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Calculus AB Educational Learning Objectives Science Academy J. Schofield 2006-07 PreCalculus Unit The learner will • Solve and graph linear and quadratic inequalities • Define and apply the concept of absolute value • Relate absolute value to distance • Apply the distance formula • Write equations of circles • Find x and y intercepts of functions • Find points of intersection for various functions and evaluate the feasibility of these intersections • Test the symmetry of a graph with respect to an axis and the origin • Write and graph linear equations using the general form • Interpret slope as a rate in a real-life application • Write equations of lines parallel and perpendicular to given lines • Use function notation to represent and evaluate a function • Find the domain and range of a function • Sketch the graph of a function • Identify different types of transformations of functions • Find expressions for composite functions • Test for even and odd functions • Evaluate trigonometric functions using special triangles and the unit circle • Graph the basic trigonometric functions • Solve trigonometric equations • Fit linear, quadratic and trigonometric models to real-life data sets Limits and Their Properties Unit The learner will • Estimate limits using graphical and numerical approaches • Learn different ways a limit can fail to exist • Study and use the ε−δ definition of a limit • Evaluate limits using properties of limits • Develop and use a strategy for finding limits • Evaluate a limit using division and rationalization techniques • Evaluate a limit using the Squeeze Theorem • Determine continuity at a point and on an open interval • Determine one-sided limits and continuity on a closed interval • Use properties of continuity • Understand and apply the Intermediate Value Theorem • Determine infinite limits from the left and right • Find and sketch the vertical asymptotes of the graph of a function • Use limits to justify vertical asymptotes • Find limits as x approaches infinity • Find and sketch the horizontal asymptotes of the graph of a function Differentiation Unit The learner will • • • • • • • • • • • • • • • • • Find the slope of the tangent line to a curve at a point Use the limit definition to find the derivative of a function Understand the relationship between differentiability and continuity Find the derivative of a function using the basic differentiation rules Find the derivative of sine and cosine functions Use derivatives to find rates of change Find the derivative of a function using the Product Rule Find the derivative of a function using the Quotient Rule Find the derivative of the six basic trigonometric functions Find higher order derivatives Find the derivative of a composite function using the Chain Rule Find the derivative of a function using the General Power Rule Simplify the derivative of a function using algebra Distinguish between functions written in implicit and explicit forms Use implicit differentiation to find the derivative of a function Find a related rate Use related rates to solve real-life problems Applications of Differentiation Unit The learner will • • • • • • • • • • • • • • • • • • Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval Understand and apply the Mean Value Theorem including Rolle’s Theorem Determine intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema Determine intervals on which a function is concave upward or concave downward Find and justify points of inflection Apply the Second Derivative Test to find relative extrema Determine limits at infinity Determine horizontal asymptotes Analyze and sketch the graph of a function Solve applied minimum and maximum problems Approximate a zero of a function using Newton’s Method Understand the concept of a tangent line approximation Compare the value of the differential with the actual change Estimate a propagated error using a differential Find the differential of a function Integration Unit The learner will • • • • • • • • • • • • • • • • • • Write the general solution of a differential equation Use indefinite integral notation for antiderivatives Use basic integration rules to find antiderivatives Find a particular solution of a differential equation Use sigma notation to write and evaluate a sum Understand the concept of an area Approximate the area of a plane region Find the area of a plane region using limits Understand the definition of a Riemann Sum Evaluate a definite integral using limits Evaluate a definite integral using the properties of definite integrals Evaluate a definite integral using the Fundamental Theorem of Calculus Understand and use the Mean Value Theorem for Integrals Find the average value of a function over a closed interval Understand and use the Second Fundamental Theorem of Calculus Use a change of variable to evaluate definite and indefinite integrals Evaluate definite integrals of even and odd functions Approximate a definite integral using Trapezoidal and Simpson’s Rules Transcendental Functions Unit The learner will • Develop and use properties of the natural logarithmic function • Understand the definition of the number e • Find derivatives of functions involving the natural logarithmic function • Integrate a rational function using the log rule • Integrate trigonometric functions • Verify that one function is the inverse function of another • Determine whether a function has an inverse • Find the derivative of an inverse function • Develop properties of the natural exponential function • Differentiate natural exponential functions • Integrate natural exponential functions • Define exponential functions that have other bases than e • Differentiate and integrate exponential functions that have other bases than e • Use exponential functions to model compound interest and exponential growth • Develop properties of inverse trigonometric functions • Differentiate inverse trigonometric functions • Integrate functions whose antiderivatives involve inverse trigonometric functions • Use completing the square to integrate a function Differential Equations Unit The learner will • • • • • Use initial conditions to find particular solutions of differential equations Use slope fields to approximate solutions of differential equations Use Euler’s Method to approximate solutions of differential equations Use separation of variables to solve simple differential equations Use exponential functions to model growth and decay in applied problems Applications of Integration Unit The learner will • Find the area of a region between two curves using integration • Describe integration as an accumulation process • Find the volume of a solid with known cross sections • Find the volume of a solid using the disk method • Find the volume of a solid using the washer method • Find the volume of a solid using the shell method • Compare the use of washer and shell methods • Find the arc length of a smooth curve • Find the area of a surface of revolution