Scranton School District Scranton, PA
Prerequisites: Successful completion of Elementary Analysis or Honors Elementary Analysis is a high level mathematics course offered by the Scranton School District. Topics covered in this course include analytic geometry, limits of functions, differentiation and integration of functions, and applications of differentiation and integration. 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 is beyond the scope of Common Core, the Collegeboard Curriculum Framework for AP Calculus AB 2016-2017 document was used as a guide to write this curriculum. Page 1
Year-at-a-glance Subject: Grade Level: 12 Date Completed: 3/1/15 1 st Quarter Elementary Analysis Review Topic Resources AP Calculus Standard Chapter P Limits Chapter One, Chapter Three 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 1.1D1 Page 2
2 nd Quarter Limits (continued) Continuity Topic Resources AP Calculus Standard LO 1.1A(a), LO 1.1A(b) Chapter One, Chapter Three LO 1.1B, LO 1.1C, LO 1.1D Chapter One EK 1.1A2, EK 1.1B1, LO 1.2A, LO 1.2B Differentiablilty Derivatives Derivative Rules Geometric Applications of Derivatives Chapter Two Chapter Two Chapter Two Chapter Two LO 2.2B, EK 2.2B1, EK 2.2B2 LO 2.1A, EK 2.1A5 EK 2.1B1 LO 2.1C, EK 2.1C4 EK 2.1C2, EK 2.1A5 LO 2.1A Page 3
3 rd Quarter Implicit Differentiation Higher Order Derivatives Related Rates Motion Extreme Values Using Derivatives to Analyze Graphs Optimization Problems Topic Resources AP Calculus Standard LO 2.1C, EK 2.1C5 Chapter Two LO 2.3B, EK 2.3B1, EK 2.3B2 Chapter Two Chapter Two Throughout Exercises and Supplemental Materials Chapter Three Chapter Three Chapter Three LO 2.1D LO 2.3C, EK 2.3C2 LO 2.3C, EK 2.3C1 LO 2.3C, EK 2.3C3 LO 2.2A, EK 2.2A1 LO 2.3C, EK 2.3C3 Page 4
4 th Quarter Antiderivatives Topic Resources AP Calculus Standard LO 3.1A, EK 3.1A1, EK 3.1A2, Chapters Four and Five LO 3.3B(a), EK 3.3B3, EK 3.3B5 The Definite Integral Concept Evaluate Definite Integrals Chapter Four Chapter Four 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 The Definite Integral Chapter Four LO 3.3A, EK 3.3A1, EK 3.3A3 Applications of Integrals Chapter Seven LO 3.4D, EK 3.4D1 Final Review Review Packet Page 5
Review Limits General Topic Academic Standard(s) LO 1.1A(a) LO 1.1A(b) Essential Knowledge, Skills & Vocabulary Calculus Library of Functions Linear functions Functions as models of change Transformation of functions Solving equations(algebraically and on the calculator) Express limits symbolically using correct notation. Interpret limits expressed symbolically. Resources & Activities Assessments Suggested Time Chapter P 1.2 Teacher prepared tests, quizzes, etc. 35 20 Intuitive definition of Limits LO 1.1B EK 1.1B1 Estimate limits of functions. Numerical and graphical information can be used to estimate limits. Visualizing 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. 1.2 1.2, 3.5 Page 6
The algebra of limits: LO 1.1C EK 1.1C1 EK 1.1C2 Determine limits of functions. Limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules. The limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem (Sandwich Theorem). 1.3, 1.4 Substitution (continuous functions) Intuitive discussion of removable versus nonremovable discontinuities Factoring (removable discontinuities) Page 7
Limits Involving Infinity LO 1.1D EK 1.1D1 Deduce and interpret behavior of functions using limits Asymptotic and unbounded behavior of functions can be explained and described using limits Asymptotic behavior (horizontal and vertical asymptotes) Continuity LO 1.2A Analyze functions for intervals of continuity or points of discontinuity. 1.5 3.5 1.4 Continuity at a point Continuous Functions Discontinuous functions: Removable discontinuities Jump discontinuities LO 1.2B Determine the applicability of important Calculus theorems using continuity. Continuous functions Intermediate Value Theorem 1.4 Page 8
Derivative Concepts LO 2.1A EK 2.1A5 EK 2.1B1 Identify the derivative of a function as the limit of a difference quotient. 2.1 38 The derivative can be represented graphically, numerically, analytically, and verbally. The derivative at a point can be estimated from information given in tables or graphs. Definition of the derivative (difference quotient) Derivative at a point Page 9
Derivative Rules LO 2.1C Calculate derivatives. Constant rule Constant multiple rule 2.2, 2.3 Sum and difference Power rule Product and quotient rules Proper form of derivatives (factored) Chain rule EK 2.1C4 The chain rule provides a way to differentiate composite functions Derivatives of composite functions 2.4 Derivatives using repeated use of the chain rule Derivatives of Various Functions EK 2.1C2 Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, and trigonometric functions. 2.2, 2.3, 2.4 Page 10
Derivatives Geometric Applications LO 2.1A Identify the derivative of a function as the limit of a difference quotient. Throughout Chapter 2 Exercises 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 Using the derivative to calculate points of horizontal tangencies Higher Order Derivatives LO 2.1D Determine higher order derivatives. Second and higher order derivatives 2.3, 2.4 Page 11
Differentiability LO 2.2B EK 2.2B1 EK 2.2B2 Recognize the connection between differentiability and continuity. A continuous function may fail to be differentiable at a point in its domain. If a function is differentiable at a point, then it is continuous at that point. 2.1 Why the derivative may fail to exist Numerical Derivatives EK 2.1A5 The derivative can be represented graphically, numerically, analytically, and verbally. Supplemental Material Implicit Derivatives LO 2.1C EK 2.1C5 Finding derivatives on the graphing calculator Calculate derivatives. The chain rule is the basis for implicit differentiation. Explicit versus implicit definitions of functions Implicit differentiation process 2.5 Page 12
Derivatives Geometric Applications LO 2.3B EK 2.3B1 EK 2.3B2 Solve problems involving the slope of the tangent line. The derivative at a point is the slope of the line tangent to a graph at that point on the graph. 2.5 The tangent line is the graph of a locally linear approximation of the function near the point of tangency. Using implicit differentiation write equations of tangent and normal lines to functions Using implicit differentiation calculate points of horizontal tangencies and equations of vertical asymptotes 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 Page 13
Related Rates LO 2.3C EK 2.3C2 Solve problems involving related rates and optimization. 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 What are related rates of change Related rate equations Related rate problem strategies Page 14
Motion LO 2.3C EK 2.3C1 Solve problems involving related rates and optimization and motion. The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration. Position, velocity, and acceleration Throughout Exercises Supplemental Materials Page 15
Extreme Values LO 2.3C 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. 3.1, 3.3 25 Absolute (global) extrema Relative (local) extrema Definition of critical value Page 16
Using derivatives to analyze graphs LO 2.2A EK 2.2A1 Use derivatives to analyze properties of a function. 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. 3.3, 3.4, 3.6 The first derivative test Increasing and decreasing (intervals) The second derivative test Concavity and inflection points Curve Sketching Page 17
Optimization Problems LO 2.3C 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. Writing and optimizing functions 3.7 Page 18
Antiderivatives LO 3.1A EK 3.1A1 EK 3.1A2 LO 3.3B(a) EK 3.3B3 EK 3.3B5 Recognize antiderivatives of basic functions. An antiderivative of a function f is a function g whose derivative is f. 4.1, 5.2, 5.3, 5.4, 5.5, 5.7 30 Differentiation rules provide the foundation for finding antiderivatives. Calculate Antiderivatives 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. Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square, substitution of variables. Indefinite Integral rules Power rules Trigonometric rules Exponential and logarithmic rules Page 19
The Definite Integral Concept LO 3.2A(a) EK 3.2A1 EK 3.2A3 Interpret the definite integral as a limit of a Riemann sum. 4.2, 4.3 5 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. 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. Page 20
The Definite Integral Concept (continued) LO 3.2B EK 3.2B1 EK 3.2B2 LO 3.2C Approximate a definite integral. Definite integrals can be approximated for functions that are represented graphically, numerically, algebraically, and verbally. Scranton School District 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 nonuniform partitions. Calculate a definite integral using areas and properties of definite integrals. Page 21
Evaluate Definite Integrals LO 3.3B(b) EK 3.3B2 Evaluate definite integrals. If f is continuous on the interval [a, b] and F is an antiderivative of f. then b a Scranton School District f(x)dx = F(b) F(a). Evaluation by hand and on the calculator Properties of definite integrals Page 22
The Definite Integral LO 3.3A EK 3.3A1 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 4.3, 4.4 12 The Fundamental Theorem of Calculus FTC 1 FTC 2 Page 23
Applications of Integrals LO 3.4D EK 3.4D1 Scranton School District Apply definite integrals to problems involving area. Areas of certain regions in the plane can be calculated with definite integrals. Area (with respect to either axis) Area Between a curve and an axis Area Between two curves 5 Final Review Review Packet 10 Page 24