Wellston City Schools Calculus 2006-2007 Curriculum Calendar Grading Period 1:Week 1: Review 11 th grade standards Learn to represent functions using: *Words *Tables of values *Graphs *Formulas Present the Vertical Line Test for curves in the plane Discuss piecewise defined functions, including the absolute value function Study symmetry of functions, including even and odd functions Discuss increasing and decreasing functions GP 1:Week 2: Describe the process of mathematical modeling Discuss the following types of function used in mathematical models: *Linear *Polynomial *Power and root *Rational *Algebraic *Trigonometric *Exponential and logarithmic *Transcendental GP 1: Week 3: Learn to create new functions from existing ones by Translations, consisting of vertical and horizontal shifting Vertical and horizontal stretching and reflecting Algebraic combinations Composition GP 1:Week 4: Illustrate by example that the use of graphing technology has both benefits and pitfalls. Study the behavior of exponential functions, including their definition and graphs Review the laws of exponents Introduce the number e as a special exponential base GP 1:Week 5: Give conditions under which a function has an inverse: Definition of one-to-one Horizontal Line Test Find and graph the inverse of a one-to-one function Apply this to exponential functions in order to define logarithms, including Properties of logarithms The natural logarithm The change of base formula
GP 1:Week 6: Discuss the limitations of expressing curves in the form y = f(x) Introduce parametric equations and give examples Use two problems to explain the need for the notion of limit: The tangent problem The velocity problem GP 1:Week 7: Discuss limits of functions in general, including a general definition Find limits both numerically and graphically Introduce and discuss one-sided limits GP 1:Week 8: Introduce *Limit Laws *Direct Substitution *The Squeeze Theorem Use these to calculate limits Introduce continuity, including *Types of discontinuity *One-sided continuity *Continuity on an interval Identify categories of continuous function, including combinations of continuous functions Apply the Intermediate Value Theorem GP 1:Week 9: OHIO GRADUATION TESTING Grading Period 2:Week 1: Examine the global behavior of functions giving special attention to horizontal and vertical asymptotes Study infinite limits at infinity GP 2:Week 2: Use limits to find exact values of slopes of tangent lines velocities other rates of change Introduce the derivative of a function Interpret the derivative as the slope of a tangent and a rate of change GP 2:Week 3: View the derivative f (x) as a function of x
Study graphs of f (x) and f(x) together Study differentiability and continuity Introduce higher-order derivatives GP 2:Week 4: Learn formulas for the derivatives of *Constant functions *Power functions *Exponential functions Find new derivatives from old: Constant multiples Sums and differences GP 2: Week 5: Learn formulas for the derivatives of the product and quotient of two functions whose derivatives are known GP 2:Week 6: Apply the derivative, interpreted as a rate of change, to various fields of study, including *Physics *Chemistry *Biology GP 2:Week 7: Derive formulas for the derivatives of the six trigonometric functions Learn the Chain Rule for finding derivatives of composite functions. Apply the Chain Rule to the case of parametric curves GP 2:Week 8: GP 2:Week 9: REVIEW SEMESTER TEST Grading Period 3:Week 1: Learn to find dy/dx in situations where y is defined only implicitly as a function of x Discuss orthogonal trajectories Apply implicit differentiation to the special case of the inverse trigonometric functions Use implicit differentiation to find the derivatives of the logarithmic functions Introduce logarithmic differentiation Study the number e as a limit
GP 3: Week 2: Apply the Chain Rule to problems in which two or more rates are related. GP 3:Week 3: Solve problems requiring the minimum or maximum value of a quantity Study absolute vs. local maxima/minima of a function Introduce the Extreme Value Theorem and Fermat s Theorem, as well as critical points. GP 3:Week 4: Apply the Mean Value Theorem to finding where functions are increasing and decreasing Discuss the first derivative test and second derivative test for local max/minima GP 3:Week 5: Beginning with a graph produced by a graphing calculator or computer, refine it using calculus, making sure that we reveal all important aspects of the curve. Introduce the various types of indeterminate forms Find limits of indeterminate forms using l Hospital s Rule GP 3:Week 6: Apply our earlier work on maximum and minimum values to finding the optimum value of some variable Apply optimization to business and economics, including topics of *cost and marginal cost *revenue and marginal revenue *profit and marginal profit GP 3:Week 7: Introduce Newton s Method for solving equations numerically Understand significance of starting point for Newton s Method Apply Newton's Method to examples. GP 3: Week 8: Introduce antiderivatives Describe the set of antiderivatives of a given function Present antidifferentiation formulas Introduce direction fields and apply to rectilinear motion GP 3:Week 9: OHIO GRADUATION TESTING
Grading Period 4:Week 1: Attempt to solve the area problem by using rectangles to approximate the area under a curve Give a more exact definition of area Introduce sigma notation Discuss the distance problem GP 4: Week 2: Define definite integral using Riemann sums Begin evaluating definite integrals Approximate definite integrals using the Midpoint Rule Develop properties of the definite integral Introduce the Evaluation Theorem Discuss indefinite integrals Apply the Evaluation Theorem as the Net Change Theorem. GP 4:Week 3: Study differentiation and integration as inverse processes Introduce parts 1 and 2 of the Fundamental Theorem of Calculus GP 4:Week 4: Introduce the Substitution Rule for finding antiderivatives Apply the Substitution Rule to definite integrals Use this to study integrals of symmetric functions GP 4:Week 5: Present the formula for integration by parts Apply the formula to finding various antiderivatives Use trigonometric identities to integrate certain combinations of trigonometric functions Learn the methods of trigonometric substitution and partial fractions GP 4:Week 6: Learn to find antiderivatives using tables and computer algebra systems Learn to approximate definite integrals using the Trapezoidal Rule and Simpson s Rule, in addition to the Midpoint Rule and give error bounds for these approximations
GP 4:Week 7: Define improper integrals of Types 1 and 2 Study convergence and divergence of improper integrals Use comparison to help decide whether an improper integral converges GP 4:Week 8: GP 4:Week 9: REVIEW SEMESTER TEST