Calculus

Course Description

In Calculus, the student brings together all the skills learned in Algebra through Precalculus and applies them to the study of limits.  Students will find themselves in a course traditionally taken by first and second  semester college students.  During the first semester, the student will engage in a complete analysis of limits of ratios (derivatives).  During the second semester, the student will perform the same analysis on limits of sums (integrals).

Grade Levels: 11th - 12th

Related Priority Standards (State &/or National): K-12 Mathematics Missouri Learning Standards

Essential Questions: 

• How can I evaluate limits, the definition of continuity, and the Intermediate Value Theorem?
• How can I find derivatives, equations of tangent lines and test for differentiability?
• How can I find derivatives of specific types of functions?
• How can I apply derivative techniques to analyze graphs of functions and select applications?
• How can I apply derivative techniques to select applications to deepen my understanding of how derivatives can be used further in other problem-solving processes?
• How can I find antiderivatives?
• How can I use definite integration as an application to determine area?
• How can I use definite integration as an application to determine area between curves, volume, average values of functions and accumulating amounts?

Enduring Understanding/Big Ideas:

• Students will determine expressions and values using mathematical operations, procedures and rules.
• Students will translate mathematical information from a single representation or across multiple representations in order to develop processes to problem solve.
• Students will recognize mathematical reasoning requires justification of both process and solution.
• Students will use correct notation, language and mathematical convention to classify concepts and communicate results or solutions.

Course Level Scope & Sequence (Units &/or Skills):

Unit 1: Limits

• Evaluating limits using graphs or tables
• Limits at Infinity
• Formal Definition of a Limit
• Evaluating Limits Algebraically
• Justifying Limits that do not exist
• Limits of Trigonometric Functions
• Continuity
• Intermediate Value Theorem
• Tangent/Velocity

Unit 2: Derivatives

• Definition of a Derivative
• Derivative Rules (Power, Product, Quotient)
• Derivatives of Trig Functions
• Chain Rule
• Differentiability
• Higher Order Derivatives

Unit 3: Inverse Functions

• Exponential Functions & Their Derivatives
• Derivatives of Logarithmic Functions

Unit 4: Applications of Derivatives (part 1)

• Maximum & Minimum Function Values
• Mean Value Theorem
• First Derivative Test (finding Relative Extrema)
• Using the First Derivative to determine where a function increases or decreases
• Second Derivative Test (finding Relative Extrema)
• Using the Second Derivative to determine the concavity of a function and inflection points
• Graphs of Functions & Their Derivatives

Unit 5: Applications of Derivatives (part 2)

• Rectilinear Motion
• Implicit Differentiation
• Related Rates
• Linear Approximations
• L’Hopital’s Rule
• Newton’s Method

Unit 6: Antiderivatives

• Antiderivatives
• Antiderivatives by Substitution
• Differential Equations

Unit 7: The Definite Integral

• Sigma Notation
• Approximating Area (Riemann Sums)
• Exact Area using Limit of Riemann Sums
• The Definite Integral
• The Fundamental Theorems of Calculus

Unit 8: Area & Volume

• Area Between Curves
• Volumes by Slicing
• Volumes using Disk/Washer
• Average Value of a Function

Course Resources & Materials: Calc (Math) Medic, Desmos

Date Last Revised/Approved: 2011