Mikel Haug » AP CALCULUS AB: 4th PERIOD

AP CALCULUS AB: 4th PERIOD

AP Calculus AB Overview and Syllabus

 

Course Description Calculus AB is a college level course covering derivatives, integrals, limits, approximation, and applications and modeling. The course emphasizes a multi-representational approach to calculus, with concepts, results and problems being expressed graphically, numerically, analytically and verbally. The connections among these representations also are important. This course is intended to be challenging and demanding.  Broad concepts and widely applicable methods are emphasized. Although facility with manipulation and computational competence are important components, they are not the core of this course. Technology is used to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Through the use of unifying themes of derivatives, integrals, limits, approximation, and applications and modeling, the course becomes a cohesive whole rather than a collection of unrelated topics.  Prerequisites are a knowledge of algebra, geometry, trigonometry, analytic geometry, and elementary functions.

Course Goals and/or Major Student Outcomes:  After completing this course, students should: 

  • be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal and understand the connections among these representations.
  • understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems.
  • understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems.
  • understand the relationship between the derivative and the definite integral as expressed in both parts of the fundamental theorem of calculus.
  • be able to communicate mathematics and explain solutions to problems both verbally and in written sentences.
  • be able to model a written description of a physical situation with a function, a differential equation or an integral.

 

Course Objectives:  The objectives of this course are to cultivate a basic understanding of the concepts of calculus, provide experience in the use of calculus methods, and show calculus methods may be applied to practical applications.

 

 

GRADING POLICIES:  For the yearly average, the two semesters are averaged. 

  • Test/Summative Assessment Average is weighted 35%.
  • Quiz/Formative Assessment Average is weighted 50%.
  • Daily Participation average is weighted 15%.
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Error Analyses and Tutorials:  Students have the opportunity to complete and error analysis for formative assessments which consists of reworking the missed problems correctly on a separate sheet of paper, boxing the answer, and writing a 2 to 3 sentence description of what their error was and how they corrected it.  This should be stapled to the front of the graded test and I will give back half of what was taken off.

 

I am available for tutorials every morning at 7:30, during WIN with notice, and most afternoons as late as need be.

AP Calculus is a challenging course.   I will work hard to help you learn the material and I expect you to work hard to learn it.

 

Course Outline

 

Unit 1: 

 

Students should learn how to evaluate limits and continuity graphically, numerically, and analytically. They should have an intuitive understanding of the limiting process, be able to calculate limits and using algebra and estimate limits from graphs or tables of data.  Students should describe one sided limits and asymptotic behavior in terms of limits involving infinity.  Students should develop an intuitive understanding of continuity (The function values can be made as close as desired by taking sufficiently close values of the domain.), discussing continuity of a function at a point and over a domain.   Students should describe continuity in terms of limits.  Students should acquire and cultivate a geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)

 

Unit 2: 

 

Students should have an intuitive understanding the interpretation of a derivative as an instantaneous rate of change and recognize derivatives presented graphically, numerically, and analytically. They should understand how to calculate derivatives as difference quotients and be able to describe the relationship between differentiability and continuity.  Students should be able to find the slope of a curve at a point including emphasized, points at which there are vertical tangents and points at which there are no tangents.  Students should write the equation for tangent and normal lines to a curve at a point by calculating the derivative and use local linear approximation.  They will derive the instantaneous rate of change as the limit of average rate of change and approximate rates of change from graphs and tables of values.

 

Students should describe equations involving derivatives including verbal representations translated into equations involving derivatives and vice versa.  Students should acquire and cultivate a geometric understanding of graphs and corresponding characteristics of graphs of ƒ and ƒ∙.  Students should develop an understanding of the relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ∙.  Student should develop an understanding of The Mean Value Theorem and its geometric interpretation.

 

Students should extend and apply their knowledge to higher order derivatives noting the characteristics of the graphs of ƒ, ƒ∙, and ƒ ∙including the relationship between the concavity of ƒ and the sign of ƒ ∙ and points of inflection as places where concavity change

 

Unit 3:  

 

Students should use derivatives to solve a variety of problems presented graphically, numerically, and analytically.  Students should be able to compute derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions; be able to apply derivative rules for sums, products, and quotients of functions, and use the chain rule and implicit differentiation

 

They should understand how to calculate derivatives as difference quotients and be able to describe the relationship between differentiability and continuity.  Students should be able to find the slope of a curve at a point including emphasized, points at which there are vertical tangents and points at which there are no tangents.  Students should write the equation for tangent and normal lines to a curve at a point by calculating the derivative and use local linear approximation.  They will derive the instantaneous rate of change as the limit of average rate of change and approximate rates of change from graphs and tables of values.

 

Students should extend and apply their knowledge of derivatives to solve optimization problems, both absolute (global) and relative (local) extrema.  They should also be able to interpret the derivative as a rate of change in varied applied contexts including velocity, speed, and acceleration.  Students will model and solve problems involving rates of change, including related rates problems.

 

Students should use of implicit differentiation to find the derivative of an inverse function.  They should develop a geometric interpretation of differential equations via slope fields and comprehend the relationship between slope fields and solution curves for differential equations.

 

 

Unit 4: 

 

Students should develop an awareness of families of antiderivatives as reversing derivatives. 

 

Students will explore and interpret the concept of the definite integral, computing Riemann sums using left, right, and midpoint evaluation points, finding the definite integral as a limit of Riemann sums over equal subdivisions, and finding the definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

Students will be able to estimate the solution to an integral problem through calculation of finite sums, using both Riemann sums with left, right, and midpoint evaluation points over equal subdivisions, and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, graphically, and by a table of values.

 

The student will gain proficiency applying the rules for definite integrals; the average value theorem; and MVT for definite integrals.  The student will determine and evaluate antiderivatives following directly from derivatives of basic functions and by substitution of variables (including change of limits for definite integrals).  The student will also apply the basic properties of integrals.

 

The student will understand how the Fundamental Theorem relates derivatives to integrals, and will master the use of the Fundamental Theorem to evaluate definite integrals and to represent a particular antiderivative.  The student will apply the Fundamental Theorem to the analytical and graphical analysis of functions.

 

Unit 5: 

 

Students should be aware integrals are used in a variety of applications to model physical, biological, or economic situations.  Students should be able to adapt their knowledge and techniques to solve other similar application problems with an emphasis on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral.  Specific applications should include finding the area of a region, the volume of a solid with known cross sections, the volume of a solid of revolution, the average value of a function, the distance traveled by a particle along a line, and accumulated change from a rate of change.  Students should be able to find specific antiderivatives using initial conditions, including applications to motion along a line.

 

 

Unit 6: 

 

The student will be able to approximate the solution to any differential equation of two variables graphically using slope fields.  The student will solve separable differential equations and use them in modeling (including the study of the equation and exponential growth.  The student will be able to find specific antiderivatives using initial conditions.