## AP Calc

### Course Description:

This course prepares students for the AP Calculus AB Exam as well as university level calculus courses. The study of calculus is divided into two main branches: differential and integral calculus. Differential calculus is the study of changing quantities, and integral calculus involves techniques for finding sums using infinitesimally small quantities. We will explore the relationship between these two branches and will find many applications as we prepare for the AP exam.

AP Calculus utilizes a culmination of algebraic, geometric, and mathematical reasoning skills gleaned from earlier high school math courses. Many calculus problems require some rigor, but the work is interesting and rewarding. This course is usually the first opportunity for students to use the concept of infinity in a formal, mathematical way, and these new tools confer a gratifying sense of power.

The mathematical and analytical training that students receive in this course will enable them to:

- Model situations using functions represented symbolically and graphically, and to understand the relationships between these representations.
- Determine the derivatives of functions and understand the meaning in terms rate of change, and to use derivatives and linear approximations to solve problems.
- Understand the relationship between the derivative and the definite integral.
- Communicate mathematics orally and in well-written sentences to explain solutions to problems.
- Use functions, differential equations, and integrals to model given physical situations.
- Use appropriate technology to explore problems, experiment with solutions, interpret results, and verify conclusions.
- Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
- Cultivate an appreciation for the beauty of calculus as both an invention and a discovery.

### Assessment and Evaluation:

- Tests – 50%
- Quizzes – 30%
- Homework – 20%

### Comments on Assessment & Evaluation:

- Students are required to organize and maintain all notes, homework, and worksheets for this class in a clearly labeled notebook or binder. Homework assignments must be completed on pages that can be removed from the binder and added back after grading. This notebook will serve as a helpful reference tool if it is properly maintained.
- The date for unit tests will be announced about 1 week in advance. Quizzes will often be unannounced.

### Materials:

- Course Textbook:
*Calculus: Graphical, Numerical, Algebraic, AP Edition*, by B. Waits, F. Demana, and D. Kennedy (2007 Pearson). - TI-84 (or similar approved model) graphing calculator
- Notebook/binder with pockets for storing handouts
- Graph paper and notebook paper
- Pen, pencil, eraser, ruler and other general stationery items

### Use of Technology:

Every student in the class has his or her own TI-84 (or similar Texas Instruments model) graphing calculator. Students usually purchase their own calculators. Calculators are provided for students for whom this is a financial burden. Most homework problems are clearly identified as being “calculator allowed” or “non-calculator” problems. Students are encouraged to develop a clear sense of when it is appropriate to use a calculator and when a calculator is not appropriate. Tests are divided into calculator and non-calculator sections.

Graphing calculators are used to help students to:

- Graph functions.
- Explore domain and range, and set appropriate windows for inspection.
- Calculate numerical solutions and derivatives.
- Analyze and interpret results.
- Justify and explain results from graphs and equations.

Explicit instructions for usage of these calculators come from example presentations in class. Further examples can be found in the text and online.

### The Well-Rounded Calculus Problem Solver:

In order to help students gain a comfortable understanding of calculus and develop a well-rounded approach, we must explore many problems and work with functions represented in a variety of ways. Making connections among the various representations is part of the learning process. This comes from solving problems, and our tack will mainly involve a combination of:

- Numerical analysis (where data points are known, but not an equation)
- Graphical analysis (where a graph is known, but again, not an equation)
- Analytic/algebraic analysis (traditional equation and variable manipulation)
- Verbal/written methods of representing problems and justifying solutions

### Course Outline:

Unit One: Prerequisites For Calculus – approximately 3 days

1.1 Lines

1.2 Functions and Graphs

1.3 Exponential Functions

1.5 Functions and Logarithms

1.6 Trigonometric Functions

Learning Outcomes:

- Students will be able to write an equation and sketch a graph of a line given necessary information.
- Students will be able to identify the domain and range of a function.
- Students will be able to interpret, determine, and write formulas for piecewise defined functions.
- Students will be able to solve problems involving exponential growth and decay.
- Students will be able to determine the algebraic and graphical representations of functions and their inverses.
- Students will be able to generate the graphs of the trigonometric functions and explore various transformations of these graphs.

Unit Two: Limits and Continuity – approximately 7 days

2.1 Rates of Change and Limits

2.2 Limits Involving Infinity

2.3 Continuity

2.4 Rates of Change and Tangent Lines

Learning Outcomes:

- Students will be able to calculate average and instantaneous speeds.
- Students will be able to define and calculate limits.
- Students will be able to identify horizontal and vertical asymptotes based on a limit definition.
- Students will be able to identify the intervals upon which a function is continuous.
- Students will be able to apply the Intermediate Value Theorem.
- Students will be able to apply the definition of the slope of a curve to calculate slopes.
- Students will be able to find the equations of the tangent line and normal line to a curve at a given point.
- Students will be able to find the average rate of change of a function.

Unit Three: Derivatives – approximately 15 days

3.1 Derivative of a Function

3.2 Differentiability

3.3 Rules for Differentiation

3.4 Velocity and other Rates of Change

3.5 Derivatives of Trigonometric Functions

3.6 Chain Rule

3.7 Implicit Differentiation

3.8 Derivatives of Inverse Trigonometric Functions

3.9 Derivatives of Exponential and Logarithmic Functions

Learning Outcomes:

- Students will be able to calculate slopes and derivatives using the definition of the derivative.
- Students will be able to graph f from the graph of f’, graph f’ from the graph of f, and graph the derivative of a function given numerically with data.
- Students will be able to determine where a function is not differentiable.
- Students will be able to approximate derivatives numerically and graphically.
- Students will be able to use the rules of differentiation to calculate derivatives, including second and higher order derivatives.
- Students will be able to use derivatives to analyze motion along a line.
- Students will be able to differentiate the six basic trigonometric functions.
- Students will be able to differentiate composite functions using the Chain Rule.
- Students will be able to find derivatives using implicit differentiation.
- Students will be able to find derivatives involving the inverse trigonometric functions.
- Students will be able to calculate derivatives of exponential and logarithmic functions.

Unit Four: Applications of Derivatives – approximately 12 days

4.1 Extreme Values of a Function

4.2 Mean Value Theorem

4.3 Connecting f ‘ and f” with the graph of f

4.4 Modeling and Optimization

4.5 Linearization and Newton’s Method

4.6 Related Rates

Learning Outcomes:

- Students will be able to determine the local and global extreme values of a function.
- Students will be able to apply the Mean Value Theorem and find intervals on which a function is increasing or decreasing.
- Students will be able to use the First and Second Derivative Tests to determine the local extreme values of a function.
- Students will be able to determine the concavity of a function and locate points of inflection.
- Students will be able to solve application problems involving finding maximum or minimum values of a function.
- Students will be able to estimate change in a function using differentials.
- Students will be able to solve related rate problems.

Unit Five: The Definite Integral – approximately 8 days

5.1 Estimating with Finite Sums

5.2 Definite Integrals

5.3 Definite Integrals and Antiderivatives

5.4 Fundamental Theorem of Calculus

5.5 Trapezoidal Rule

Learning Outcomes:

- Students will be able to approximate the area under the graph of a function by using rectangle approximation methods.
- Students will be able to interpret the area under a graph as a net accumulation of a rate of change.
- Students will be able to express the area under a curve as a definite integral and as a limit of Riemann sums.
- Students will be able to compute the area under a curve using a numerical integration procedure.
- Students will be able to apply the rules for definite integrals and find the average value of a function over a closed interval.
- Students will be able to apply the Fundamental Theorem of Calculus.
- Students will be able to understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
- Students will be able to approximate the definite integral by using the Trapezoidal Rule.

Unit Six: Differential Equations and Mathematical Modeling – approximately 11 days

6.1 Antiderivatives and Slope Fields

6.2 Integration by Substitution

6.3 Integration by Parts

6.4 Exponential Growth and Decay

6.5 Population Growth

Learning Outcomes:

- Students will be able to construct antiderivatives using the Fundamental Theorem of Calculus.
- Students will be able to find antiderivatives of polynomials, e
^{kx}, and selected trigonometric functions and linear combinations of these functions. - Students will be able to construct slope fields and interpret slope fields as visualizations of differential equations.
- Students will be able to compute indefinite and definite integrals by method of substitution.
- Students will be able to solve a separable differential equation.
- Students will be able to use integration by parts to evaluate indefinite and definite integrals.
- Students will be able to solve problems involving exponential growth and decay in application questions.
- Students will be able to solve problems involving exponential or logistic population growth.

Unit Seven: Applications of Definite Integrals – approximately 5 days

7.1 Integrals as Net Change

7.2 Areas in a Plane

7.3 Volumes

Learning Outcomes:

- Students will be able to solve problems in which rate is integrated to find the net change over time in application problems.
- Students will be able to use integration to calculate areas of regions in a plane.
- Students will be able to use integration to calculate volumes of solids.
- Students will be able to calculate volume by cross-sections of a known geometric figure.

### Estimated schedule for unit coverage:

Dates |
Topics |

August 27 – September 21 | Limits and Continuity |

September 24 – November 2 | Derivatives |

November 5 – December 7 | Applications of Derivatives |

December 10 – January 18 | The Definite Integral |

January 21 – February 22 | Differential Equations |

February 25 – March 29 | Applications of Definite Integrals |

April 1 – May 7 | Review and Practice for AP Exam |

May 8 | AP Calculus AB Exam |

May 9 – June 12 | Advanced Topics |