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
- 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
- Intermediate Value Theorem
Unit 2: Derivatives
- Definition of a Derivative
- Derivative Rules (Power, Product, Quotient)
- Derivatives of Trig Functions
- Chain Rule
- 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 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