* 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. The midline is the x-axis, and the second picture shows a reflection over that.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. Then, reflecting the function over a vertical line halfway through a period maps the function back to itself. This is a horizontal line over which one can reflect the function. Vertical and horizontal shifts change the midline and y-intercept of a trig function.Įach of the trigonometric functions has a midline. Finally, adding a coefficient to the variable stretches the function horizontally, changing the period. Adding a coefficient to the function stretches or compresses the function vertically. Each is also either $2\pi$ or $\pi$ periodic, meaning the graph repeats every $2\pi$ or $\pi$ radians.Īdding a vertical or horizontal shift changes the midline and y-intercept respectively. In the graph above, the tangent graph is green and the cotangent graph is blue.Īfter these shapes become familiar, graphing transformations of these functions follows.Įach function has a midline at the x-axis and a fixed y-intercept. Tangent and cotangent have a periodic shape similar to an $x^3$ graph. In the graph above, cosecant is the green graph and secant is the blue graph. In the graph above, sine is green and cosine is blue.Ĭosecant and secant have alternating positive and negative periodic, parabolic graphs. To graph trig functions, begin with the base graph for each of the six trig functions. Know base graphs, know how to shift vertical and horizontal, know how to change period, know effect of vertical dilation.
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