Course Overview

This course is the equivalent of an introductory college-level calculus course. Calculus helps scientists, engineers, and financial analysts understand the complex relationships behind real-world phenomena. Students learn to evaluate the soundness of proposed solutions and apply mathematical reasoning to real-world models. Students also learn to understand change geometrically and visually (by studying graphs of curves), analytically (by studying and working with mathematical formulas), numerically (by seeing patterns in sets of numbers), and verbally. Students prepare for the AP Exam and further studies in science, engineering, and mathematics. This is the second semester of MTH500.

Course Outline

SEMESTER ONE

Unit 1: The Basics

Students prepare to study calculus by reviewing some basic precalculus concepts from algebra and trigonometry. They learn what calculus is, why it was invented, and what it's used for.

• Pre-Calculus Review
• Introduction to Calculus
• Using a Graphing Calculator
• Combining Functions
• Composite and Inverse Functions
• Graphical Symmetry
• Patterns in Graphs

Unit 2: Applications of the Integral

The topics referred to below are those listed in the College Board's Calculus AB topic outline. This unit addresses Topic I: Functions, Graphs, and Limits. Students learn two important concepts that underlie all of calculus: limits and continuity. Limits help students understand differentiation (the slope of a curve) and integration (the area inside a curved shape). Continuity is an important property of functions.

• Finding Limits Analytically
• Asymptotes and Limits
• Relative Magnitudes for Limits
• When Limits Do and Don't Exist
• Continuity
• Intermediate and Extreme Value Theorems

Unit 3: The Derivative

This unit addresses Topic II: Derivatives. Students learn how to calculate a derivative, the slope of a curve at a specific point. They learn techniques for finding derivatives of algebraic functions (such as y = x2) and trigonometric functions (such as y = sin x).

• Derivatives at a Point
• The Derivative
• The Power Rule
• Sums, Differences, Products, and Quotients
• Graphs of Functions and Derivatives
• Continuity and Differentiability
• Rolle's and Mean Value Theorems
• Higher-Order Derivatives
• Concavity
• Chain Rule
• Implicit Differentiation

Unit 4: Rates of Change

This unit focuses on Second Derivatives and Applications of Derivatives within Topic II: Derivatives. Students learn how to use calculus to model and analyze changing aspects of our world.

• Extrema
• Optimization
• Tangent and Normal Lines
• Tangent Line Approximation
• Rates and Derivatives
• Related Rates
• Rectilinear Motion

Unit 5: Semester Review and Test

Students review what they have learned so far and take the semester exam.

• Semester 1 Review
• Semester 1 Exam

SEMESTER TWO

Unit 1: The Integral

This unit focuses on Topic III: Integrals. Students learn numerical approximations to definite integrals, interpretations and properties of definite integrals, the Fundamental Theorem of Calculus, and techniques of antidifferentiation. They learn how to find areas of curved shapes.

• Riemann Sums
• Area Approximations
• The Definite Integral
• Properties of Integrals
• Graphing Calculator Integration
• Applications of Accumulated Change
• Antiderivatives
• Composite Functions
• The Fundamental Theorem of Calculus
• Definite Integrals of Composite Functions
• Analyzing Functions and Integrals

Unit 2: Applications of the Integral

This unit focuses on Topic III: Integrals. Students learn to use integrals and antiderivatives to solve problems.

• Area
• Volumes of Revolution
• Cross Sections
• More Rectilinear Motion
• Other Applications of the Definite Integral

Unit 3: Inverse and Transcendental Functions

This unit focuses on Topic II: Derivatives and Topic III: Integrals. Students learn to calculate and use derivatives, antiderivatives, and integrals of exponential functions (such as y = 3x where the input variable is an exponent), logarithmic functions (the inverses of exponential functions), and inverse trigonometric functions (such as y = secant(x)).

• Inverse Trig Functions
• Review of Logarithmic and Exponential Functions
• Transcendentals and 1/x
• Derivatives of Logs and Exponentials
• Analysis of Transcendental Curves
• Integrating Transcendental Functions
• Applications of Transcendental Integrals

Unit 4: Separable Differential Equations and Slope Fields

This unit focuses on Topic II: Derivatives; specifically, on Equations Involving Derivatives. Students investigate differential equations, and solve the equations using a technique called "separating the variables."

• Slope Fields
• Differential Equations as Models
• Exponential Growth and Decay
• More Applications of Differential Equations

Unit 5: AP Exam Review and Final Exam

Students review what they have learned and prepare for the AP Exam with practice tests that simulate the AP test experience.

• Calculus as a Cohesive Whole
• Review of Topics
• Practice Final Exams

Unit 6: Calculus Project

Teachers may choose to assign a final project.

• Project Days