I. Functions, Graphs, and Limits
Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
Limits of functions (including onesided limits)  An intuitive understanding of the limiting process PreCalculus Review
 Calculating limits using algebra PreCalculus Review
 Estimating limits from graphs or tables of data PreCalculus Review
Asymptotic and unbounded behavior  Understanding asymptotes in terms of graphical behavior PreCalculus Review
 Describing asymptotic behavior in terms of limits involving infinity PreCalculus Review
 Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth) PreCalculus Review
Continuity as a property of functions  An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.) PreCalculus Review
 Understanding continuity in terms of limits PreCalculus Review
 Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) PreCalculus Review (IVT is NEW)
Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form. Chapter 10/11 II. Derivatives
Concept of the derivative
 Derivative presented graphically, numerically, and analytically PreCalculus Review
 Derivative interpreted as an instantaneous rate of change PreCalculus Review
 Derivative defined as the limit of the difference quotient PreCalculus Review
 Relationship between differentiability and continuity PreCalculus Review
Derivative at a point  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. PreCalculus Review
 Tangent line to a curve at a point and local linear approximation PreCalculus Review
 Instantaneous rate of change as the limit of average rate of change PreCalculus Review
 Approximate rate of change from graphs and tables of values PreCalculus Review
Derivative as a function  Corresponding characteristics of graphs of ƒ and ƒ' PreCalculus Review
 Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ' PreCalculus Review
 The Mean Value Theorem and its geometric interpretation PreCalculus Review (MVT is NEW)
 Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. PreCalculus Review
Second derivatives  Corresponding characteristics of the graphs of ƒ, ƒ', and ƒ'' PreCalculus Review
 Relationship between the concavity of ƒ and the sign of ƒ' PreCalculus Review
 Points of inflection as places where concavity changes PreCalculus Review
Applications of derivatives  Analysis of curves, including the notions of monotonicity and concavity PreCalculus Review
 Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration Chapter 10/11
 Optimization, both absolute (global) and relative (local) extrema PreCalculus Review (Optimization is NEW)
 Modeling rates of change, including related rates problems PreCalculus Review (Special Review of this Topic)
 Use of implicit differentiation to find the derivative of an inverse function PreCalculus Review
 Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration PreCalculus Review
 Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations Chapter 6
 Numerical solution of differential equations using Euler’s method Chapter 6
 L’Hospital’s Rule, including its use in determining limits Chapter 8 and convergence of improper integrals and series Chapter 9A
Computation of derivatives  Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, PreCalculus Review and inverse trigonometric functions Chapter 5
 Derivative rules for sums, products, and quotients of functions PreCalculus Review
 Chain rule and implicit differentiation PreCalculus Review
 Derivatives of parametric, polar, and vector functions. Chapter 10/11
III. Integrals
Interpretations and properties of definite integrals
 Definite integral as a limit of Riemann sums Chapter 4
 Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: Chapter 4
 Basic properties of definite integrals (examples include additivity and linearity) Chapter 4
Applications of integrals  Appropriate integrals are used in a variety of applications to model physical, biological, 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 on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region Chapter 4 (including a region bounded by polar curves) Chapter 10/11 , the volume of a solid with known cross sections Chapter 7 , the average value of a function Chapter 4 , the distance traveled by a particle along a line Chapter 4 , the length of a curve Chapter 7 (including a curve given in parametric form) Chapter 10/11 and accumulated change from a rate of change. Chapter 4
Fundamental Theorem of Calculus  Use of the Fundamental Theorem to evaluate definite integrals Chapter 4
 Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined Chapter 4
Techniques of antidifferentiation  Antiderivatives following directly from derivatives of basic functions Chapter 4
 Antiderivatives by substitution of variables (including change of limits for definite integrals) Chapter 4 , parts Chapter 8 , and simple partial fractions (nonrepeating linear factors only) Chapter 8
 Improper integrals (as limits of definite integrals) Chapter 8
Applications of antidifferentiation  Finding specific antiderivatives using initial conditions, including applications to motion along a line Chapter 4
 Solving separable differential equations and using them in modeling (including the study of the equation y' = ky and exponential growth) Chapter 6
 Solving logistic differential equations and using them in modeling Chapter 6
Numerical approximations to definite integrals  Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. Chapter 4
IV. Polynomial Approximations and Series
Concept of series  A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence and divergence. Chapter 9A Series of constants
 Motivating examples, including decimal expansion. Chapter 9A
 Geometric series with applications. Chapter 9A
 The harmonic series. Chapter 9A
 Alternating series with error bound. Chapter 9A
 Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of pseries. Chapter 9A
 The ratio test for convergence and divergence. Chapter 9A
 Comparing series to test for convergence or divergence. Chapter 9A
Taylor series
 Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve). Chapter 9B
 Maclaurin series and the general Taylor series centered at x = a. Chapter 9B
 Maclaurin series for the functions e^{x} , sin x, cos x, and 1/(1  x) Chapter 9B
 Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series. Chapter 9B
 Functions defined by power series. Chapter 9B
 Radius and interval of convergence of power series. Chapter 9B
 Lagrange error bound for Taylor polynomials. Chapter 9B
