Easy Worksheet™ AP Calculus

AP Calculus Objectives

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 1 sided limits)
- An intuitive understanding of the limiting process
- Calculating limits using algebra
- Estimating limits from graphs or tables of data
Asymptotic and Unbounded Behavior
- Understanding asymptotes in terms of Graphical Behavior
- Describing asymptotic behavior in terms of limits involving infinity.
- Comparing relative magnitudes of functions and their rates of change.(For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)
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.)
- Understanding continuity in terms of limits.
- Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem.)
(BC TEST) Parametric, Polar, and Vector Functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.

Derivatives

Concept of Derivative
- Derivative presented graphically, numerically, and analytically
- Derivative interpreted as an instantaneous rate of change
- Derivative defined as the limit of difference quotient
- Relationship between differentiability and continuity
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.
- Tangent lines to a curve at a point and local linear approximation.
- Instantaneous rate of change as the limit of average rate of change.
- Approximate rate of change from graphs and tables of data.
Derivative as a function.
- Corresponding characteristics of the graphs of f and f'.
- Relationship between the increasing and decreasing behavior of f and the sign of f'
- The Mean Value Theorem and its geometric consequences.
- Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and visa versa.
Second Derivatives.
- Corresponding characteristics of the graphs of f , f' , and f''.
- Relationship between the concavity of f and the sign of f''.
- Points of inflection as places where concavity changes.
Applications of Derivatives
- Analysis of curves, including the notions of monotonicity and concavity.
- (BC TEST) Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.
- Optimization, both absolute (global) and relative (local) extrema.
- Modeling rates of change, including related rates problems.
- Use of implicit differentiation to find the derivative of an inverse function.
- Interpretation of the derivative as the rate of change in varied applied contexts, including velocity, speed and acceleration.
- Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
- (BC TEST) Numerical solution of differential equations using Euler's method.
- (BC TEST) L'Hospital's Rule, including its use in determining limits and convergence of improper integrals and series.
Computation of Derivative
- Knowledge of the derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
- Basic rules for the derivative of sums, products, and quotients of functions.
- Chain rule and implicit differentiation.
- (BC TEST) Derivatives of parametric, polar, and vector functions

Integrals

Interpretation and properties of definite integrals
- Definite integrals as the limit of Riemann sums.
- Definite integrals of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
  ∫ ^b f'(x) dx = f(b) - f(a)
  _a
- Basic properties of definite integrals (examples include additivity and linearity)
Applications of integrals
- Appropriate integrals are used 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 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 using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves for BC test), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and (BC test) the length of a curve (including a curve given in parametric form).
Fundamental Theorem of Calculus
- Use the Fundamental Theorem to evaluate definite integrals.
- Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation
- Antiderivatives following directly from derivatives of basic functions
- Antiderivatives by substitution of variables (including change of limits for definite integrals), (BC TEST) parts, and (BC TEST) simple partial fractions (nonrepeating linear factors only)
- (BC TEST) Improper Integrals (as limits of definite integrals)
Applications of antidifferentiation
- Finding specific antiderivatives using initial conditions, including applications to motion along a line
- Solving separable differential equations and using them in modeling. (in particular, studying the equation y' = ky and exponential growth)
- (BC Test) Solving logistic differential equations and using them in modeling
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, geometrically, and by tables of values

(BC Test) Polynomial Approximation and Series

(BC Test) Concept of series
- (BC Test) 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 or divergence.
(BC Test) Series of constants
- (BC Test) Motivating examples, including decimal expansion
- (BC Test) Geometric series with applications
- (BC Test) The harmonic series
- (BC Test) Alternating series with error bound
- (BC Test) Terms of series as areas of rectangles and their relationship to the improper integrals, including the integral test and its use in testing the convergence of p-series
- (BC Test) The ratio test for convergence and divergence
- (BC Test) Comparing series to test for convergence or divergence
(BC Test) Taylor Series
- (BC Test) Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve)
- (BC Test) Maclaurin series and the generalized Taylor series centered at x = a
- (BC Test) Maclaurin series for the functions e^x, sin x, cos x, and ¹/_{(1 - x)}
- (BC Test) Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series
- (BC Test) Functions defined by power series
- (BC Test) Radius and interval of convergence of power series
- (BC Test) Lagrange error bound for Taylor polynomials