Grand Prairie Independent School District



Summer BC InstructionsCongratulations on completing AB Calculus. This is a big accomplishment and one you should be very proud of. By completing it during your Junior year, you have given yourself a unique opportunity. You are now qualified to take BC Calculus, and you have the chance to receive 6 hours of college credit with successful passing of the BC Calculus Exam at the end of the school year. Having a second year of Calculus in high school is really an advantage. This second year of Calculus is an extension of AB Calculus. You will be reinforcing the concepts you learned and expanding your knowledge of the use and application of Calculus. I suspect the year will be far more rewarding than this past year, because you already know the basics of Calculus, and this will make the additional material so much easier to master.The pace of the year will be different from last. Since we will be going back over the same material you already have shown a mastery for, we will progress through this material very quickly. This will enable us to spend more time than ever before on the unique aspects of BC Calculus.I am including a calendar of the pacing of the material this year. It will look familiar since most of the BC topics are a repeat of those same topics from AB. Review the calendar to see how it looks different from last year. Think about it. Returning to the topics you mastered last year will ensure those topics are real strengths for the coming year. If there were topics that gave you trouble, you will have the time to make sure they are understood this year. However, the pace will be very quick as we move through these previous topics. While this may appear as an easier path than last year, you must realize mastery of previous topics and a demonstration of that mastery will offer a different kind of challenge. Nevertheless, you will realize quickly that your knowledge of Calculus will be vastly enhanced.In order to stay ahead of the pace, you must take some time this Summer to get started with the curriculum. The farther you progress, the less stress you will feel as we begin the year. Immediately upon return, we will take an AB assessment. This will be in the form of a mock AB exam. Unlike last year, this exam will be graded and be the first major grade of the semester. It will also serve as the remediation device to show you where you need to concentrate. You will also be given a Calculus SYMKC – BC form to complete. This, too, will be for a major grade. I know there will be topics you have yet to see, but I still want you to demonstrate your memorization of the terms used to describe each new topic. We will fill in your understanding as we move along.Have a great summer, and I look forward to a fantastic Calculus year coming up. I have included the code for the BC Kahn Academy course and the way to begin with your work.Go to coaches and type in:GF9UQ2WUSee You in August.Mr. LaughlinBC20182019LIMITS AND CONTINUITY??Limits from graphs??Creating tables for approximating limits??Limits from tables??One-sided limits from graphs??One-sided limits from tables??Connecting limits and graphical behavior??Continuity at a point??Continuity over an interval??Continuity and common functions??Limits of composite functions??Direct substitution9/1?Direct substitution with limits that don't exist??Limits by factoring??Limits using conjugates??Limits of trigonometric functions??Limits using trig identities??Squeeze theorem??Infinite limits and graphs??Analyze unbounded limits??Limits at infinity of rational functions??Limits at infinity of quotients with square roots??Limits at infinity of quotients with trig??Classify discontinuities??Analyzing functions for discontinuities: algebraic??Removable discontinuities??Conclusions from direct substitution (finding limits)??Next steps after indeterminate form (finding limits)9/10?Strategy in finding limits?8/31???DERIVATIVES INTRODUCTION??Derivative as slope of curve9/15?Derivative & the direction of a function??Secant lines & average rate of change??Estimate derivatives??Secant lines & average rate of change with arbitrary points??Secant lines & average rate of change with arbitrary points (with simplification)??Derivative as a limit9/19?Differentiability at a point: graphical??Differentiability at a point: algebraic??The derivative & tangent line equations9/21?Approximation with local linearity9/279/7???DERIVATIVE RULES??Basic derivative rules: find the error??Basic derivative rules: table??Power rule intro??Differentiate polynomials??Tangents of polynomials??Negative powers differentiation??Fractional powers differentiation??Radical functions differentiation intro??Derivatives of sin(x) and cos(x)??Differentiate products??Product rule with tables??Differentiate quotients??Quotient rule with tables??Differentiate rational functions??Identify composite functions??Differentiate composite functions (chain rule)??Chain rule with tables??Differentiate radical functions??Derivatives of tan(x), cot(x), sec(x), and csc(x)??Differentiate trigonometric functions??Differentiating functions: Find the error??Manipulating functions before differentiation10/129/21???ADVANCED DERIVATIVES??Differentiating using multiple rules: strategy??Differentiating using multiple rules??Second derivatives??Implicit differentiation10/25?Differentiate related functions??Derivatives of inverse functions??Derivatives of inverse trigonometric functions10/29?Exponential functions differentiation intro??Differentiate exponential functions??Logarithmic functions differentiation intro??Differentiate logarithmic functions??Parametric functions differentiation??Vector-valued functions differentiation??Second derivatives (parametric functions)??Second derivatives (vector-valued functions)??Differentiate polar functions??Tangents to polar curves10/3010/5???EXISTENCE THEOREMS??Conditions for IVT and EVT: graph??Conditions for IVT and EVT: table??Intermediate value theorem??Conditions for MVT: graph??Conditions for MVT: table??Mean value theorem11/610/12???USING DERIVATIVES TO ANALYZE FUNCTIONS??L'H?pital's rule: 0/0??L'H?pital's rule: ∞/∞??Justification using first derivative??Find critical points11/8?Increasing & decreasing intervals??Relative minima & maxima??Absolute minima & maxima (closed intervals)??Absolute minima & maxima (entire domain)11/10?Concavity intro??Inflection points intro??Justification using second derivative??Inflection points from graphs of first & second derivatives11/14?Second derivative test??Analyze concavity??Find inflection points??Visualizing derivatives??Connecting f, f', and f'' graphically11/1610/19???APPLICATIONS OF DERIVATIVES??Applied rates of change??Analyzing related rates problems: expressions??Analyzing related rates problems: equations12/1?Related rates intro??Related rates (multiple rates)??Related rates (Pythagorean theorem)??Related rates (advanced)12/6?Optimization??Planar motion (differential calc)??Motion along a curve (differential calc)12/1211/2???2nd Semester?????ACCUMULATION AND RIEMANN SUMS??Definite integrals intro??Definite integral by thinking about the function's graph??Left & right Riemann sums??Over- and under-estimation of Riemann sums??Midpoint & trapezoidal sums??Summation notation intro??Summation notation??Riemann sums in summation notation??Definite integral as the limit of a Riemann sum??Definite integral properties 1??Definite integral properties 2??Definite integral properties (no graph)??Functions defined by integrals1/1811/16???ANTIDERIVATIVES AND THE FUNDAMENTAL THEOREM OF CALCULUS??Antiderivatives and indefinite integrals??Finding derivative with fundamental theorem of calculus??Finding derivative with fundamental theorem of calculus: chain rule??Finding definite integrals with fundamental theorem of calculus1/24?Reverse power rule??Reverse power rule: negative and fractional powers??Reverse power rule: sums & multiples??Reverse power rule: rewriting before integrating??Indefinite integrals: e? & 1/x1/26?Indefinite integrals: sin & cos??Definite integrals: reverse power rule??Definite integrals: common functions??Definite integrals of piecewise functions??Improper integrals1/30??-substitution: defining ????-substitution: indefinite integrals???-substitution: definite integrals??Integration by parts2/2?Integration by parts: definite integrals??Integration with partial fractions??Average value of a function??Interpreting behavior of ? from graph of ?'=?2/612/7???DIFFERENTIAL EQUATIONS??Check solutions to differential equations??Write differential equations??Separable differential equations: find the error??Separable differential equations??Identify separable equations2/13?Finding specific antiderivatives??Separable equations: specific solutions??Slope fields & equations??Slope fields & solutions2/15?Euler's method??Differential equations: exponential model equations??Differential equations: exponential model word problems??Differential equations: logistic model word problems2/1912/14???APPLICATIONS OF DEFINITE INTEGRALS??Interpreting definite integrals in context??Analyzing problems involving definite integrals??Accumulation of change??Problems involving definite integrals (algebraic)??Analyzing motion problems (integral calculus)??Motion problems (with integrals)??Planar motion (with integrals)??Area between a curve and the x-axis??Area between two curves given end points??Area between two curves??Horizontal areas between curves??Area bounded by polar curves??Arc length??Volumes of solids of known cross-section2/23?Disc method??Washer method??Shell method3/2012/21???SERIES??Sequences review??Sequence convergence/divergence??Finite geometric series??Partial sums intro??Partial sums & series??Infinite geometric series3/20?nth term test3/30?Integral test??p-series??Direct comparison test??Limit comparison test??Ratio test??Alternating series??Alternating series remainder??Interval of convergence??Integrate & differentiate power series??Taylor & Maclaurin polynomials??Lagrange error bound??Maclaurin series of sin(x), cos(x), and e???Function as a geometric series??Integrals & derivatives of functions with known power series3/301/25 ................
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