COURSE OUTLINE BC Calculus. Unit I: Limits and Continuity (4 days one assessment)

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1 COURSE OUTLINE BC Calculus Unit I: Limits and Continuity (4 days one assessment) Graphical behavior of functions to be familiar with Introduction to limits Graphical approach to limits Computational approach to limits Numeric approach to limits (estimating by using a table of values) Algebraic approach to limits Limits involving Infinity Limits involving Trigonometric functions Asymptotic behavior of functions using limits Comparison of relative magnitudes of functions and their rates of change Concept and definition of continuity Graphical approach to continuity Types of discontinuity Intermediate Value Theorem Extreme Value Theorem Continuity of Trigonometric functions Unit II: Derivatives (10 days two assessments) Average rate of change/slope of a secant line Instantaneous rate of change/slope of a tangent line Instantaneous rate of change as the limit of the average rate of change Definition of the derivative as a limit Sketching the derivative of a function from the graph of the function Techniques of differentiation power rule, sum/difference, product rule and quotient rule Derivatives of Trigonometric Functions Approximation of the derivative of a function at a point numerically Approximation of the derivative of a function at a point graphically Approximation of a rate of change graphically and numerically Differentiability versus Continuity Chain rule Slope of a curve at a point Writing the equation of the tangent line to a curve Use of tangent line to approximate the function at a point

2 Local Linear Approximations of a function Implicit Differentiation Derivatives of exponential and logarithmic functions Logarithmic Differentiation Derivatives of Inverse Trigonometric Functions Higher Order Derivatives Unit III: Applications of Derivatives (10 days two assessments) Related Rates 0 0 Indeterminate forms ( 0/0, /,,0 i,1,, and 0 ) and L Hopital s Rule Intervals of Increasing/Decreasing of a Function Intervals of concavity of a function First Derivative test for local extrema Second Derivative test for local extrema Curve sketching using the first and second derivative Sketching a function using the graph of the derivative Analyze the relationships between the graphs of f, f ' and f '' Absolute extrema of a function Applied Maximum and Minimums Motion along a line: particle moving left/right, speeding up and slowing down Velocity, Speed and Acceleration of an object Newton s Method Rolle s Theorem and the Mean Value Theorem Unit IV: Antidifferentiation and Integration (6 days one assessment) Geometric approach to antidifferentiation Net Change Properties of the indefinite Integral Basic indefinite integration Integration by u-substitution Midpoint, Right and Left endpoint approximation for area under the curve Riemann Sum as a Definite Integral Trapezoidal approximation Numerical approximation of the definite integral using the midpoint, left endpoint, right endpoint and trapezoidal approximation represented algebraically, numerically, and graphically

3 Fundamental Theorem of Calculus Accumulation Function Derivative of the integral of a continuous function Interpretation of a Definite Integral of a rate of change of a quantity over an interval Antidifferentiation from the graph of the derivative Properties of a Definite Integral Average value of a function Motion along a line using antidifferentiation Distance traveled vs. displacement Definite Integrals with u-substitution Natural Logarithm defined by an integral Initial value problems Unit V: Applications of the Definite Integral (6 days one assessment) Area between a curve and an axis Area between two curves Volumes of a solid of revolution when revolved about an axis by disks and washers Volumes of a sold of revolution when revolved about a vertical or horizontal line Volume of a solid with a cross sectional area perpendicular to an axis Volume of a solid of revolution by cylindrical shells Arc Length Area of a Surface of Revolution Work Unit VI: More Techniques of Antidifferentiation (6 days two assessments) Integration by Parts Trigonometric Integrals Trigonometric Substitutions Partial Fractions Improper Integrals (as limits of definite integrals including L Hopitals) Unit VII: Differential Equations (4 days one assessment) Exploration Solutions of Differential Equations by Separation of Variables

4 Differential Equations with and without initial conditions Sketching slope fields given a differential equation Sketching a particular solution to a differential equation with a given slope field Numerical Approximation Euler s method Modeling differential Equations exponential growth/decay, Newton s law of Cooling Logistic Model of Differential Equations Unit VIII: Sequences and Series (8 days 2 assessments) Sequences: terms, infinite, convergence of a sequence Monotone Sequences: increasing, decreasing, bounds of a sequence Infinite Series: * Arithmetic series, Geometric series, alternating series, harmonic series, p- series, and power series * Motivating examples, including decimal expansion nth partial sum * Convergence/Divergence of an infinite Series * Divergence test (terms do not approach zero) * Integral test * Comparison test * Ratio test * Root test * Limit comparison test * Alternating Series test * Ratio test for Absolute Convergence Error bound for an Alternating Series Taylor polynomial approximation of a function (see attached) Maclaurin and Taylor Series centered at x=a x Maclaurin Series for e,sin x,cos x, and 1 1 x. Absolute and Conditional Convergence Interval of Convergence Lagrange Error bound for Taylor Polynomials Differentiating and Integrating Power Series Manipulation of Taylor series using substitution, differentiation and antidifferentiation Formation of new series from known series Modeling functions using the Maclaurin and Taylor Series

5 Unit IX: Polar and Parametric - Planar Curves only (2 days one assessment) Slope of the tangent line to a polar curve Slope of the polar curve at a point Arc length of a polar curve First and second derivative of a parametric curve Arc length of a parametric curve Area of a region bounded by a polar curve Unit X: Vector (3 days one assessment) Vector operations Geometric representation of vectors Differentiation of vector valued functions Slope of the tangent line to a vector valued function Velocity Vector Acceleration Vector Speed of a particle Distance traveled Review