AP Calculus Curriculum Guide Dunmore School District Dunmore, PA

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1 AP Calculus Dunmore School District Dunmore, PA

2 AP Calculus Prerequisite: Successful completion of Trigonometry/Pre-Calculus Honors Advanced Placement Calculus is the highest level mathematics course offered by the Dunmore School District. It is very rigorous and taught at the college level. Topics covered in this course include analytic geometry, limits of functions, differentiation and integration of functions, and applications of differentiation and integration. The Advanced Placement Calculus course prepares students to take the Advanced Placement Calculus Examination in May of their senior year, thus affording these students with the opportunity to do college level work and earn college credit while still in high school. This course will challenge even the most capable of mathematical minds. The work covered in this course will help the student develop analytical reasoning skills and disciplined study habits necessary for success in college. Students pursuing college majors requiring advanced mathematics courses will benefit from this advanced mathematics training. Since the content of the Advanced Placement Calculus AB course is beyond the scope of Common Core, the Collegeboard Curriculum Framework document was used as a guide to write this curriculum. AP Calculus Page 1

3 Year-at-a-glance Subject: AP Calculus Grade Level: 12 Date Completed: 5/22/17 1 st Quarter Topic Resources AP Standards Trigonometry/Pre-Calculus Review Summer Packet Chapter P Limits Chapter 1, Chapter 3 LO 1.1A(a), LO 1.1A(b) LO 1.1B, LO 1.1C, LO 1.1D EK 1.1A2, EK 1.1B1, EK 1.1C1, EK 1.1C2, EK 1.1C3, EK 1.1D1 Continuity Differentiability Chapter 1 Chapter 2 LO 1.2A, LO 1.2B LO 2.2B, EK 2.2B1, EK 2.2B2 AP Calculus Page 2

4 Derivatives Dunmore School District Chapter 2, Chapter 5 (natural logarithmic and exponential functions) LO 2.1A, EK 2.1A5 EK 2.1B1 AP Calculus Page 3

5 2 nd Quarter Derivative Rules Topic Resources AP Standards LO 2.1C, EK 2.1C4 EK 2.1C2, EK 2.1A5 Geometric Applications of Derivatives Implicit Differentiation Higher Order Derivatives Chapter 2 Chapter 2 Chapter 2 Chapter 2 LO 2.1A LO 2.1C, EK 2.1C5 LO 2.3B, EK 2.3B1, EK 2.3B2 LO 2.1D AP Calculus Page 4

6 Related Rates Motion Extreme Values Implications of Derivatives Dunmore School District Chapter 2 Throughout Exercises and Supplemental Materials Chapter 3 Chapter 3 LO 2.3C, EK 2.3C2 LO 2.3C, EK 2.3C1 LO 2.3C, EK 2.3C3 LO 2.4A, EK 2.4A1 AP Calculus Page 5

7 3 rd Quarter Topic Resources AP Standards Using Derivatives to Analyze Graphs Connecting f and f with the Graph of f(x) Optimization Problems More Applications of Derivatives Chapter 3 Chapter 3, Supplemental Materials Chapter 3 Chapter 3 LO 2.2A, EK 2.2A1 LO 2.2A, EK 2.1D1, EK 2.2A1 LO 2.3C, EK 2.3C3 LO 2.3B, EK 2.3B1, EK 2.3B2 AP Calculus Page 6

8 Antiderivatives The Definite Integral Concept Evaluate Definite Integrals The Definite Integral Dunmore School District Chapters 4 and 5 Chapter 4 Chapter 4 Chapter 4 LO 3.1A, EK 3.1A1, EK 3.1A2, LO 3.3B(a), EK 3.3B3, EK 3.3B5 LO 3.2A(a), EK 3.2A1, EK 3.2A3 LO 3.2B, EK 3.2B1, EK 3.2B2, LO 3.2C LO 3.3B(b), EK 3.3B2 LO 3.3A, EK 3.3A1, EK 3.3A2, EK 3.3A3 AP Calculus Page 7

9 4 th Quarter Applications of Integrals I Topic Resources AP Standards LO 2.3E, EK 2.3E2, LO 2.3F, EK 2.3F1, LO 3.5A, EK 3.5A1 EK 3.5A2, LO 3.5B, EK 3.5B1 Applications of Integrals II AP Review Post AP Topics Review and Final Exam Chapter 6 Chapter 7 Review Packet designed material LO 3.4A, EK 3.4A1, EK 3.4A2, LO 3.4B, EK 3.4B1, LO 3.4C, EK 3.4C1, LO 3.4D, EK 3.4D1, EK 3.4D2, LO 3.4E, EK 3.4E1 AP Calculus Page 8

10 General Topic AP Calculus Standards Essential Knowledge, Skills & Vocabulary Review (Summer Assignment) Calculus Library of Functions Linear functions Resources & Activities Summer Packet Assessments Suggested Time (In Days) 3 days Functions as models of change Transformation of functions Solving equations (algebraically and on the calculator) Limits LO 1.1A(a) Express limits symbolically using correct notation. LO 1.1A(b) Interpret limits expressed symbolically. LO 1.1B Estimate limits of functions. Introduction to limits: an intuitive interpretation Intuitive definition estimating limits from graphs and tables Understanding indeterminate forms of limits days EK 1.1B1 Numerical and graphical information can be used to estimate limits. AP Calculus Page 9

11 Limits at a Point EK 1.1A2 The concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits. The Algebra of Limits LO 1.1C Determine limits of functions. EK 1.1C1 Limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules. Properties of limits Substitution (continuous functions) Intuitive discussion of removable versus nonremovable discontinuities Factoring (removable discontinuities) 1.2, , 1.4 Chapter 8 EK 1.1C2 The limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem (Sandwich Theorem). EK 1.1C3 Limits of the indeterminate forms, 0 0 and may be evaluated using L Hospital's Rule. AP Calculus Page 10

12 Limits Involving Infinity LO 1.1D Deduce and interpret behavior of functions using limits EK 1.1D1 Asymptotic and unbounded behavior of functions can be explained and described using limits Limits at Asymptotic and unbounded behavior Understanding asymptotes in relationships to graphs Algebraic techniques for evaluating limits at infinity Various non-existent limits Limits involving trigonometric functions Proof of special trigonometric limit: sin x lim = 1 x 0 x Continuity LO 1.2A Analyze functions for intervals of continuity or points of discontinuity. LO 1.2B Determine the applicability of important Calculus theorems using continuity. Continuity Continuity at a point Discontinuous functions Removable discontinuity Jump discontinuity Continuous functions 1.4 Intermediate Value Theorem AP Calculus Page 11

13 Derivative Concepts LO 2.1A Identify the derivative of a function as the limit of a difference quotient. EK 2.1A5 The derivative can be represented graphically, numerically, analytically, and verbally. Definition of the derivative (difference quotient) Derivative at a point Finding the derivative using the definition Proof: Power Rule for derivatives days EK 2.1B1 The derivative at a point can be estimated from information given in tables or graphs. Derivative Rules LO 2.1C Calculate derivatives. Derivative rules: Constant rule Constant multiple rule Sum and difference rules Power rules Product and quotient rules Proper form of derivatives 2.2, 2.3 Derivatives of sine and cosine functions and the other trigonometric functions AP Calculus Page 12

14 Chain Rule EK 2.1C4 The chain rule provides a way to differentiate composite functions. Derivatives of composite functions Derivatives using repeated use of the chain rule 2.4 Derivatives of Various Functions EK 2.1C2 Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric. Derivatives of exponential and logarithmic functions Derivatives of inverse trigonometric functions 2.2, 2.3, , 5.4, 5.6 Derivatives Geometric Applications LO 2.1A Identify the derivative of a function as the limit of a difference quotient. Geometric applications of the derivative and rates of change Average rates of change versus instantaneous rates of change Using the derivative to find information necessary to write the equations of tangent lines and normal lines Throughout Chapter 2 Exercises Using the derivative to calculate points of horizontal tangencies AP Calculus Page 13

15 Approximating derivatives from tables and graphs Derivatives at a point One-sided derivatives Higher Order Derivatives LO 2.1D Determine higher order derivatives. Differentiability LO 2.2B Recognize the connection between differentiability and continuity. Second and higher order derivatives Various forms of derivatives/limit of a difference quotient Why the derivative may fail to exist Local linearity 2.3, 2.4 EK 2.2B1 A continuous function may fail to be differentiable at a point in its domain. 2.1 EK 2.2B2 If a function is differentiable at a point, then it is continuous at that point. AP Calculus Page 14

16 Numerical Derivatives EK 2.1A5 The derivative can be represented graphically, numerically, analytically, and verbally. Finding derivatives on the graphing calculator Supplemental Material Implicit Derivatives LO 2.1C Calculate derivatives. Derivatives Geometric Applications EK 2.1C5 The chain rule is the basis for implicit differentiation. LO 2.3B Solve problems involving the slope of a tangent line. EK 2.3B1 The derivative at a point is the slope of the line tangent to a graph at that point on the graph. Explicit versus implicit definitions of functions Implicit differentiation process Using implicit differentiation to write equations of tangent and normal lines to functions Using implicit differentiation to calculate points of horizontal tangencies and equations of vertical asymptotes EK 2.3B2 The tangent line is the graph of a locally linear approximation of the function near the point of tangency. AP Calculus Page 15

17 Numerical Derivatives EK 2.1A5 The derivative can be represented graphically, numerically, analytically, and verbally. Finding and evaluating implicit derivatives on the graphing calculator Supplemental Material Related Rates LO 2.3C Solve problems involving related rates, optimization, and rectilinear motion. EK 2.3C2 The derivative can be used to solve related rates problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known. What are related rates of change Related rate equations Related rate problem strategies 2.6 Motion LO 2.3C Solve problems involving related rates, optimization, rectilinear motion. EK 2.3C1 The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration. Position, velocity, acceleration, and particle motion Finding position, velocity, and acceleration from graphs and tables Throughout Exercises Supplemental Materials AP Calculus Page 16

18 Extreme Values LO 2.3C Solve problems involving related rates, optimization, rectilinear motion. EK 2.3C3 The derivative can be used to solve optimization problems, that is, finding a maximum or minimum value of a function over a given interval. Definition of critical value Relative Extrema and curve sketching Absolute Extrema 3.1, days Implications of the Derivatives LO 2.4A Apply the Mean Value Theorem to describe the behavior of a function over an interval. EK 2.4A1 If a function f is continuous over the interval [a, b] and differentiable over the interval (a,b), the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval. Rolle s Theorem Mean Value Theorem 3.2 Supplemental Material AP Calculus Page 17

19 Using Derivatives to Analyze Graphs LO 2.2A Use derivatives to analyze properties of a function. EK 2.2A1 First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection. Increasing and decreasing functions Minimum and Maximum Value Points Pointed and rounded Graphs First Derivative Test for Extrema Applied Maximum and Minimum Applications to Describe Concepts Maximizing Profit and Minimizing Distance with Respect to Time Construction Projects Medicine 3.3, 3.4, 3.6 Analysis of graphs using the first and second derivatives Concavity Second Derivative Test Points of Inflection AP Calculus Page 18

20 Connecting of f and f with the Graph of f(x) LO 2.2A Use derivatives to analyze properties of a function. EK 2.1D1 Differentiating f' produces the second derivative f", provided the derivative of f' exists; repeating this process produces higher order derivatives of f. Connecting the graphs of f and f with the graph of f(x) Throughout Chapter 3 Exercises Supplemental Material EK 2.2A1 First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection. Optimization Problems LO 2.3C Solve problems involving related rates, optimization, rectilinear motion. EK 2.3C3 The derivative can be used to solve optimization Writing and optimizing functions Steps for Optimization 3.7 AP Calculus Page 19

21 More Applications of Derivatives problems, that is, finding a maximum or minimum value of a function over a given interval. LO 2.3B Solve problems involving the slope of a tangent line. EK 2.3B1 The derivative at a point is the slope of the line tangent to a graph at that point on the graph. Local Linearity Differentials Tangent line approximations days EK 2.3B2 The tangent line is the graph of a locally linear approximation of the function near the point of tangency. Antiderivatives LO 3.1A Recognize antiderivatives of basic functions. Antidifferentiation/Indefinite Integral rules Power rule 25 days EK 3.1A1 An antiderivative of a function f is a function g whose derivative is f. EK 3.1A2 Constant Rule Trigonometric rules Exponential and logarithmic rules 4.1, 5.2, 5.3, 5.4, 5.5, 5.7 AP Calculus Page 20

22 Differentiation rules provide the foundation for finding antiderivatives. LO 3.3B(a) Calculate antiderivatives EK 3.3B3 The notation f(x)dx = F(x) + C means that F (x) = f(x) and f(x) is called an indefinite integral of the function f. Inverse trigonometric rules Applications of Antidifferentiation Particular Antiderivative Obtaining Cost from Marginal Cost Obtaining Distance Formula from Velocity EK 3.3B5 Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square, substitution of variables. The Definite Integral Concept LO 3.2A(a) Interpret the definite integral as a limit of a Riemann sum. EK 3.2A1 A Riemann sum, which requires a partition of an Sigma or Summation Notation Riemann Sums/Area Under a Curve Rectangle Approximation of the Area (Left, Right, Midpoint) Trapezoidal Approximation 4.2, days AP Calculus Page 21

23 interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition. Sums of unequal partition width EK 3.2A3 The information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral. LO 3.2B Approximate a definite integral. EK 3.2B1 Definite integrals can be approximated for functions that are represented graphically, numerically, algebraically, and verbally. EK 3.2B2 Definite integrals can be approximated using a left Riemann sum, a right AP Calculus Page 22

24 Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or non-uniform partitions. Evaluate Definite Integrals LO 3.2C Calculate a definite integral using areas and properties of definite integrals. LO 3.3B(b) Evaluate definite integrals. EK 3.3B2 If f is continuous on the interval [a, b] and F is an antiderivative of f. then b f(x)dx = F(b) a F(a). Evaluation by hand and on the calculator Properties of definite integrals 4.4 The Definite Integral LO 3.3A Analyze functions defined by an integral. EK 3.3A1 The definite integral can be used to define new functions. EK 3.3A2 Mean Value Theorem for Integrals The Fundamental Theorem of Calculus FTC 1 FTC 2 Average Value Theorem 4.3, days AP Calculus Page 23

25 If f is a continuous function on the interval (a,b], then d dx x a ( f(t)dt) = f(x) where x is between a and b. EK 3.3A3 Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as x g(x) = f(t)dt. a Applications of Integrals I LO.2.3E Verify solutions to differential equations. EK 2.3E2 Derivatives can be used to verify that a function is a solution to a given differential equation. LO 2.3F Estimate solutions to differential equations. Integration by Substitution Exponential Growth and Decay Slopefields General Differential Equations Newton s Law of Cooling 6.1, 6.2, 10 days EK 2.3F1 Slope fields provide visual clues to the behavior of solutions to AP Calculus Page 24

26 first order differential equations. LO 3.5A Analyze differential equations to obtain general and specific solutions. EK 3.5A1 Antidifferentiation can be used to find specific solutions to differential equations with given initial conditions, including applications to motion along a line, and exponential growth and decay. EK 3.5A2 Some differential equations can be solved by separation of variables. LO 3.5B Interpret, create and solve differential equations from problems in context. EK 3.5B1 AP Calculus Page 25

27 The model for exponential growth and decay that arises from the statement "The rate of change of a quantity is proportional to the size of the quantity" is dy dx = ky. Applications of Integrals II LO 3.4A Interpret the meaning of a definite integral within a word problem. Integral as accumulator of net change Area (with respect to either axis) 15 days EK 3.4A1 A function defined as an integral represents an accumulation of a rate of change. EK 3.4A2 The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval. LO 3.4B Apply definite integrals to problems Area between a curve and an axis Area Between two curves Volumes of solids of revolution (with respect to either axis) Disc method Washer Method Shell method Volumes of solids with known cross-sections (with respect to either axis) 7.1, 7.2, 7.3 AP Calculus Page 26

28 involving the average value of a function. EK 3.4B1 The average value of a function f over an interval [a, b] is 1 b a b a f(x)dx. LO 3.4C Apply definite integrals to problems involving motion. EK 3.4C1 For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle's displacement over the interval of time, and the definite integral of speed represents the particle's total distance traveled over the interval of time. LO 3.4D Apply definite integrals to problems involving area and volume. AP Calculus Page 27

29 EK 3.4D1 Areas of certain regions in the plane can be calculated with definite integrals. EK 3.4D2 Volumes of solids with known cross sections, including discs and washers, can be calculated with definite integrals. LO 3.4E Use the definite integral to solve problems in various contexts. EK 3.4E1 The definite integral can be used to express information about accumulation and net change in many applied contexts. Review Review Packet 6 days Post AP Topics Review and Final Exam designed enrichment topics and projects 17 days 8 AP Calculus Page 28