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AP Calculus Pacing Guide
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Two-week summary of major instructional content and skills • 36 instructional weeks
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Week(s)Unit / TextMajor ContentMajor Skills and Assessments
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Weeks 1–2Unit 1: Pre-RequisitesBasic Function Review; Solving Equation and Inequalities; Review of Logarithms; Evaluating Trig Functions and Thier Graphs; Summer Packet Review and Test (8 days)Recall all basic functions from Algebra II and Precalculus along with the properties of pos/neg, inc/dec, continuity, Domain/Range, and Extrema; A single day crash course on solving ALL equations and inequalities including quadratics, logs/ecp, and absulute value functions; Recall all preevious skills pertaining to logarithms and change of base formulas; Recall the unit circle, important trig identies, and properties of sine/cosine wave funtions; Assessments/Review: Basic Function Review; Review of Logarithms; Summer Packet Review and Test
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Weeks 3–4Unit 2: Limits and Rates of ChangeCan Change Occur in an Instant?; Defining Limits and Limit Notation; Estimating Limits from Graphs; Estimating Limits from tables; Finding Algebraic Property Limits; Finding Limits Using Algebraic Manipulation; Limit Procedures and Trig Limits (9 days)Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant; Represent limits analytically using correct notation; Estimate the limits from a graph; Estimate the vlaue of a limit from a given table; Use algebra t simplfy an expression then evaluate thhe limit through substitution; Simplfy complex algebraic probalems in oder to simplify a limit; Use trignometric identies to estimate a limit
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Weeks 5–6Unit 2: Limits and Rates of ChangeSqueeze Theory; Representation of Limits; Types of Discontinuity; Confirming Continuity Over an Interval; Defining Continuity at a Point; Removing Discontinuity; Infinite Limits and Veritcal Asymptotes; Limits AT Infinity and Horizontal Asymptotes; Intermediate and Extreme Value Theorem; Unit 1 Test and review (10 days)Impliment an upper and lower boundary to squeeze or sandwich out the vlaue of limit; Represent limit in algebric and set builder form; Use limiit notation to identify the three forms of disconiuty; Use a lmit definition in order to show whether a graph is contnious at a point and/or over an interval; The defnition of contuity with limits to determine whether a function is continuous at a point; Find the value of limit as a hole in a graph and then construct a piecewise function to fill that removable discontuity; Use a limit notation to identify a VA on a graph; Use a limit notation to identify a HA on a graph; Apply a limit definition to the extreme value theorem; Assessments/Review: Unit 1 Test and review
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Weeks 7–8Unit 3: Derivatives and DifferentiabilityAverage and Instantaneous Rate of Change; Limit Definition of a Derivative; Estimating Derivatives at a Point; Differentiability and Continuity; Power Rule; Constant, Sum, Difference, and Constant Multiple Rule; Derivatives of cos, sin, e^x and ln(x); The Product Rule (10 days)Recall the information to calculate an average change and apply a limit to identify an instaneaneous rate of change; The limit defnition of a derative in order to determine IROC; Graphically connecting that slope of tangent line will estimatee the value of IROC; Definie a smooth continuous curve is completely differentiable; Derive the Power rule using the limit definition of a derivative then applyy to rule to find the derivatives of a function; Apply the Rules of derivatives to expand to finding the derivatice of power functions and polynomials; The definitions of the derivative for cos, sin, exponentials and logs; Derive and apply the rules for the product of functions to find the derivative of that produt
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Weeks 9–10Unit 3: Derivatives and Differentiability / Unit 4: More DerivativesThe Quotient Rule; Deratives of Tan, Cot, Sec, and Csc; Unit 2 Test and Review; The Chain Rule (9 days)The LoDHi rule in orer to find the derivative of rational functions or the quotient of two fucnitons; The derivatives of the other 4 basic trig functions in order to expand the ammount of curves we can find the derivative of; Use the chain rule for derivatives to find the slopes of functions that are constructed via composition; Assessments/Review: Unit 2 Test and Review
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Weeks 11–12Unit 4: More Derivatives / Unit 5: Contextual and Analytical Applications of DifferentiationImplicit Differentiation; Differentiating Inverse Functions; Derivatives of Inverse Trig Functions; Procedures for Calculating Derivatives; Calculating Higher Order Derivativces; Unit 3 Review and Test; Interpretting A Derivative in Context (9 days)Differentiate using the rules for functions that are implicitly defined in terms of x and y and not just in terms of x; Use the inverse rule of functions to find the slope of a functions inverse at its equivalent output value from the first input; Memorize the rules for finding the derivatives of the 6 inverse trigonometry functions; Determine the differences between each "flavor" of function in order to apply different derivative rules to differentiate and find slopes of tangent lines; To find 2nd, 3rd, 4th, and higher derivatives of both explicitly defined and implicitly defined functions; Recognize a derivative shows rate of change or clope of a tangent line on a graph; Assessments/Review: Unit 3 Review and Test
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Weeks 13–14Unit 5: Contextual and Analytical Applications of DifferentiationStraight-Line Motion; Rates of Changes (Outside of Motion); Related Rates; Applying Related Rates (11 days)Pratice applications of Instaneasous rate of change in straight line motion and particle motion problem; See exmaples of IROC problems and application of derivatives, in Area/Volume changes, finances, and queing situations; Students wil be able to complete differential application problems in volving changes in area and volume as a changee in time (Related Rate); Complete application word problems using tools of solving related rates
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Weeks 15–16Unit 5: Contextual and Analytical Applications of DifferentiationLinearization; L'Hospital's Rule; Unit 4 Review and Test; Mean Value Theorem; Extreme Value Theorem and the Global Argument; Determining Increasing/Decreasing Intervals; The First Derivative Test; The Candidates Test (10 days)Use a tangent line approximation to evaluate under or over eestimates of irrational numbers; Use derivitives in conjustion with L'Hops rule to evaluate indeterinate limits; Apply a Mean Value theorem to find the value on a function that has the same instaneaouse rate of change as the Average rate of cchange; Apply the Extreme value theorem to determine where absolute min or max values are on a continous function over a closed interval; Use a first derivative in order to determine where a function is increasing or descreasing (slope is positive or negative); Use a first derivative test in order to determine where local min/max exist on the interior of a function; Combine the global argument with a FDT in order to find the location of a min/max that is absolute via a candidate test; Assessments/Review: Unit 4 Review and Test; The First Derivative Test; The Candidates Test
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Weeks 17–18Unit 5: Contextual and Analytical Applications of DifferentiationDetermining Concavity; The Second Derivative Test; Sketching Graphs and Their Derivatives; Connecting f to f' to f"; Optimization; Unit 5 Review and Test (11 days)Studnets will use a second derivative to determine where a function is concave up or concave down; A second derivative test at a Critical value of the first derivative, also to determine whether a function as a min/max or the interior; Information from the first and second derivatives in order to sketch functions and their Derivatives is necessary; All information from derivaitve tests to determine infroamtion about f, f', and f" from f, f', or f" (this is not a typo); The first derivative test in aplication problems to find minimum and maximum values that will solvee or optimize certain situations. (Example: If I have 500feet of fencing, what is the maximum Area that I can inclose if one side of the yard is up against a river and doesnnt need a fence; Assessments/Review: The Second Derivative Test; Unit 5 Review and Test
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Weeks 19–20Unit 6: Indefinite and AccumulationAccumulation of Change; Approximating Area with Reimann Sums; Summation Notation and the Definite Integral; The Fundamental Theorem of Calculus (10 days)Connect the the area under a velocity graph is a n accumlation of area eual to position change; The recangle approximation METHODS (left, right, middle, aver, and trapezoidal) in order to estimate area under a curve; Take the limit of an infinte sum inorder to devolp notation for integration also know as the inifinte sum of infinitely many skinny pieces; Apply thee Fundamental thheorem of calculus to evaluate exact values of definite bounded inteegrals
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Weeks 21–22Unit 6: Indefinite and AccumulationAccumulation Functions Involving Area; Properties of Definite Integrals; FTOC and Definite Integrals; AntiDerivatives and Indefinite Integrals (8 days)Aplly the integral defniition inorder to calculate exact areas under curved shapes; Manipulate, change, and slve for values of definite integrals or values of areas using the properties of integrals; Determine the second part of the FTOC in order to connect integration to the definition of a derivative as opposites of each other; Be be able to use the concept of antidifferentiation in order to find the value of an indefinite integral plus a constant value
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Weeks 23–24Unit 6: Indefinite and AccumulationIntegration by U-Substitution; Long Division and Completing The Square to Integrate; Techniques for Antidifferentiation; Unit 6 Review and Test (10 days)Leanr the process of u-substitutions in order to expand the ammount of functions they can take the indefinite ingral of; For more complicated integrals, algebraiac methods of CTS and LD can be used to simplify and/or break up an intergral into smaller pieces that can then be inteegrated indefnitely; Work on pattern and problem recognition skills in order to quickly determine an approach to every ""flavor" of integration problem; Assessments/Review: Unit 6 Review and Test
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Weeks 25–26Unit 7: Differential Equations and Slope FieldsModeling With DiffEQs; Verifying solutions to DiffEQs; Sketching Slope Fields; Analyzing with Slope Fields; Finding General Solutions w/ Separation of Variables; Finding a Particular Solution; Exponential Models With DiffEQs; Unit 7 Review and Test (9 days)Model real world equations as differential equations (equations that have a derivative in them); Take possible solutions to first and second order DE's w find their derivatives and then substitute them back into the problem in order to verify that solutions work; Use a slope field in order to model the family of solutions that can solve a DIffEQ; Use the slope field to extract particular solutions and determine Initial conditions that satisfy a particular solution; Use the separation of variables tchnique to integrate both sides of a diffEQ in order to find a general (family) of solution that solve the equation; Use an initail condition aftr finding a general solution in order to find a particular, also known as a specific, solution; Solve the simple y'=ky diffEQ to open up the concept of an exponential solution to problems, like bacteria growth, half-life, and Newton's Law of Cooling Problems; Assessments/Review: Unit 7 Review and Test
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Weeks 27–28Unit 7: Differential Equations and Slope Fields / Unit 8: Applications of IntegralsSenior Trip; Average Value of a Function; Connecting Position, Velocity and Acceleration; Using Accumulation Functions w/ Integrals (9 days)B able to take the definite integral of a function and divide by the difference of its bound to calculate the average value of a funcction; Apply properties of integrals and integration to solve particle motio problems in the opposite dirrection as the did with their derivative tools; Use inteegrals in order to find and solve accumlation problems. (the value of fucntion is what you started with plus what you hae accumulated over an interval
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Weeks 29–30Unit 8: Applications of IntegralsArea Between Two Curves; Area Between Two curves expressed as y; Area Between two curves with multiple Intersections; Volume w/ square and rectangle cross sections; Volume w/ triangle and semcircle cross sections; Volume: Disc Method; Volume: Disc Method with other rotational axes; Volume: Washer Method; Volume: Washer Method with other rotational axes; Unit 8 Review and Test (10 days)Calculate the area bounded inbetween two curves; Calcuate the area bounded by two curves with bounds on the left and right rather then above and below; Be ble to find the area bounded between two curves even when two three or more "lobes" area created via their intersections; Create solids and find the volumes produced by an accumaltion of recatangles; Create solids and find their volumes when the solid is prodcuced by an accumlation of triangles or semicircles; Studnets will be able to use integration and the method of Discs to calculate solis from funcitons rotated around an axis; Studnets will be able to use integration and the method of Discs to calculate solis from funcitons rotated around an axis that is not the x-axis; Combine the Disc method with the Area between two curves method in order to calculator volumes of convex shapes that are the result of rotating the area produced betweeen two curves around an axis; Combine the Disc method with the Area between two curves method in order to calculator volumes of convex shapes that are the result of rotating the area produced betweeen two curves around an axis or thn the x or y axis; Assessments/Review: Unit 8 Review and Test
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Weeks 31–32Unit 9: AP Test Review and Bonus TopicsReview and Practice Limits MCQ; Limits FRQ; Review and Practice Derivatives MCQ; Derivatives FRQ; Review and Practice Integrals MCQ; Integrals FRQ; Test Prep and Review; AP Calculus AB Test Date (11 days)An extensiv crash course on limits and their propertiees and practice Multiple Choice Items; An extensiv crash course on limits and their propertiees and practice Free Response Items; An extensiv crash course on derivatives and their propertiees and practice Multiple Choice Items; An extensiv crash course on derivatives and their propertiees and practice Free Response Items; An extensiv crash course on integration and their propertiees and practice Multiple Choice Items; An extensiv crash course on integration and their propertiees and practice Free Response Items; Any exta time can be dediccated to individual Test Prep; Day of the Chapter test; Assessments/Review: Review and Practice Limits MCQ; Limits FRQ; Review and Practice Derivatives MCQ; Derivatives FRQ; Review and Practice Integrals MCQ; Integrals FRQ; Test Prep and Review; AP Calculus AB Test Date
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Weeks 33–34Unit 9: AP Test Review and Bonus Topics6.11 - Integration by Parts; 8.13 - Arc Length and Distance Traveled; 7.9 - Logistic Modeling with DiffEQ's (11 days)Learn an advanced calculaus skil to solve an integration by parts integral or using the table meethod (tic tac toe or hindu method) Students will also learn the skills necessary to solve for the unknown Integral with this method; Calculate arc length and distance traveled by applying an advanced calculus formula and applying integration; Model real-world situation with a logist model or a linear diffEQ which follows a near similar pattern to exponential diffEQ's. These problems will lead to solutions of rumor course and disease spread through a population. they cna also model population growth and decay situations that contain environmental controls
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Weeks 35–36Unit 9: AP Test Review and Bonus Topics7.5 - Eulers method of Approximation; Final Project choices, improper integrals, sequences and series, indeterminate forms, and cardinality are options for final topis to finish the last week of school (10 days)Use Eulers approximation to approximate a DiffEQ without separation of variables by creating a polynomial chain that gets more accurate depending on delta 'x' or 'y' and how many steps or iterations are used; Final Project choices, improper integrals, sequences and series, indeterminate forms, and cardinality are options for final topis to finish the last week of school; Assessments/Review: Final Project choices, improper integrals, sequences and series, indeterminate forms, and cardinality are options for final topis to finish the last week of school
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