HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT A P C a l c u l u s ( B C ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS Limits and Continuity Make sense of problems and persevere in solving them. Derivatives Reason abstractly and quantitatively. More Derivatives Construct viable arguments and critique the reasoning of others. Applications of Derivatives Model with mathematics. The Definite Integral Use appropriate tools strategically. Differential Equations and Mathematical Modeling Attend to precision. Applications of Definite Integrals Look for and make use of structure. Sequences, L’Hôpital’s Rule, and Improper Integrals Look for and express regularity in repeated reasoning. Infinite Series Parametric, Vector, and Polar Functions PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS Average and Instantaneous Calculate average and instantaneous 2.1 Speed • Definition of Limit • speeds. Define and calculate limits for Rates of Change Properties of Limits • One-sided function value and apply the properties of and Limits and Two-sided Limits • limits. Use the Sandwich Theorem to find Sandwich Theorem certain limits indirectly. Finite Limits as x → ± ∞ and College Board Find and verify end behavior models for their Properties • Sandwich 2.2 Calculus (BC) various function. Calculate limits as and Theorem Revisited • Limits Involving Standard I to identify vertical and horizontal Limits as x → a • Chapter 2 Infinity asymptotes. End Behavior Models • Limits of functions “Seeing” Limits as x → ± ∞ Limits and (including one-sided Continuityy limits) Identify the intervals upon which a given (9 days) Asymptotic and function is continuous and understand the unbounded behavior Continuity as a Point • meaning of continuous function with and Continuous Functions • without limits. Remove removable 2.3 Continuity as a property Algebraic Combinations • discontinuities by extending of modifying a Continuity of functions Composites • Intermediate function. Apply the Intermediate Value Value Theorem for Continuous Theorem and the properties of algebraic Functions combinations and composites of continuous functions. Apply directly the definition of the slope of Average Rates of Change • 2.4 a curve in order to calculate slopes. Find Tangent to a Curve • Slope of a Rates of Change the equations of the tangent line and normal Curve• Normal to a Curve • and Tangent Lines line to a curve at a given point. Find the Speed Revisited average rate of change of a function. PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS Definition of Derivative • Notation 3.1 Calculate slopes and derivatives using the • Relationships Derivative of a definition of derivative. Graph f from the Between the Graphs of f and f ʹ• Function graph of f’ and graph f’ from the graph of f. Graphing the Derivative from Data • One-sided Derivatives College Board How f ʹ (a) Might Fail to Exist • Calculus (BC) Be able to find where a function is not Differentiability Implies Local Standards I and II differentiable and distinguish between Linearity • Numerical 3.2 corners, cusps, discontinuities, and vertical Derivatives on a Calculator • Differentiability Continuity as a property tangents. Be able to approximate Differentiability Implies of functions derivatives numerically and graphically. Continuity • Intermediate Value Theorem for Derivatives Conceptpt of the Chapter 3 derivative Derivative Rules for Positive Use the rules of differentiation to calculate Integer Powers, Multiples, 3.3 Derivatives Derivative at a point derivatives, including second and higher Sums, Differences, Products Rules for order derivatives. Use the derivative to and Quotients • Negative Differentiation (11 days) Derivative as a function calculate the instantaneous rate of change. Integer Powers of x • Second and Higher Order Derivatives Second derivatives Instantaneous Rates of Change • 3.4 Computation of Use derivatives to analyze straight line Motion Along a Line • Velocity and Other derivatives motion and solve other problems involving Sensitivity to Change Rates of Change rates of change. • Derivatives in Economics Applications of derivatives Derivatives of the Sine 3.5 Function and Cosine Function Derivatives of Use the rules for differentiating the six • Simple Harmonic Motion • Trigonometric basic trigonometric functions. Jerk • Derivatives of the Other Functions Basic Trigonometric Functions PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS Derivative of a Composite Differentiate composite functions using the Function • Use of the Chain 4.1 Chain Rule. Find the slopes of Rule (“Outside-Inside”) • Chain Rule parametrized curves. Slopes of Parametrized Curves • Power Chain Rule College Board Calculus (BC) SStandards I and II Continuity as a property of functions Implicitly Defined Functions • 4.2 Find derivatives using implicit Tangents & Normal Lines • Implicit Concept of the differentiation, and the Power Rule for Derivatives of Higher Order • Chapter 4 Differentiation derivative Rational Powers of x. Rational Powers of Differentiable Functions More Derivative at a point Derivatives Derivative as a function (12 days) 4.3 Derivatives of Inverse Derivatives of Second derivatives Functions • Derivative of Calculate derivatives of functions involving Inverse Arcsine, Arctangent and the inverse trigonometric functions. Trigonometric Computation of Arcsecant • Derivatives of the Functions derivatives Other Three Applications of derivatives 4.4 Derivatives of Derivative of ex , ax, ln x, loga x Calculate derivatives of exponential and Exponential and • Power Rule for Arbitrary Real logarithmic functions. Logarithmic Powers Functions PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS Absolute (Global) Extreme 5.1 Determine the local or global extreme Values • Local (Relative) Extreme Values of values of a function. Extreme Values • Finding Functions Extreme Values (Critical Point) Mean Value Theorem • Physical 5.2 Apply the Mean Value Theorem and find Interpretation • Increasing and Mean Value the intervals on which a function is Decreasing Functions • Other Theorem increasing or decreasing. Consequences and Antiderivatives Use the First and Second Derivative Tests First Derivative Test for Local 5.3 to determine the local extreme values of a Extrema • Concavity • Points of Connecting f’ and function. Determine the concavity of a Inflection • Second Derivative Chapter 5 f’’ with the Graph function and locate the points of inflection Test for Local Extrema • of f College Board Calculus by analyzing the second derivative. Graph Learning about Functions from (BC) Standard II Applications f using information about f’ and/or f’’. Derivatives of DDerivatives i ati Applications of Examples from Mathematics, derivatives (15 days) 5.4 Solve application problems involving Business, Industry, and Modeling and finding minimum or maximum values of Economics • Optimization functions. Modeling Discrete Phenomena with Differentiable Functions Linear Approximation • 5.5 Find linearizations and differentials, and Differentials and Estimating Linearization and estimate the change in a function using Change with Differentials • Differentials differentials. Absolute, Relative, and Percent Change • Newton’s Method 5.6 Related Rates Equations • Solve related rates problems. Related Rates Solution Strategy PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS Approximate the area under the graph of a 6.1 nonnegative continuous function by using Distance Traveled • Rectangular Estimating with rectangle approximation methods. Interpret Approximation Method (RAM) Finite Sums the area under a graph as a net accumulation of a rate of change. Riemann Sums • Terminology Express the area under a curve as a definite and Notation of Integration • 6.2 College Board Calculus integral and as a limit of Riemann sums. Definite Integral and Area • Definite Integrals (BC) Standard III Compute the area under a curve using Constant Functions • Integrals varied numerical integration procedures. on a Calculator • Discontinuous Interpretations and Integrable Functions properties of definite intintegrals l Chapter 6 Properties of Definite Integrals • 6.3 Applications of integrals Average Value of a Function • Apply rules for definite integrals and find The Definite Definite Integrals Mean Value Theorem for the average value of a function over a Integral and Fundamental Theorem Definite Integrals • Connecting closed interval. Antiderivatives of Calculus Differential and Integral (13 days) Calculus Techniques of antidifferentiation Fundamental Theorem of Numerical Calculus (Part 1) • Graphing the Apply the Fundamental Theorem of 6.4 approximations to x Calculus. Understand the relationship Function f ( t)dt • Fundamental definite integrals ∫a between the derivative and definite integral Theorem of Fundamental Theorem of a expressed in both parts of the Calculus Calculus (Part 2) • Area Fundamental Theorem of Calculus. Connection • Analyzing Antiderivatives Graphically Approximate the definite integral by using 6.5 Trapezoidal Approximations • the Trapezoidal Rule and by using Trapezoidal Rule Simpson’s Rule Simpson’s Rule. PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS Construct antiderivatives using the Fundamental Theorem of Calculus. Solve initial value problems in the form 7.1 dy Differential Equations • Initial = f (x) , y0 = f ( x0 ) . Construct slope Slope Field and dx Value Problems • Slope Fields • Euler’s Method fields and interpret slope fields as Euler’s Method visualizations of different equations.