* Quiz – The Fundamental Theorem of Calculus * Application of definite integrals including area, volume, position/velocity/acceleration and accumulation functions * Antiderivatives and Indefinite Integration * Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically and by tables of values * Finding specific antiderivatives using initial conditions, including applications to motion along a line * Find antiderivatives including the use of substitution * Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined * Use of the Fundamental Theorem of Calculus to evaluate definite integrals * Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: * Definite integral as a limit of Riemann sums Module 4: Integration Suggested Pace: 4 weeks * Oral Review: Discussion about using the calculator to find the critical values of a function by examining the graph of the function and the graph of the function’s derivative. Major Assignments and Assessments * Problem sets * Concavity and the second derivative test * Rolle’s Theorem and the Mean Value Theorem * Optimization – absolute and relative extrema * Analysis of curves including monotonicity and concavity * Mean Value Theorem and geometric consequences * Points of inflection as places where concavity changes * Relationship between the concavity of f and the sign of f’ * Corresponding characteristics of graphs of f, f’, and f’’ * Relationship between the increasing and decreasing behavior of f and the sign of f’ * Corresponding characteristics of graphs of f and f’ * Test – Differentiation Module 3: Applications of Differentiation Suggested Pace: 6 weeks Discussion about the limitations of the calculator to find the numerical derivative (for example, f ‘(0) for f (x) = |x|). * Oral Review: Discussion about using a calculator to find the value of a derivative at a point, and how to graph the derived function using a calculator. ![]() * Quiz – Definition and computation of derivatives * Basic differentiation rules and rates of change * The derivative and the tangent line problem * Modeling rates of change and solving related rates problems * Equations involving derivatives and problems using their verbal descriptions * Chain rule and implicit differentiation * Approximate rate of change from graphs and tables of values * Instantaneous rate of change as the limit of average rate of change * Derivative interpreted as instantaneous rate of change * Basic rules for the derivatives of sums, products, and quotients of functions * Knowledge of derivatives of power and trigonometric functions * Graphic, numeric and analytic interpretations of the derivative * Derivative defined as the limit of the difference quotient * Test – Limits and Continuity Module 2: Differentiation Suggested Pace: 5 weeks ![]() * Elluminate Session: Discussion about conditions of continuity. Discussion about the limitation of a graphing calculator to show discontinuities in functions and the value of using a calculator to support conclusions found analytically. * Oral Review: Discussion about using the Calculator to experiment and produce a table of values to examine a function and estimate a limit as x approaches a point and as x grows without bound. ![]() * Finding limits graphically and numerically * Understanding graphs of continuous or non-continuous functions geometrically * Understanding continuity in terms of limits * Describing asymptotic behavior in terms of limits involving infinity * Calculating limits using algebraic methods * Intuitive understanding of limit process * Quiz – Functions, Graphs, and Rates of Change Module 1: Limits and Continuity Suggested Pace: 2 weeks * Oral Review: Discussion about using Calculator zoom features to examine a graph in a good viewing window and calculator operations to find the zeros of a graph and the point of intersection of two graphs * Comparing relative magnitudes of functions – contrasting exponential, logarithmic and polynomial growth ![]() * Using the Cartesian coordinate system to graph functions * Understanding the properties of real numbers and the number line Major Topics and Concepts Module 0: Preparation for Calculus Suggested Pace: 2 weeks This course includes a study of limits, continuity, differentiation, and integration of algebraic, trigonometric and transcendental functions, and the applications of derivatives and integrals. Walk in the footsteps of Newton and Leibnitz! An interactive text and graphing software combine with the exciting on-line course delivery to make Calculus an adventure.
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