Summer BC Instructions

Congratulations 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 khanacademy.org/coaches and type in: GF9UQ2WU

See You in August.

Mr. Laughlin

BC 2018 2019

LIMITS AND CONTINUITYLimits from graphsCreating tables for approximating limitsLimits from tablesOne-sided limits from graphsOne-sided limits from tablesConnecting limits and graphical behaviorContinuity at a pointContinuity over an intervalContinuity and common functionsLimits of composite functionsDirect substitution 9/1Direct substitution with limits that don't existLimits by factoringLimits using conjugatesLimits of trigonometric functionsLimits using trig identitiesSqueeze theoremInfinite limits and graphsAnalyze unbounded limitsLimits at infinity of rational functionsLimits at infinity of quotients with square rootsLimits at infinity of quotients with trigClassify discontinuitiesAnalyzing functions for discontinuities: algebraicRemovable discontinuitiesConclusions from direct substitution (finding limits)Next steps after indeterminate form (finding limits) 9/10Strategy in finding limits 8/31

DERIVATIVES INTRODUCTIONDerivative as slope of curve 9/15Derivative & the direction of a functionSecant lines & average rate of changeEstimate derivativesSecant lines & average rate of change with arbitrary pointsSecant lines & average rate of change with arbitrary points (with simplification)Derivative as a limit 9/19Differentiability at a point: graphicalDifferentiability at a point: algebraic

The derivative & tangent line equations 9/21Approximation with local linearity 9/27 9/7

DERIVATIVE RULESBasic derivative rules: find the errorBasic derivative rules: tablePower rule introDifferentiate polynomialsTangents of polynomialsNegative powers differentiationFractional powers differentiationRadical functions differentiation introDerivatives of sin(x) and cos(x)Differentiate productsProduct rule with tablesDifferentiate quotientsQuotient rule with tablesDifferentiate rational functionsIdentify composite functionsDifferentiate composite functions (chain rule)Chain rule with tablesDifferentiate radical functionsDerivatives of tan(x), cot(x), sec(x), and csc(x)Differentiate trigonometric functionsDifferentiating functions: Find the errorManipulating functions before differentiation 10/12 9/21

ADVANCED DERIVATIVESDifferentiating using multiple rules: strategyDifferentiating using multiple rulesSecond derivativesImplicit differentiation 10/25Differentiate related functionsDerivatives of inverse functionsDerivatives of inverse trigonometric functions 10/29Exponential functions differentiation introDifferentiate exponential functionsLogarithmic functions differentiation introDifferentiate logarithmic functionsParametric functions differentiation

Vector-valued functions differentiationSecond derivatives (parametric functions)Second derivatives (vector-valued functions)Differentiate polar functionsTangents to polar curves 10/30 10/5

EXISTENCE THEOREMSConditions for IVT and EVT: graphConditions for IVT and EVT: tableIntermediate value theoremConditions for MVT: graphConditions for MVT: tableMean value theorem 11/6 10/12

USING DERIVATIVES TO ANALYZE FUNCTIONSL'Hôpital's rule: 0/0L'Hôpital's rule: ∞/∞Justification using first derivativeFind critical points 11/8Increasing & decreasing intervalsRelative minima & maximaAbsolute minima & maxima (closed intervals)Absolute minima & maxima (entire domain) 11/10Concavity introInflection points introJustification using second derivativeInflection points from graphs of first & second derivatives 11/14Second derivative testAnalyze concavityFind inflection pointsVisualizing derivativesConnecting f, f', and f'' graphically 11/16 10/19

APPLICATIONS OF DERIVATIVESApplied rates of changeAnalyzing related rates problems: expressionsAnalyzing related rates problems: equations 12/1Related rates introRelated rates (multiple rates)Related rates (Pythagorean theorem)

Related rates (advanced) 12/6OptimizationPlanar motion (differential calc)Motion along a curve (differential calc) 12/12 11/2

2nd Semester

ACCUMULATION AND RIEMANN SUMSDefinite integrals introDefinite integral by thinking about the function's graphLeft & right Riemann sumsOver- and under-estimation of Riemann sumsMidpoint & trapezoidal sumsSummation notation introSummation notationRiemann sums in summation notationDefinite integral as the limit of a Riemann sumDefinite integral properties 1Definite integral properties 2Definite integral properties (no graph)Functions defined by integrals 1/18 11/16

ANTIDERIVATIVES AND THE FUNDAMENTAL THEOREM OF CALCULUSAntiderivatives and indefinite integralsFinding derivative with fundamental theorem of calculusFinding derivative with fundamental theorem of calculus: chain ruleFinding definite integrals with fundamental theorem of calculus 1/24Reverse power ruleReverse power rule: negative and fractional powersReverse power rule: sums & multiplesReverse power rule: rewriting before integratingIndefinite integrals: eˣ & 1/x 1/26Indefinite integrals: sin & cosDefinite integrals: reverse power ruleDefinite integrals: common functionsDefinite integrals of piecewise functionsImproper integrals 1/30𝘶-substitution: defining 𝘶𝘶-substitution: indefinite integrals𝘶-substitution: definite integrals

Integration by parts 2/2Integration by parts: definite integralsIntegration with partial fractionsAverage value of a functionInterpreting behavior of 𝑔 from graph of 𝑔'=ƒ 2/6 12/7

DIFFERENTIAL EQUATIONSCheck solutions to differential equationsWrite differential equationsSeparable differential equations: find the errorSeparable differential equationsIdentify separable equations 2/13Finding specific antiderivativesSeparable equations: specific solutionsSlope fields & equationsSlope fields & solutions 2/15Euler's methodDifferential equations: exponential model equationsDifferential equations: exponential model word problemsDifferential equations: logistic model word problems 2/19 12/14

APPLICATIONS OF DEFINITE INTEGRALSInterpreting definite integrals in contextAnalyzing problems involving definite integralsAccumulation of changeProblems involving definite integrals (algebraic)Analyzing motion problems (integral calculus)Motion problems (with integrals)Planar motion (with integrals)Area between a curve and the x-axisArea between two curves given end pointsArea between two curvesHorizontal areas between curvesArea bounded by polar curvesArc lengthVolumes of solids of known cross-section 2/23Disc methodWasher methodShell method 3/20 12/21

SERIESSequences reviewSequence convergence/divergenceFinite geometric seriesPartial sums introPartial sums & seriesInfinite geometric series 3/20nth term test 3/30Integral testp-seriesDirect comparison testLimit comparison testRatio testAlternating seriesAlternating series remainderInterval of convergenceIntegrate & differentiate power seriesTaylor & Maclaurin polynomialsLagrange error boundMaclaurin series of sin(x), cos(x), and eˣFunction as a geometric series

Integrals & derivatives of functions with known power series 3/30 1/25