Calculus Honors Curriculum Guide Dunmore School District Dunmore, PA

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1 Calculus Honors Dunmore School District Dunmore, PA

2 Calculus Honors Prerequisite: Successful completion of Trigonometry/Pre-Calculus Honors Major topics include: limits, derivatives, integrals. Instruction is presented in a variety of ways including graphically, numerically, analytically, and verbally. Students will be required to use graphing calculators to help solve problems. Since the content of a Calculus course is beyond the scope of Common Core, the Collegeboard Curriculum Framework document was used as a guide to write this curriculum. Calculus Honors Page 1

3 Year-at-a-glance Subject: Calculus Honors Grade Level: 12 Date Completed: 5/22/17 1 st Quarter Topic Resources Standards Preparation for Calculus (Part I) HSF.IF.C.8.a, A Chapter P Preparation for Calculus (Part II) Chapter P HSF.IF.C.7.b, HSF.TF.B.5 Limits Continuity Chapter 1, Chapter 3 Chapter 1 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 LO 1.2A, LO 1.2B Calculus Honors Page 2

4 Differentiability Dunmore School District Chapter 2 LO 2.2B, EK 2.2B1, EK 2.2B2 Calculus Honors Page 3

5 2 nd Quarter Derivatives Topic Resources Standards LO 2.1A, EK 2.1A5 EK 2.1B1 Chapter 2, Chapter 5 (natural logarithmic and exponential functions) Derivative Rules Geometric Applications of Derivatives Implicit Differentiation Chapter 2 Chapter 2 Chapter 2 LO 2.1C, EK 2.1C4 EK 2.1C2, EK 2.1A5 LO 2.1A LO 2.1C, EK 2.1C5 LO 2.3B, EK 2.3B1, EK 2.3B2 Calculus Honors Page 4

6 Higher Order Derivatives Related Rates Dunmore School District Chapter 2 Chapter 2 LO 2.1D LO 2.3C, EK 2.3C2 Calculus Honors Page 5

7 3 rd Quarter Topic Resources Standards Motion Extreme Values Implications of Derivatives Throughout Exercises and Supplemental Materials Chapter 3 Chapter 3 LO 2.3C, EK 2.3C1 LO 2.3C, EK 2.3C3 LO 2.4A, EK 2.4A1 Using Derivatives to Analyze Graphs Chapter 3 LO 2.2A, EK 2.2A1 Calculus Honors Page 6

8 Connecting f and f with the Graph of f(x) Optimization Problems More Applications of Derivatives Dunmore School District Chapter 3, Supplemental Materials Chapter 3 Chapter 3 LO 2.2A, EK 2.1D1, EK 2.2A1 LO 2.3C, EK 2.3C3 LO 2.3B, EK 2.3B1, EK 2.3B2 Calculus Honors Page 7

9 4 th Quarter Antiderivatives Topic Resources Standards LO 3.1A, EK 3.1A1, EK 3.1A2, LO 3.3B(a), EK 3.3B3, EK 3.3B5 The Definite Integral Concept Evaluate Definite Integrals The Definite Integral Chapters 4 and 5 Chapter 4 Chapter 4 Chapter 4 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 Calculus Honors Page 8

10 Applications of Integrals I Applications of Integrals II Review and Final Exam Dunmore School District Chapter 6 Chapter 7 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 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 Calculus Honors Page 9

11 General Topic Standards Essential Knowledge, Skills & Vocabulary Preparation for Calculus I HSF.IF.C.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Graphs and Models Linear Models and Rates of Change Resources & Activities Assessments Suggested Time (In Days) P.1, P.2 5 days A Write or identify a linear equation when given: the graph of a line, two points on a line, the slope and point on the line, and parallel and perpendicular lines. Preparation for Calculus II HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. HSF.TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Functions and Their Graphs Fitting Models to Data P.3, P.4 5 days Calculus Honors Page 10

12 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. 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 Properties of limits Substitution (continuous functions) Factoring (removable discontinuities) 1.2, , 1.4 Chapter 8 Calculus Honors Page 11

13 theorems of limits and algebraic rules. 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 and may be 0 evaluated using L Hospital's Rule. 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 Calculus Honors Page 12

14 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 Continuous functions Intermediate Value Theorem days Differentiability LO 2.2B Recognize the connection between differentiability and continuity. Why the derivative may fail to exist Local linearity 5 days 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. Calculus Honors Page 13

15 Derivative Concepts LO 2.1A Identify the derivative of a function as the limit of a difference quotient. Definition of the derivative (difference quotient) Derivative at a point 15 days EK 2.1A5 The derivative can be represented graphically, numerically, analytically, and verbally. Finding the derivative using the definition 2.1 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 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 Calculus Honors Page 14

16 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 Calculus Honors Page 15

17 Using the derivative to calculate points of horizontal tangencies Approximating derivatives from tables and graphs Derivatives at a point One-sided derivatives Higher Order Derivatives Numerical Derivatives LO 2.1D Determine higher order derivatives. EK 2.1A5 The derivative can be represented graphically, numerically, analytically, and verbally. Second and higher order derivatives Various forms of derivatives/limit of a difference quotient Finding derivatives on the graphing calculator 2.3, 2.4 Supplemental Material 5 days 5 days Implicit Derivatives LO 2.1C Calculate derivatives. EK 2.1C5 The chain rule is the basis for implicit differentiation. Explicit versus implicit definitions of functions Implicit differentiation process days Calculus Honors Page 16

18 Derivatives Geometric Applications 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. 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 2.5 EK 2.3B2 The tangent line is the graph of a locally linear approximation of the function near the point of tangency. 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 5 days Related Rates LO 2.3C Solve problems involving related rates, optimization, rectilinear motion. Related rate problems 5 days 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. 2.6 Calculus Honors Page 17

19 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 4 days Extreme Values LO 2.3C Solve problems involving related rates, optimization, rectilinear motion. Definition of critical value Relative Extrema and curve sketching 34 days 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. Absolute Extrema 3.1, 3.3 Calculus Honors Page 18

20 Implications of the Derivatives Using Derivatives to Analyze Graphs 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. 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. Rolle s Theorem Mean Value Theorem Increasing and decreasing functions Minimum and Maximum Value Points Pointed and rounded graphs First Derivative Test for Extrema Analysis of graphs using the first and second derivatives Concavity Second Derivative Test Points of Inflection 3.2 Supplemental Material 3.3, 3.4, 3.6 Calculus Honors Page 19

21 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. Calculus Honors Page 20

22 Optimization Problems LO 2.3C Solve problems involving related rates, optimization, rectilinear motion. Writing and optimizing functions Steps for Optimization 3 days More Applications of Derivatives 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. 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. Calculus Honors Page 21

23 Antiderivatives LO 3.1A Recognize antiderivatives of basic functions. Antidifferentiation/Indefinite Integral rules Power rule 15 days EK 3.1A1 An antiderivative of a function f is a function g whose derivative is f. EK 3.1A2 Differentiation rules provide the foundation for finding antiderivatives. Constant rule Trigonometric rules Exponential and logarithmic rules Inverse trigonometric rules 4.1, 5.2, 5.3, 5.4, 5.5, 5.7 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. EK 3.3B5 Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square, substitution of variables. Calculus Honors Page 22

24 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 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. Sigma or Summation Notation Basic Interpretations: Riemann Sums/Area Under a Curve Rectangle Approximation of the Area (Left, Right, Midpoint) Trapezoidal Approximation Sums of unequal partition width 4.2, days 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 Calculus Honors Page 23

25 graphically, numerically, algebraically, and verbally. EK 3.2B2 Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or non-uniform partitions. LO 3.2C Calculate a definite integral using areas and properties of definite integrals. Evaluate 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 a f(x)dx = F(b) F(a). Evaluation by hand and on the calculator Properties of definite integrals 4.4 Calculus Honors Page 24

26 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 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. The Fundamental Theorem of Calculus Average Value Theorem 4.3, days 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 Calculus Honors Page 25

27 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. Integration by Substitution Exponential Growth and Decay General Differential Equations 6.1, 6.2, 5 days LO 2.3F Estimate solutions to differential equations. EK 2.3F1 Slope fields provide visual clues to the behavior of solutions to 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 Calculus Honors Page 26

28 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 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. EK 3.4A1 A function defined as an integral represents an accumulation of a rate of change. Area (with respect to either axis) Area Between a curve and an axis Area Between two curves Volumes of solids of revolution (with respect to either axis) Disc method 7.1, 7.2, days Calculus Honors Page 27

29 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. Shell method Washer Method Volumes of solids with known cross-sections (with respect to either axis) LO 3.4B Apply definite integrals to problems 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. Calculus Honors Page 28

30 Review and Final Exam LO 3.4D Apply definite integrals to problems involving area and volume. 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. 8 days Calculus Honors Page 29