Mathematics Department

Essential Knowledge and Skills

Students in Calculus Honors study functions, graphs, limits, derivatives and integration. The course is “applications based” where a concept is discussed and presented in such a way that the students’ learning is based on previous mathematical experience. New concepts draw upon that experience and involve the student actively in the development of each new topic. The students’ learning experience is then enhanced with relevant and factual applications. The learning experience concludes with a formalization of the concept, often a calculus technique or formula. Students will work with functions in a variety of ways: graphical, numerical, analytical and verbal. Students will graph functions, determine limits, calculate the derivative as the limit of the difference quotient and use derivatives to solve a variety of problems. Students will apply the Fundamental Theorem of Calculus. Students will determine the definite integral as a limit and also use integrals solve a variety of problems.

Indicators of Student Learning Upon the completion of this course, students will:

Problem Solving

• Select appropriate method for problem solving.

• Utilize different problem solving strategies including; analytical process to organize information, solve the problem and analyze the results.

• Represent and solve routine and non-routine problems verbally, numerically graphically, geometrically and algebraically

Functions, Graphs and Limits

• Calculate limits and estimate limits from a graph.

• Determine asymptotes and their effects on limits at infinity

• Determine continuity of functions and their graphs and their impact on limits

Derivatives

• Calculate the derivative as the limit of the difference quotient

• Determine the relationship between differentiability and continuity

• Find the derivative of polynomial, rational, trigonometric, logarithmic and exponential functions using the power rule, product rule, quotient rule, and chain rule as well as higher order derivatives.

• Determine the instantaneous rate of change and the average rate of change analytically and graphically

• Analyze functions by determining the extrema and intervals where the function is increasing and decreasing and determining the concavity, points of inflection and corresponding characteristics on the graph of the function.

• Find rates of change and related rates using implicit differentiation

• Use the Fundamental Theorem of Calculus to evaluate definite integrals and for analytic and graphical analysis of functions.

• Find anti-derivatives using techniques of integration including area, the definite integral, power rule, substitution, recognition, integration by parts, integration using tables, improper integrals and multivariable integration.

• Analyze exponential models, including exponential growth and decay, compound interest and probability.

• Calculate the volume of a solid of revolution by means of cross sections using the disk, washer or shell method.

Communication

• Express differential and integral calculus concepts using correct terminology and mathematical symbols in oral and written form.

• Use the language of mathematics to clearly express mathematical ideas.

• Use communication strategies to facilitate retention of information such as repeating information, constructing mnemonics and taking notes.

• Synthesize information to answer questions, solve problems, and communicate ideas.

• Connect prior knowledge and new information to expand understanding of topics.

• Share ideas and information in small and whole class discussion, visual presentations, written response and multi-media presentations.

Technology

• Use a graphing calculator efficiently and accurately for computation, graphing, and modeling a wide variety of mathematical concepts.

• Use technology to solve problems, graph functions, experiment, interpret results, and verify conclusions.

• Select and use appropriate technology to solve problems analytically and graphically.

• Select and apply technology tools such as three-dimensional graphing software, online resources and information for problem-solving and real-world simulations.

• Collaborate with peers to use technology to compile and produce projects, models, and other creative works.

Assessment

Upon the completion of this course, students will:

• Demonstrate competency in problem solving by solving differential equation and integrals using appropriate calculus techniques correctly to determine results and then checking the reasonableness of those results.

• Demonstrate competency in functions, graphs and limits by analyzing the graph and determining its continuity, extrema, intervals where the function is increasing and decreasing and determining the concavity, points of inflection and corresponding characteristics on the graph of the function.

• Demonstrate competency in finding the derivative of functions using the power rule, product rule, quotient rule, chain rule, higher order derivatives and determining instantaneous rates of change, average rates of change using implicit differentiation.

• Demonstrate competency in integration using appropriate integration techniques such as definite integration, substitution, recognition, integration by parts, integration using tables, improper integrals and multivariable integration and determining area, volume, growth and decay using integration.

• Demonstrate competency in communication by using correct terminology, expressing ideas clearly, explaining analytical process, presenting ideas and information in visual presentations, written response, and multi-media presentations.

• Demonstrate competency in technology by modeling and graphing functions, solids of revolution and real-world simulations using computer software, on line resources and a graphing utility.