Bell Ringer Review Lesson Exit Ticket  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  APC.3
APC.4 APC.5 APC.6 APC.7 APC.8 APC.9 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 crosssection;
 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 pseries; 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
