chain rule for derivatives?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets

• Instantaneous rate of change
• Tangent line
• Discontinuity
• Quotient rule
• Chain rule
• Derivative
• Corner
• Power rule
• Higher order derivative
• Jerk
• Differentiation
• Cusp
• Sum/Difference rules
• Composition of functions
• Normal line
• Vertical tangent
• Product rule
• Decomposition of functions

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments

College Board AP Calculus Standards

• FUN-3.A Calculate derivatives of familiar functions.
• FUN-3.D Calculate derivatives of implicitly defined functions.
• FUN-3.E Calculate derivatives of inverse and inverse trigonometric functions.
• LIM-3.A Interpret a limit as a definition of a derivative.
   

• Derivatives of implicitly defined functions.
• Inverse trig functions and their derivatives.
• Derivatives of exponential and log functions.
   

• Recognizing opportunities to apply derivative rules can simplify differentiation.
• Reasoning with definitions, theorems, and properties can be used to determine a limit.

• Find derivatives using implicit differentiation.
• Calculate derivatives of exponential and logarithmic functions. 
• Find derivatives of trigonometric functions and their inverses.
• Find derivatives using the power rule for rational powers of x.
   

• How would you find the derivative of nonfunctions? 
• How can implicit differentiation be used to find the rates of change in relationships that are not explicitly solved for one variable in terms of another?

• What are the strategies used to find the derivatives of non-algebraic functions? 
• How do we complete implicit differentiation?
• How do we develop the rules for the derivatives of inverse sine and tangent functions?
• What are the rules for the derivatives of inverse trig functions, exponential functions, and log functions?
• How do we apply the derivative rules to find the equations of tangent lines and normal lines?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Implicity defined functions
• Exponential function
• Implicit differentiation
• Logarithmic function
• Power rule for rational powers
• Inverse trig functions

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments
   

College Board AP Calculus Standards

• CHA-3.A Interpret the meaning of a derivative in context.
• CHA-3.B Calculate rates of change in applied contexts.
• CHA-3.C Interpret rates of change in applied contexts.
• CHA-3.D Calculate related rates in applied contexts.
• CHA-3.E Interpret related rates in applied contexts.
• CHA-3.F Approximate a value on a curve using the equation of a tangent line.

• Extreme value theorem.
• Mean value theorem.
• Related rates.
   

• Derivatives allow us to solve real-world problems involving rates of change.
• Derivatives give us precise information about how the curve bends and exactly where it turns.
• Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.

• Apply the mean value theorem.
• Apply the extreme value theorem.
• Determine local or absolute extreme values of a function.
• Use first and second derivative tests.
• Solve related rates problems.

• How can derivatives be used to investigate the characteristics of functions?
• Finding maximum and minimum values of functions, called optimization, is an important issue in real-world problems. How can derivatives help us find these maximums and minimums? 
   

• What are the strategies used to find extreme values of functions?
• What is the difference between a critical point and a stationary point?
• How do you apply the mean value theorem to find where functions are increasing and decreasing?
• How do you differentiate an equation with respect to t?
• How do you set up a related rate problem?
• How do you interpret the solution of a related rate problem?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Absolute extrema
• Decreasing
• Related rates
• Local extrema
• Mean value theorem
• Relative extrema
• Critical point
• Extreme value theorem
• Global extrema
• Increasing
• Monotonic function
• Stationary point

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments

College Board AP Calculus Standards

• FUN-1.B Justify conclusions about functions by applying the Mean Value Theorem over an interval.
• FUN-1.C Justify conclusions about functions by applying the Extreme Value Theorem.
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives.
• FUN-4.B Calculate minimum and maximum values in applied contexts or analysis of functions.
• FUN-4.C Interpret minimum and maximum values calculated in applied contexts.
• FUN-4.D Determine critical points of implicit relations.
• FUN-4.E Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.

• Velocity and acceleration of an object.
• Concavity and points of inflection.
• Optimization.
   

• Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.
• A function’s derivative can be used to understand some behaviors of the function.
• Derivatives allow us to solve rectilinear motion problems involving position, speed, velocity, and acceleration.
• Derivatives allow us to solve real-world problems involving optimization.
   

• Determine local and absolute extrema.
• Use First and Second Derivative Tests to determine the local extreme values of a function.
• Determine the concavity of a function and locate the points of inflection by analyzing the second derivative.
• Apply the Mean Value Theorem.
• Solve application problems involving finding minimum or maximum values of a function. 
• Solve optimization and related rates problems.
   

• What is the suggested strategy for solving optimization (min-max) problems? 
• How can first and second derivatives be used to investigate the characteristics of functions?
• How are position, velocity, and acceleration interconnected, and how can calculus be used to analyze and predict motion?
   

• What strategies can be used to find the velocity and acceleration of an object, based on a position equation?
• What is the difference between velocity and speed?
• How do you use the first derivative test to find local extrema?
• How do you use the second derivative test to find local extrema?
• How do you determine concavity of a function?
• What strategies can you use to find points of inflection?
• How do you set up an optimization problem?
• How do you interpret the solution of an optimization problem?
• How does calculus make a contribution to economic theory?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Acceleration
• Concavity
• Maximizing profit
• Optimization
• Displacement
• Inflection point
• Minimizing cost
• Speed
• First derivative test
• Marginal cost
• Velocity
• Second derivative test
• Marginal revenue

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments
   

College Board AP Calculus Standards

• CHA-4.A Interpret the meaning of areas associated with the graph of a rate of change in context.
• CHA-4.B Determine the average value of a function using definite integrals.
• LIM-5.A Approximate a definite integral using geometric and numerical methods.
• LIM-5.B Interpret the limiting case of the Riemann sum as a definite integral.
• LIM-5.C Represent the limiting case of the Riemann sum as a definite integral.
• FUN-5.A Represent accumulation functions using definite integrals.
• FUN-6.A Calculate a definite integral using areas and properties of definite integrals.
• FUN-6.B Evaluate definite integrals analytically using the Fundamental Theorem of Calculus.
• FUN-6.C Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

• Riemann Sums.
• Definite Integrals.
• Antiderivatives.
• Fundamental Theorem of Calculus.

• Definite integrals allow us to solve problems involving the accumulation of change over an interval.
• Definite integrals can be approximated using geometric and numerical methods.
• The Fundamental Theorem of Calculus connects differentiation and integration.
• Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

• Estimate the area under a curve using Reimann sums.
• Express the limit of summation notation as a definite integral.
• Apply properties of definite integrals.
• Find the average value of a function, compared to finding the average rate of a function.
• Apply the Fundamental Theorem of Calculus.
   

• What information can be obtained by computing the area beneath a function? 
• How can definite integrals be used to calculate accumulated quantities and solve real-world problems involving area, volume, and other applications?
   

• How do you use Reimann sums to approximate the area under a curve?
• How do you rewrite a Reimann sum in the form of a definite integral?
• What are the strategies for applying the properties of definite integrals?
• What is the difference between the average rate of a function and the average value of a function?
• How do you use a definite integral to define accumulator functions?
• How do you evaluate definite integrals using antiderivatives?
• What is the purpose of the Fundamental Theorem of Calculus?

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Accumulation
• Average value
• Lower bound
• MRAM
• Partition
• Accumulator function
• Definite integral
• Upper bound
• Riemann sum
• Total area
• Antidifferentiation
• Fundamental Theorem of Calculus
• LRAM
• Sigma notation
• Area under a curve
• Integrand
• RRAM
• Subinterval

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments

College Board AP Calculus Standards

• FUN-6.D For integrands requiring substitution or rearrangements into equivalents forms: a) determine indefinite integrals and b) evaluate definite integrals.
• FUN-7.A Interpret verval statements of problems as differential equations involving a derivative expressions.
• FUN-7.B Verify solutions to differential equations.
• FUN-7.C Estimate solutions to differential equations.
• FUN-7.D Determine general solutions to differential equations.
• FUN-7.E Determine particular solutions to differential equations.
• FUN-7.F Interpret the meaning of a differential equation and its variables in context.
• FUN-7.G Determine general and particular solutions for problems involving differential equations in context.

• Slope Fields.
• Differential equations.
• Initial value problems.
• Integration by substitution.

• Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.
• Solving differential equations allows us to determine functions and develop models.
   

• Construct Slope Fields.
• Construct antiderivatives using the Fundamental Theorem of Calculus.
• Solve Initial Value Problems.
• Use Euler's Method for graphing a solution to an initial value problem.
• Compute Definite and Indefinite Integrals by the method of substitution.

• What is the significance of the general solution of a differential equation, and how does it represent the family of functions that satisfy the equation?
• How can integration by substitution simplify the process of finding antiderivatives, and why is it considered the reverse of the chain rule

• What is a differential equation?
• How do you use a slope field to represent a family of solutions for a differential equation?
• What strategies can you use to solve an initial value problem?
• How do you complete the process for Euler’s method to estimate a specific value for a differential equation?
• How is the substitution method for integration related to the chain rule for differentiation?
• How do you apply the substitution method for integration?

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Differential equation
• Local linearity
• Euler’s method
• Slope field
• General solution to a differential equation
• Substitution method of integration
• Particular solution

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments

College Board AP Calculus Standards

• LIM-5.A Approximate a definite integral using geometric and numerical methods.
• CHA-4.C Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion.
• CHA-4.D Interpret the meaning of a definite integral in accumulation problems.
• CHA-5.A Calculate areas in the plane using the definite integral.
• CHA-5.B Calculate volumes of solids with known cross sections using definite integrals.
• CHA-5.C Calculate volumes of solids of revolution using definite integrals.

• Trapezoidal Rule.
• Simpson’s Rule.
• Displacement as an integral of velocity.
• Integrals to represent accumulated work.
• Areas between curves in a two-dimensional plane.
• Volumes in a three-dimensional plane.

• Definite integrals can be approximated using geometric and numerical methods.
• Definite integrals allow us to solve problems involving the accumulation of change over an interval.
• Definite integrals allow us to solve problems involving the accumulation of change in area or volume or an interval.

• Solve problems in which a rate is integrated to find the net change over time in a variety of applications.
• Set up and evaluate integrals to find accumulated work.
• Use integration to calculate areas of regions in a plane.
• Use integration (by slices or shells) to calculate volumes of solids.
• Use integration to calculate surface areas of solids of revolution. 

• How does the definite integral represent net change in a quantity, and how can it be used to analyze and model cumulative change over an interval?
• How can calculus be used to calculate areas and volumes of regions and solids, and what methods are best suited for different geometric configurations?
   

• How do you use the trapezoidal rule and Simpson’s rule to estimate a definite integral?
• What strategies can be used to find the net change in a position, velocity, acceleration model?
• How do we set up and evaluate an integral in order to find accumulated work?
• How do we use integrals to find the areas between curves in a two-dimensional plane?
• How do we set up and evaluate an integral in order to find the volumes of solids with known cross-sections?
• How do we set up and evaluate an integral in order to find the volumes of solids of revolution?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Trapezoidal rule
• Foot-pound
• Area between curves
• Volume by cylindrical shells
• Simpson’s rule
• Joule
• Solid of revolution
• Volume by slicing
• Subinterval
• Hooke’s law
• Volume of a solid
• Volume by disks
• Displacement
• Force constant
• Cross section
• Volume by washers

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments