# AP Calculus AB

## Description

Calculus AB course is an honors level math class, preparing students for the AP exam. Its structure grounds the study of calculus in real-world scenarios and integrates it with the four STEM disciplines.

The first semester covers functions, limits, derivatives and the application of derivatives.

The course goes on to cover differentiation and anti-differentiation, applications of integration, inverse functions, and techniques of integration.

Students receive simulated AP tests to prepare them to succeed in the May AP exam.

## Course Topics

Category | Objectives |
---|---|

Graph Basics | Learn about the types and various parts of graphs; look at analytic and geometric information on graphs. |

The Basics of Functions | Topics include notation, transformation, domain, range, inverse functions and composition of functions. |

How to Graph Functions | Study graphing and compounding functions, factoring, exponentials, the natural log, logarithms, implicit function and horizontal/vertical asymptotes. |

Limits of Functions | Take a look at notation in limits, one-sided limits, continuity, the Squeeze Theorem and the properties of limits. |

Continuity of Functions | Find out about continuity, discontinuities in functions and graphs, regions of continuity, the Intermediate Value Theorem and the Continuous Functions Theorems. |

Exponentials and Logarithms | Study exponential functions, exponential decay vs. growth and logarithmic properties. |

Exponents and Polynomials | Explore the five main exponent properties, simplification of expressions with exponents, rational exponents, working with polynomials and using synthetic or long division with polynomials. |

Properties of Derivatives | Find out what it means to be ‘differentiable’, how to use limits to calculate the derivative and when to use the Quotient Rule for differentiation problems. |

The Derivative at a Point | Delve into velocity or slopes and rate of change, interpretations of the slope and intercept of linear models, slopes and tangents on graphs and how to find instantaneous rate of change of a function. |

The Derivative as a Function | Learn to graph the derivative from any function, how to translate verbal descriptions into equations with derivatives and how to use the Mean Value Theorem. |

Second Derivatives | Topics include the Second Derivative Formula, concavity and inflection points. |

Using Derivatives | Study monotonicity, concavity, maximum/minimum values on graphs, separation of variables for solving differential equations and related rates. |

Computing Derivatives | Find out about the Chain Rule, differentiating factored polynomials, higher order derivatives and calculations for derivatives of exponential equations. |

Properties of Definite Integrals | Learn to find the limits of Riemann Sums, how to use them for functions and graphs and about the linear properties of definite integrals. |

Integration Applications | Explore dynamic motion, integration, calculation of volumes using single integrals and volumes of revolution using integration. |

Using the Fundamental Theorem of Calculus | Practice evaluating definite integrals with the Fundamental Theorem and using the Theorem to calculate anti-derivatives. |

Integration and Integration Techniques | Study anti-derivatives, integrals of trigonometric functions, use of substitution, factorization of fractions with quadratic denominators and solving improper integrals. |

Approximating Definite Integrals | Learn about the Average Value Theorem, linear properties of definite integrals and the Trapezoid Rule. |

Using Scientific Calculators for Calculus | Practice solving equations by graphing on your scientific calculator and understanding radians, degrees, trigonometry functions and exponentials on a scientific calculator. |

## Approach

**Throughout the course, students are required to use multiple approaches to ensure their understanding of course content. In each lesson, students are required to use their graphing calculators to solve problems. In addition, students are required to graph problems by hand in order to supplement their understanding of calculus concepts. **

**Students will use their graphing calculators experimentally to determine possible solutions to problems. They will also learn to use their calculators to justify their conclusions and to support the solutions that they have developed using an analytical approach. **

**Application problems requiring the use of analytical techniques in differential and integral calculus are included in each unit. Students will solve problems requiring numerical solutions both by hand and with their calculators. Students are expected to check the reasonableness of their numerical solutions by using other approaches as well. **

**Every unit test mirrors the format of the AP exam. In each unit, students will answer multiple-choice questions and complete two free-response questions. Some of these questions will require a calculator and some will not. Students will also complete two full-length practice exams at the end of the course.**

### Projects include:

**Designing a Roller Coaster Project:** Students will design the first drop of a new roller coaster. First, they will conduct research and determine the optimal slopes for the ascent and descent portions. Then, they will write and solve equations that connect the two segments with a parabola and ensure that the track remains smooth. Next, they will sketch the lines representing the ascent and descent and the parabola that connects the two. Finally, they will use a calculator to plot the parabola using the equations they developed, and compare that plot to their sketch. Students will be required to present their findings both graphically and numerically. They must provide an annotated diagram of their design, as well as step-by-step written solutions and justifications to all of the problems to be solved.

**Soda Can Optimization Project:** Students will solve an optimization problem to find the most economical dimensions for a soda can, given a specific volume that the can must hold. Students will provide step-by-step written solutions, diagrams, and justifications to all of the problems to be solved to achieve the can’s optimization. In addition, students will conduct research to identify real-world examples and non-examples of their optimized can design and write a report about their findings. Students will present their findings to the class.

**Wind Turbine Project:** In this project, students will use calculus to determine the energy of a potential offshore wind turbine in the Gulf of Mexico. Using NOAA buoy data, students will extrapolate a table to determine the potential wind energy available, and then calculate the energy produced by a hypothetical wind turbine. Students will provide the calculations they used to generate their answers, as well as a table that models the energy produced over a set interval (power as a function of time).