CURRICULUM MAP: 19039.map
Calculus (HON) 471
25.1 MATHEMATICS - ALG REASONING: PATTERNS & FUNCTS
-- Students will identify, describe, create and generalize numeric, geometric, and statistical patterns with tables, graphs, words, and symbolic rules.
-- Students will make and justify predictions based on patterns.
-- Students will identify the characteristics of functions and relations including domain and range.
-- Students will apply the concepts of limits to sequences and asymptotic behavior of functions.
-- Students will recognize and explain the meaning of the slope and x- and y-intercepts as they relate to a context, graph, table or equation.
-- Students will relate the graphical representation of a function to its function family and find equations, intercepts, maximum or minimum values, asymptotes and line of symmetry for that function.
25.1 MATHEMATICS - ALG REASONING: PATTERNS & FUNCTS
-- Students will recognize that the slope of the tangent line to a curve represents the rate of change.
-- Students will determine equivalent representations of an algebraic equation or inequality to simplify and solve problems.
25.2 MATHEMATICS - NUMERICAL & PROP REASONING
-- Students will select and use appropriate methods for computing to solve problems in a variety of contexts.
-- Students will develop and use a variety of strategies to estimate values of formulas, functions and roots; to recognize the limitations of estimation; and to judge the implications of the results.
25.3 MATHEMATICS - GEOM & MEASUREMT
-- Students will use geometric properties to solve problems in two and three dimensions.
-- Students will describe how a change in measurement of one or more parts of a polygon or solid may affect its perimeter, area, surface area and volume and make generalizations for similar figures.
-- Students will visualize three-dimensional objects from different perspectives and analyze cross-sections, surface area, and volume.
-- Students will use Cartesian, navigational, polar, and spherical systems to represent, analyze, and solve geometric and measurement problems.
1) What are limits? How are they analyzed? Why are they important to the study of calculus?
2) What is a derivative? What techniques are used to find derivatives? How is a derivative related to a tangent line? How can the derivative concept be applied to real world problems?
3) What is integration, and how does it relate to differentiation? What are techniques used to find an anti-derivative, and what does the answer represent? How can the integral concept be applied to real world problems?
This course focuses on the following concepts:
Analysis of Graphs
Limits of Functions
Asymptotic and Unbounded Behavior
Continuity as a Property of Functions
Concept of Derivative
Derivative at a Point
Derivative as a Function
Higher Order Derivatives
Techniques of Differentiation
Mean Value Theorem
Applications of Derivatives
Interpretations and Properties of Definite Integrals
Applications of Integrals
Fundamental Theorem of Calculus
Techniques of Antidifferentiation
Applications of Antidifferentiation
Numerical Approximations to Definite Integrals
Students will develop the ability to:
-- Determine the relationship between the geometric and analytic representations of functions and how to use calculus to explain and predict both local and global behavior,
-- Explain the limiting process,
-- Estimate limits from graphs or tables of data,
-- Calculate limits using algebra,
-- Explain asymptotes in terms of graphical behavior,
-- Describe asymptotic behavior in terms of limits involving infinity,
-- Explain what makes a function continuous or discontinuous,
-- Relate continuity to limits,
-- Present the derivative graphically and analytically,
-- Relate the derivative to instantaneous rate of change,
-- Use the definition of the derivative to find the derivative,
-- Explain the relationship between differentiability and continuity,
-- Find the derivative at a point,
-- Identify where there are vertical tangents and removable discontinuities,
-- Relate the graph of a function to the graph of its derivative,
-- Translate verbal descriptions into equations involving derivatives and vice versa,
-- Relate the graph of a function to its second derivative graph,
-- Use Rolle's Theorem and Mean Value Theorem in problem solving
-- Identify points of inflection,
-- Analyze curves in terms of increasing and decreasing intervals and concavity,
-- Find maximums and minimums through equation analysis,
-- Model rates of change including related rate problems,
-- Solve optimization poblems using differentiation
-- Solve exponential growth and decay problems using differential equations
-- Use a tangent line to approximate the graph of a function
-- Use L'Hopital's Rule to find the limit of a quotient
-- Differentiate implicitly,
-- Differentiate power, exponential, logarithmic, trigonometric, and inverse trigonometric functions,
--Differentiate sums, products, and quotients of functions,
-- Differentiate using the chain rule,
-- Compute Riemann sums using left, right, and midpoint evaluation points,
-- Explain the definite integral as a limit of Riemann sums over equal subdivisions,
-- Explain the definite integral as the change of the quantity over the interval,
-- Explain the properties of definite integrals,
-- Use the Fundamental Theorem of Calculus to represent and evaluate definite integrals,
-- Use derivatives of basic functions to antidifferentiate,
-- Use u-substitution of variables to antidifferentiate,
-- Solve separable differential equations and use them in modeling,
-- Use Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
All students will:
-- Have daily exposure to both mechanical, theoretical, and application-based problems,
-- Utilize the graphing calculator extensively to supplement pencil and paper techniques.
Students will be assessed by:
1. Daily homework assignments, which will acount for 10% of their final grade
2. Quizzes, approximately one every one to two weeks
3. Unit tests, comprised of 60-70% application, 30-40% computation
4. A final exam that will cover an entire semester's material, andwill account for 20% of the student's final grade
Projects involving hands-on experience with differentation and integration (e.g.related rates, optimization, area, or volume) and research into areas not specifically covered by the curriculum (e.g. math history).
1. Course Text - Calculus with Analytic Geometry - 6th Edition - Larson & Hostetler
2. Supplementary texts
3. Graphing calculator
4. Graphing calculator projector