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 Calculus B/C is an Advanced Placement (AP©) program for highly motivated students.  Calculus BC includes all Calculus AB topics with additions marked by * and +.  Many universities grant credit, advanced placement, or both to students who perform to certain levels on the AP Exam given in May.  The instructor of this course assumes that students will be preparing themselves for the AP Exam.  The course develops the student’s understanding of the concepts of calculus and provides experience with its methods and applications.  The course emphasizes a multi-representational and technology-enhanced approach to Calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally.  These standards are aligned to the 1999-2000 College Board course descriptions.

GOALS

The goals presented here focus on the reason and motivation behind the topics to be covered in this course.

STUDENTS WILL:

• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal.  They should understand the connections among these representations.
• Understand the meaning of the derivative in terms of rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems.
• Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of a rate of change and should be able to use integrals to solve a variety of problems.
• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
• Communicate mathematics both orally and in well-written sentences and be able to explain solutions to problems.
• Model a written description of a physical situation with a function, a differential equation, or an integral.
• Use technology to help solve problem, experiment, interpret results, and verify conclusions.
• Determine the reasonableness of solutions, including the sign, size, relative accuracy, and units of measurement.
• Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

FUNCTIONS, GRAPHS, AND LIMITS

STUDENTS WILL:

• Use calculus to predict and to explain the observed local and global behavior of a function.
• Calculate limits using algebra.
• Estimate limits from graphs or tables of data.
• Understand asymptotes in terms of graphical behavior.
• Describe asymptotic behavior in terms of limits involving infinity.
• Compare relative magnitudes of functions and their rates of change.  (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)
• Understand continuity in terms of limits.
• Use the Intermediate Value Theorem and Extreme Value Theorem to show geometric understanding of graphs of continuous functions.
• Analyze planar curves to include those given in parametric form, polar form, and vector form.

DERIVATIVES

 STUDENTS WILL:
 

• Define and understand the derivative as the limit of the difference quotient.
• Understand the relationship between differentiability and continuity.
• Find the slope of a curve at a point.  Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
• Find the tangent line to a curve at a point and can employ local linear approximation.
• Understand instantaneous rate of change as the limit of average rate of change.
• Approximate rate of change from graphs and tables of values.
• Understand corresponding characteristics of graphs of f and f '.
• Explain the relationship between the increasing and decreasing behavior of f and the sign of f '.
• Use the Mean Value Theorem and understand its geometric consequences.
• Translate verbal descriptions into equations involving derivatives and vice versa.
• Understand corresponding characteristics of the graphs of  f,  f ‘, and     f ”.
• Explain the relationship between the concavity of  f and the sign of  f ”.
• Understand points of inflection as places where concavity changes.
• Analyze curves, including the notions of monotonicity and concavity.
• Optimize functions, using both absolute (global) and relative (local) extrema.
• Solve related rates problems.
• Use implicit differentiation to find the derivative of an inverse function.
• Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
• Analyze planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors.
• Use slope fields and the relationship between slope fields and derivatives of implicitly defined functions to formulate geometric interpretation of differential equations.
• Use Euler’s Method to find numerical solution of differential equations.
• Use L’Hôpital’s Rule in determining convergency of improper integrals and series.
• Find derivatives of basic functions, including xr, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• Understand basic rules for the derivative of sums, products, and quotients of functions.
• Use chain rule and implicit differentiation.
• Find derivatives of parametric, polar, and vector functions.

INTEGRALS

STUDENTS WILL:
 

• Understand the concept of a Riemann sum over equal subdivisions.
• Compute Riemann sums using left, right, and midpoint evaluation points.
• Understand the definite integral as a limit of Riemann sums.
• Understand the definite integral of the rate of change of a quantity over an interval is interpreted as the change of the quantity over an interval.
• Know and can use basic properties of definite integrals.  (For example, additivity and linearity.)
• Use appropriate integrals in a variety of applications to model physical, social, or economic situations.  Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems.  Whatever applications are chosen, the emphasis is one using the integral of a rate of change to give accumulated change or using the method of setting up a approximating Riemann sum and representing its limit as a definite integral.  To provide a common foundation, students will solve specific applications, which include finding the area of a region, including a region bounded by polar curves, the volume of a solid with the known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve, including a curve given in parametric form.
• Use the Fundamental Theorem to evaluation definite integrals.
• Use the Fundamental Theorem to represent a particular antiderivative, and understand the analytical and graphical analysis of functions so defined.
• Find antiderivatives of basic functions.
• Find antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (non-repeating linear factors only).
• Solve improper integrals (as limits of definite integrals).
• Find specific antiderivatives using initial conditions, including applications to motion along a line.
• Solve separable differential equations and use them in modeling.  In particular, students study the equation         y' = ky and exponential growth.
• Solve logistic differential equations and use them in modeling.
• Use Riemann sums and the trapezoidal Rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

POLYNOMIAL APPROXIMATIONS AND SERIES

 STUDENTS WILL:
 

• Define Series as a sequence of partial sums, and convergence is defined as the limit of the sequence of partial sums.  Students use technology to explore convergence or divergence of various examples.
• Solve motivating examples, including decimal expansion.
• Recognize Geometric series with applications.
• Recognize the harmonic series.
• Recognize an Alternating series and find error bounds.
• Understand terms of series as areas of rectangles and their relationship to improper integrals, and can use the integral test in testing the convergence of p-series.
• Use the ratio test for convergence and divergence.
• Compare series to test for convergence or divergence.
• Understand Taylor polynomial approximation and can demonstrate graphically convergence.  (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.)
• Develop the general Taylor series centered at x = a.
• Develop the Maclaurin series for the functions ex, sin x, cos, x and  .
• Manipulate Taylor series and shortcuts to computing Taylor series, including differentiation, antidifferentiation, and can form new series from know series.
• Understand functions defined by power series and can find radius of convergence.
• Find Lagrange error bound for Taylor polynomials.