Topic Outline for Calculus

Expand - Collapse

• Topic Outline for Calculus
  • I. Functions, Graphs, and Limits
    • Analysis of graphs
      • the interplay between the geometric and analytic information
      • the use of calculus both to predict and to explain the oserved local and global behavior of a function
    • Limits of functions
      • 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 fo change
    • Continuity as a property of functions
      • An intuitive understanding of continuity
      • Understanding continuity in terms of limits
      • Geometric understanding of graphs of continuous functions
    • Parametric, polar, and vector functions
      • The analysis of planar curves includes those given in parametri form, polar form, and vector form
  • II. Derivatives
    • Concept of the derivative
      • Derivative presented graphically, numerically, and analytically
      • Derivative interpreted as an instantaneous rate of change
      • Derivative defined as the limit of the difference quotient
      • Relationship between differentiability and continuity
    • Derivative at a point
      • Slope of a curve at a point
      • Tangent line 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 values
    • Derivative as a function
      • Correspoinding characteristics of 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
    • 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 monotinicity and concavity
      • 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 a 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 solytion curves for differential equations
      • Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration
      • Numerical solution of differential equations using Euler's method
      • L'Hospital's Rule, including its use in determining limits and convergence of improper integrals and series
    • Computation of derivatives
      • Knowledge of 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
      • Derivatives of parametric, polar, and vector functions
  • III. Integrals
    • Interpretations and properties of definite integrals
      • Definite integral as a limit of Riemann sums
      • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the equantity over the interval
      • Basic properties of definite integrals
    • Applications of integrals
      • finding the area of a region
      • finding the volume of a solid with known cross sections
      • finding the average value of function
      • finding the distance traveled by a particle along a line
      • finding the length of a curve(including a curve given in parametric form)
    • Fundamental Theorem of Calculus
      • Use of 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
      • Antiderivatives by parts and simple partial fractions
      • Improper 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
      • Solving logistic differential equations and using them in modeling
    • Numerical approximations to definite integrals
      • Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values
  • IV. Polynomial Approximations and Series
    • Concept of series
    • Series of constants
      • Motivating examples, including decimal expansion
      • Geometric series with applications
      • The harmonic series
      • Alternating series with error bound
      • 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 p-series
      • The ratio test for convergence and divergence
      • Comparing series to test for convergence of divergence
    • Taylor series
      • Taylor polynomial approximation with graphical demonstration of convergence
      • Maclaurin series and the general Taylor series centered at x=a
      • Maclaurin series for the functions e^x, sinx, cosx, and 1/(x-1)
      • Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidefferentiation, and the formation of new series from known series
      • Functions defined by power series
      • Lagrange error bound for Taylor polynomials
  
  
  ap calculus는 calculus AB와 calculus BC로 구분되어 있습니다. 위의 내용 중 빨간색 글씨는 calculus BC에 해당하는 토픽들입니다. 좀 더 심화된 내용을 다루는 것을 알 수 있습니다. calculus AB에 응시하시는 분들은 빨간 글씨를 제외하고 보시면 되겠습니다. 위 내용의 마인드맵 버전을 보시려면 more 를 클릭하십시오.