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STATE UNIVERSITY OF NEW YORK COLLEGE OF TECHNOLOGY CANTON, NY COURSE OUTLINE MATH 122 – BASIC CALCULUS Prepared by: ALICE REED SCHOOL OF LIBERAL STUDIES & SUPPORT SERVICES DEPARTMENT OF MATHEMATICS MAY 2006 2 MATH 122—BASIC CALCULUS A. TITLE: Basic Calculus B. COURSE NUMBER: MATH 122 C. CREDIT HOURS: 4 D. WRITING INTENSIVE (OPTIONAL): N/A E. COURSE LENGTH: 15 weeks F. SEMESTERS OFFERED: Fall and Spring G. HOURS OF LECUTRE, LABORATORY, RECITATION, TUTORIAL, ACTIVITY: The course will consist of four 50-minute lecture-recitation periods per week for one semester. H. CATALOGUE DESCRIPTION: This course is an intuitive introduction to the Calculus. Topics include: Review of functions, analytical geometry of the line, properties of limits; the derivative with applications; trigonometric and other transcendental functions; and integrals with applications. Selected additional topics will be offered, as time permits, at the discretion of the instructor. Four hours lecture per week. I. PRE-REQUISITES/CO-REQUISITES: College Algebra (MATH 121) with a grade of C or better recommended or Course III with a 4th year of high school mathematics, or at least a grade of 80 on NYS Regents Math B or permission of instructor. J. GOALS (STUDENT LEARNING OUTCOMES): see attached. K. TEXTS: Members of the Mathematics Department who will be teaching the course will select the appropriate text. Audio-visual aids and computer software will be used when appropriate and available. L. REFERENCES: None M. EQUIPMENT: Smart classroom (Computer projection and access to the Internet) 3 MATH 122—BASIC CALCULUS N. GRADING METHOD Traditional grades of A-F will be assigned. O. MEASUREMENT CRITERIA/METHODS: Instructors may use a combination of quizzes, exams, and projects as measurement criteria. P. TOPICAL OUTINE: see attached sheet Q. LABORATORY OUTLINE: N/A 4 MATH 122 – BASIC CALCULUS STUDENT LEARNING OUTCOMES Students will be able to: I. Functions, Graphs and Limits a. Plot points on a coordinate plane and interpret data presented graphically b. Find the distance between two points in a coordinate plane c. Find the midpoints of line segments connecting two planes d. Find the x- and y- intercepts of graphs of equations algebraically and graphically using a graphing utility e. Write the standard forms of equations of circles, given the center and point on the circle f. Convert equations of circles from general form to standard form by completing the square, and sketch the circles g. Find the points of intersection of two graphs algebraically and graphically using a graphing utility h. Find the break-even point for business and the equilibrium points of supply and demand equations i. Find the slope between two points and use the slope-intercept and point- slope forms to graph equations j. Find equations of parallel and perpendicular lines k. Use vertical line test to determine functions l. Use function notation to evaluate functions m. Determine the domain and range of functions algebraically and graphically n. Combine functions to create other functions o. Use the horizontal line test to determine whether functions have inverse functions. If they do, find the inverse functions p. Determine whether limits exist. If they do, find the limits q. Find one-sided limits r. Use the definition of continuity to determine if a function is continuous at a point, on an open or on a closed interval II. Differentiation a. Approximate the slope of a tangent line to a graph at a point b. Interpret the slope of a graph c. Use the limit definition to find the derivative of a function and the slope of a graph at a point d. Use the derivative to find the derivative of a function and the slope of a graph at a point e. Use the graph of a function to recognize points at which the function is not differentiable f. Find the derivative using the constant rule, the power rule, the constant multiple rule, and sum and difference rules 5 MATH 122 – BASIC CALCULUS STUDENT LEARNING OUTCOMES (continued) g. Find the average rate of change of a function over an interval and the instantaneous rate of change at a point h. Find the velocity of an object that is moving in a straight line i. Find the marginal revenue, marginal cost, and marginal profit for a product j. Find the derivative using the product rule, quotient rule and chain rule k. Find higher order derivatives l. Find and use the position function to determine the velocity and acceleration of a moving object m. Find derivatives implicitly n. Solve related-rate problems III. Applications of Derivatives a. Find the critical numbers of a function b. Find the open intervals on which a function is increasing or decreasing c. Use the First Derivative Test to find the relative extrema of a function d. Find the absolute extrema of a continuous function on a closed interval e. Find the open intervals on which a function is concave upward or concave downward f. Find the points of inflection of the graph of a function g. Use the Second Derivative Test to find the relative extrema of a function h. Find the vertical and horizontal asymptotes of a function and sketch its graph i. Find the point of diminishing returns j. Solve applied optimization problems k. Solve business and economic optimization problems l. Find infinite limits and limits at infinity m. Analyze the graph of a function n. Find the differential of a function o. Use differentials to approximate changes in a function IV. Integration a. Use the definition of the natural logarithm to write exponential equations in logarithmic form, and vice versa b. Sketch the graphs of exponential and logarithmic functions c. Use the properties of logarithms to expand and condense logarithmic expressions d. Find the derivatives of natural exponential and natural logarithmic functions e. Use the basic integration rules to find indefinite integrals f. Use substitution to find indefinite integrals 6 MATH 122 – BASIC CALCULUS STUDENT LEARNING OUTCOMES (continued) g. Use the Fundamental Theorem of Calculus to evaluate definite integral h. Use substitution to find definite integrals i. Find the area of regions bounded by the graph of a function and the x-axis j. Find areas of regions bounded by two or more graphs k. Find consumer and producer surpluses l. Use the Midpoint rule to approximate values of definite integrals m. Use the Trapezoidal rule to approximate values of definite integrals 7 MATH 122 – BASIC CALCULUS DETAILED OUTLINE I. Functions, Graphs, and Limits A. The Cartesian Plane and Distance Formula B. Graphs of equations C. Lines in the plane and slope D. Functions 1. Notation 2. Evaluation 3. Domain, range, zeros 4. Linear functions 5. Graphs 6. Economic functions E. Limits 1. Estimate limits using tables and graphs 2. Find limits using algebra 3. Determine when limits exist and when they do not exist. F. Continuity 1. Definition of continuity 2. Removable and non-removable discontinuity II. Differentiation A. The derivative and the slope of a graph B. Basic rules for differentiation 1. Constant rule 2. Power rule 3. Sum and difference rules C. Rates of change 1. Velocity 2. Marginal cost, revenue, and profit D. The product and quotient rules E. The chain rule F. Higher-order derivatives 1. Acceleration G. Implicit differentiation H. Related rates 8 MATH 122 – BASIC CALCULUS DETAILED OUTLINE (continued) III. Applications of the Derivative A. Increasing and decreasing functions B. Extrema and the First-Derivative test 1. Critical points C. Concavity and the Second-Derivative test 1. Inflection points D. Asymptotes 1. Limits involving infinity E. Curve-sketching F. Optimization Problems 1. Maximum and minimum applications 2. Area 3. Business and economic problems G. Differentials and marginal analysis IV. Integration A. Exponential and logarithmic functions 1. Review of natural exponential and logarithmic properties 2. Derivatives of exponential and logarithmic functions 3. Exponential growth and decay B. Antiderivatives and indefinite integrals C. The general power rule D. Integration by substitution E. Exponential and logarithmic integrals F. Evaluate definite integrals G. Area under a curve H. The Fundamental Theorem of Calculus I. The area of a region bounded by two graphs J. Approximate area 1. Midpoint rule 2. Trapezoidal rule 9