Testing

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Lesson


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Lesson Objective(s)


Standard(s)
  • APC.1
    • Define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions, using a graphing calculator.

      • Clarification Statement

        • Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing.

  • APC.2

    • Define and apply the properties of limits of functions.

      • Limits will be evaluated graphically and algebraically.

        • Includes:

          • ​limits of a constant

          • ​limits of a sum, product, and quotient

          • ​one-sided limits

          • ​limits at infinity, infinite limits, and non-existent limits*

  • APC.3
    • Use limits to define continuity and determine where a function is continuous or discontinuous.

      • Includes:

        • ​continuity in terms of limits

        • continuity at a point and over a closed interval

        • ​application of the Intermediate Value Theorem and the Extreme Value Theorem

        • ​geometric understanding and interpretation of continuity and discontinuity

  • APC.4

    • Investigate asymptotic and unbounded behavior in functions.

      • Includes:

        • describing and understanding asymptotes in terms of graphical behavior and limits involving infinity

        • comparing relative magnitudes of functions and their rates of change

  • APC.5

    • Investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability.

      • The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.

  • APC.6

    • ​The student will investigate the derivative at a point on a curve.

      • Includes:

        • finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangents

        • using local linear approximation to find the slope of a tangent line to a curve at the point

        • ​defining instantaneous rate of change as the limit of average rate of change

        • approximating rate of change from graphs and tables of values.

  • APC.7

    • Analyze the derivative of a function as a function in itself.

      • Includes:

        • comparing corresponding characteristics of the graphs of f, f', and f''

        • ​defining the relationship between the increasing and decreasing behavior of f and the sign of f'

        • ​translating verbal descriptions into equations involving derivatives and vice versa

        • analyzing the geometric consequences of the Mean Value Theorem;

        • defining the relationship between the concavity of f and the sign of f "; and

        • ​identifying points of inflection as places where concavity changes and finding points of inflection.

  • APC.8

    • Apply the derivative to solve problems.

      • Includes:

        • ​analysis of curves and the ideas of concavity and monotonicity

        • optimization involving global and local extrema;

        • modeling of rates of change and related rates;

        • use of implicit differentiation to find the derivative of an inverse function;

        • interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and

        • differentiation of nonlogarithmic functions, using the technique of logarithmic differentiation.*

          • *AP Calculus BC will also apply the derivative to solve problems.

            • Includes:

              • ​analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors;

              • ​numerical solution of differential equations, using Euler’s method;

              • ​l’Hopital’s Rule to test the convergence of improper integrals and series; and

              • ​geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.

  • APC.9

    • Apply formulas to find derivatives.

      • Includes:

        • derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions

        • derivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functions

        • derivatives of implicitly defined functions

        • higher order derivatives of algebraic, trigonometric, exponential, and logarithmic, functions*

          • *AP Calculus BC will also include finding derivatives of parametric, polar, and vector functions.

  • APC.10

    • Use Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, graphically, and by a table of values and will interpret the definite integral as the accumulated rate of change of a quantity over an interval interpreted as the change of the quantity over the interval.

    • Riemann sums will use left, right, and midpoint evaluation points over equal subdivisions.

  • APC.11
    • ​The student will find antiderivatives directly from derivatives of basic functions and by substitution of variables (including change of limits for definite integrals).
  • APC.12​
    • The student will identify the properties of the definite integral. This will include additivity and linearity, the definite integral as an area, and the definite integral as a limit of a sum as well as the fundamental theorem.
  • APC.13​
    • The student will use the Fundamental Theorem of Calculus to evaluate definite integrals, represent a particular antiderivative, and facilitate the analytical and graphical analysis of functions so defined.
  • APC.14​
    • The student will find specific antiderivatives, using initial conditions (including applications to motion along a line). Separable differential equations will be solved and used in modeling (in particular, the equation y' = ky and exponential growth).
  • APC.15​
    • The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include
      • a)​ the area of a region;
      • b) ​the volume of a solid with known cross-section;
      • c)​ the average value of a function; and
      • d)​ the distance traveled by a particle along a line.
  • APC.16​
    • The student will define a series and test for convergence of a series in terms of the limit of the sequence of partial sums. This will include
      • a)​ geometric series with applications;
      • b)​ harmonic series;
      • c)​ alternating series with error bound;
      • d)​ 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; and
      • e)​ ratio test for convergence and divergence.
  • APC.17
    • ​The student will define, restate, and apply Taylor series. This will include
      • a)​ Taylor polynomial approximations with graphical demonstration of convergence;
      • b)​ Maclaurin series and the general Taylor series centered at x = a;
      • c)​ Maclaurin series for the functions ex, sin x, cos x, and 1/(1 – x);
      • d) ​formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series;
      • e) ​functions defined by power series;
      • f)​ radius and interval of convergence of power series; and
      • g)​ Lagrange error bound of a Taylor polynomial.

Mathematical Practice(s)
  • #1 - Make sense of problems and persevere in solving them
  • #2 - Reason abstractly and quantitatively
  • #3 - Construct viable arguments and critique the reasoning of others
  • #4 - Model with mathematics
  • #5 - Use appropriate tools strategically
  • #6 - Attend to precision
  • #7 - Look for and make use of structure
  • #8 - Look for and express regularity in repeated reasoning