AP Calculus
2019
Mrs. Ioana Vescan
Room 211
Calculus, Bass, Finney, Demana; Prentice Hall Mathematics
Course Description:
An Advanced Placement (AP) course in calculus consists of a full high school academic year of work that is comparable to a calculus course in colleges and universities. AP Calculus will follow the topics outlined by the College Board. During the last weeks of spring semester (after the official AP exam), advanced topics will be covered. The course is primarily concerned with developing students’ understanding of the concepts of calculus and providing experience with its methods and applications. 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 are demonstrated through the unifying themes of derivatives, integrals, limits, approximation, applications, and modeling. A major objective of the class is to prepare students for the AP Calculus AB exam to be given in the spring. Most universities award credit to students based upon their scores on this exam.
Outline of Required Topics: (Taken directly from College Board)
- Functions, Graphs, & Limit
- Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
- Limits of functions (including one-sided limits)
- An intuitive understanding of the limiting process
- Calculating limits using algebra
- Estimating limits from graphs or tables of data
- Asymptotic and unbounded behavior
- Understanding asymptotes in terms of graphical behavior
- Describing asymptotic behavior in terms of limits involving infinity
- Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)
- Continuity as a property of functions
- An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain)
- Understanding continuity in terms of limits
- Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)
- Derivatives
- Concept of the derivative
- Derivative presented graphically, numerically, and analytically
- Derivative interpreted as an instantaneous rate of change
- Derivative defined as the limit of the difference quotient
- Relationship between differentiability and continuity
- Derivative at a point
- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
- Tangent line to a curve at a point and local linear approximation
- Instantaneous rate of change as the limit of average rate of change
- Approximate rate of change from graphs and tables of values
- Derivative as a function
- Corresponding characteristics of graphs of f and f’
- Relationship between the increasing and decreasing behavior of f and the sign of f’
- The Mean Value Theorem and its geometric consequences
- Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa
- Applications of derivatives
- Analysis of curves, including the notions of monotonicity and concavity
- Optimization, both absolute (global) and relative (local) extrema
- Modeling rates of change, including related rates problems
- Use of implicit differentiation to find the derivative of an inverse function
- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration
- Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations
- Computation of derivatives
- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions
- Basic rules for the derivative of sums, products, and quotients of functions
- Chain rule and implicit differentiation
III. Integrals
- Interpretations and properties of definite integrals
- Definite integral as a limit of Riemann sums
- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
- Basic properties of definite integrals (examples include additivity and linearity)
- Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a small sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.
- Fundamental Theorem of Calculus
- Use of the Fundamental Theorem to evaluate definite integrals
- Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined
- Applications of antidifferentiation
- Finding specific antiderivatives using initial conditions, including applications to motion along a line
- Solving separable differential equations and using them in modeling (in particular, studying the equation y’ = ky and exponential growth)
- Numerical approximations to definite integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
- L’Hospital’s Rule including its use in determining limits and convergence of improper integrals and series.
- Calculation of volume through use of cylindrical shells in addition to disk and washer methods.
- Integration by Parts
The Role of Technology in AP Calculus:
Technology is designed to make our lives as mathematicians easier; yet, technology is not a substitute for mathematical understanding and proficiency. Students are expected, both by the instructor and by College Board, to understand the underlying mathematical concepts with and without the use of technology; as a result, tests in this course will frequently be divided in the same fashion.
This course will involve extensive use of the graphing calculator and various computer programs to give students mathematical understanding across multiple representations. Students are expected to become “best-friends” with their graphing calculators.
Required Daily Materials:
- Textbook:
- Graphing Calculator
- Pens, Pencils, and Paper.
- A Three-Ring Binder is strongly suggested for organization of class materials.
- This syllabus
Grading in an AP Course:
AP Calculus is a college level course. As a result, the manner in which the overall course grade is determined varies from a typical high school mathematics course.
- Tests – 70%
- Exams – 15%
- CW/QZ – 15%
Tests will be given roughly every two to three weeks and will always be announced in advance. Quizzes may be given with or without warning and are to be completed in (what the instructor determines to be) a reasonable amount of time. A homework quiz will be given every Friday.
*Note: Although tests count considerably more than homework in the overall course grade, homework is the most important component of this course. If a student does not complete the assigned homework, she/he will not succeed in this course. Late assignments will not be accepted. Homework assignments are designed to be challenging. (Often, we can learn more from incorrect solutions than we can from correct ones).
Collaboration and Additional Instruction:
Students are expected to participate in class every day. Participation may include, but is not limited to: discussion, presentation, cooperative work, and individual work. Being able to do mathematics as an individual is a not a sufficient condition for being mathematically successful. Students must be able to discuss the daily mathematics with each other and the instructor using respectful dialogue and correct mathematical terminology.
Collaboration between students outside of class is encouraged and necessary in advanced mathematics courses. If students have questions regarding the difference between collaboration and cheating, the course instructor should be consulted; academic dishonesty will not be tolerated. I will be available after school most days for additional assistance. Appointments are never necessary, but some warning is nice. Email is a perfectly valid way to receive assistance or confirm that the instructor will be available for help).
Calculus is a great human achievement that I hope each of you will come to appreciate this year. I am excited about teaching this course and hope you are as excited about taking it. You have been preparing yourselves for college all your lives. If you do well in this course, you will have an advantage over those admitted to college without AP experiences. I look forward to working closely with each of you throughout the academic year to ensure your success in this course and in your future mathematical education.
Ioana Vescan