BC Calculus Syllabus. Assessment Students are assessed in the following ways:

Transcription

1 BC Calculus Syllabus Assessment Students are assessed in the following ways: Unit tests Project Problem Sessions Weekly assignments done outside of class that consist of problems from released Quizzes and homework Class Participation Students are allowed to use calculators when appropriate. Calculator usage described in table at the end of the syllabus. BC Calculus Course Outline Unit 1: Limits Limits Intuitively (algebraically and graphically) Delta-Epsilon definition and proof Properties (local and global behavior) Techniques and theorems to compute limits Squeeze Theorem Trigonometric limits Continuity/Discontinuity Removable and non-removable (asymptotic behavior and holes) Intuitively and using limit definition One-sided Intermediate Value Theorem Infinite limits, right and left including asymptotes Limits at infinity L Hopital s rule Magnitude of a function Rate of change of f(x) as a limit Unit 2: Derivatives Definition of a derivative as a limit of the difference quotient Derivatives failing at x=a Derivative of the following types (both by definition and inspection through graphical, numerical and analytical methods) linear, quadratic, cubic, rational, radicals, constants, powers, trigonometric, parametrics Properties of and theorems about derivatives Sums and differences Odd and even functions Product rule Quotient rule Chain rule Power rule

2 Equations of tangent and normal lines Slope Vertical tangents Derivative at a point Instantaneous rate of change Relationship between continuity and differentiability Significance of the sign of the derivative Local linearization The numerical derivative and using technology to find derivatives Unit 3: Applications of Derivatives Average rate of change /Instantaneous rate of change Relationship between position, velocity and acceleration Implicit differentiation Related rates Rolle s Theorem Mean Value Theorem Curve sketching from a given function, from the graph of a derivative and from a table of information about the function and its derivatives Extrema and the Extreme Value Theorem Increasing/decreasing intervals, monotonic behavior Intervals where f(x) is concave up or concave down Points of inflection First derivative test for extrema Second derivative test for extrema Optimization with a major emphasis on justification Local and global Use of the tangent line Euler s Method Newton s Method Differentials Error propagation Slopefields Linear approximation and quadratic approximation with error Unit 4: Antiderivatives Separable differential equations Indefinite integrals and their properties Techniques of integration Power rule Trigonometric U-substitution with change in limits Integration by parts Partial fractions Completing the square Integration requiring trigonometric identities Trig substitution Partial fractions

3 Improper Integrals Differential equations with initial conditions Acceleration, velocity and position Approximating Riemann Sums Upper and lower sums Left-hand and right-hand sums Midpoint sums Trapezoidal sums Simpson s Rule Definite integral and the limit of sums Fundamental Theorems of Calculus Graphical interpretation Analytical interpretation Definite integral as an accumulator Functions defined by integrals Unit 5: Applications of Integration Mean Value Theorem for integrals Average value of a function Area between two curves Volume Discs / Washers Cylindrical shells Solids with known cross sections Length of arc Surface area Work done by a variable force Particle motion problems Total distance and displacement Other accumulation problems Unit 6: Logarithms, Exponentials and other Transcendentals Define ln x as a definite integral Use definition to develop properties and f (x) Derivatives of natural log Antiderivatives using natural logs Inverse functions and the relationship between their derivatives Define ex as the inverse of ln x Use properties of ln x to develop properties of ex Antiderivatives of functions with eu Derivatives of log with any base Derivatives and antiderivatives of au Logarithmic differentiation The logistic function as a growth or decay model Derivatives of inverse trig functions Integrations of functions that produce inverse trig functions

4 Unit 7: Sequences and Series Limits of sequences Definition of a convergent series Partial sums and nth Term Test for divergence of a series Geometric and Telescopic Series Integral Test and p-series Error analysis with Integral Test Direct Comparison Test Limit Comparison Test Alternating Series Test and error analysis Absolute convergence Ratio Test nth Root Test Unit 8: Power Series Macclaurin and Taylor Polynomials Macclaurin and Taylor Series Remainders when a series is terminated after a finite number of terms A lternating and Legrange Interval and radius of convergence Operations on power series- addition, differentiation, integration Unit 9: Parametric, Polar and Vectors First and second derivatives of functions defined parametrically Length of a curve, area under a curve, and surface area of a surface of revolution when a parametrically defined function is given Sketch polar curves Convert from rectangular to polar form and vice versa Length of a polar curve Area bounded by one or more polar curves Slope of a tangent to a polar curve Basic vector review Find R (t) and R (t) for a given vector function R(t) Find the velocity vector and acceleration vector for a given vector-valued function Find the vector function given the acceleration vector and velocity vector with initial conditions Find the speed of a given vector function References: Larson, Calculus, 10e Other resources AP Central Web site Apcentral.collegeboard.com Calculus models for volumes of solids Foster Manufacturing, Plano Texas

5 Acces Computer Program EAS EducAide Software, Inc. Stewart, James, Brooks, Brooks/Cole, Concepts and Contexts, Brooks/Cole Unit Time frame Evidence of Curricular Requirements Calculator usage Limits and Continuity 2 weeks Limits are used throughout the course where appropriate. In the first unit, problems are assigned that include investigating limits algebraically, graphically and with tables. Continuity is investigated through graphical and algebraic methods. Includes looking at graphs and tables. Differentiation 3 weeks Derivatives are analyzed through rate of change, inspection and tangent lines. This investigation includes discussion of the derivative as presented graphically, numerically, as well as algebraically. Includes analysis of graphs, tables, algebraic and the numeric derivative. Applications of the derivative 4 weeks Students are expected to understand the properties of the derivative and use them to analyze the behavior of a function. A major emphasis is placed upon the use of the graph to reinforce understanding of these concepts. Students are required to verbally justify conclusions drawn. Includes exploring relationships between functions and their derivatives as well as calculations. Antiderivatives 2 weeks Students explore the antiderivative from the Includes calculations perspective of the tangent approximating the graph of a function. This is done graphically through the use of slopefields and numerically through the use of Euler s method. Simple indefinite integrals are explored through solving separable differential equations. for Euler s method, slopefield programs, and the exploration of the function from the graph of its derivative. The definite integral 4 weeks The topic is introduced through the use of approximation of an accumulation of a quantity whose rate is known over a given interval of time. The Riemann sum is formally developed along with the properties of the definite integral. There is a major emphasis on approximation and reasonableness of answers. Includes calculations involved in trapezoidal sums, Riemann sums (left, right and midpoint) Simpson s and the use of the calculator s definite integral calculation. Transcendental functions 3 weeks Introduction to these functions begins with the expression of ln x as a function defined by an integral and developed through the use of the Fundamental Theorems. Applications addressed include growth, decay, and logistic models. Includes graphical exploration of the logistic model and continuous growth. Techniques of Integration 3 weeks Students develop an understanding of the common techniques of integration. Applications of the DefiniteIntegral 3 weeks The applications done enable the students to develop and reinforce an understanding of the Riemann summing process. The physical applications enable the students to

6 understand the accumulation process. Includes numeric integration and calculations. Sequences and series 5 weeks Students initially explore the convergence and divergence of series of constants and calculate their sums when possible. When computation of a sum is not possible approximation and error analysis are explored. Series are analyzed to determine which common tests are appropriate for determining convergence. Then the students explore functions expressed as power series. Students develop techniques for building new power series from known series. Includes calculations, partial sums, and graphical comparison of a power series and the function to which it converges. Parametric, Polar and Vector-Valued functions 2 weeks Students explore calculus concepts as applied to functions defined as polar, parametric and vector-valued. Includes calculations, graphical exploration, numeric derivatives.