Derivatives

Derivatives

Using derivatives to describe the rate of change of one variable with respect to another variable allows students to understand change in a variety of contexts. In AP Calculus, students build the derivative using the concept of limits and use the derivative primarily to compute the instantaneous rate of change of a function. Applications of the derivative include finding the slope of a tangent line to a graph at a point, analyzing the graph of a function (for example, determining whether a function is increasing or decreasing and finding concavity and extreme values), and solving problems involving rectilinear motion. Students should be able to use different definitions of the derivative, estimate derivatives from tables and graphs, and apply various derivative rules and properties. In addition, students should be able to solve separable differential equations, understand and be able to apply the Mean Value Theorem, and be familiar with a variety of real-world applications, including related rates, optimization, and growth and decay models.

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Lessons

Unit 1: Introduction to Derivatives
- 2.10 Ad-Out - Instantaneous Velocity
- 2.11 That’s So Derivative - Introduction to Differential Calculus
- 2.12 The Derivative as a Function - Multiple Representations
- 2.13 Get Real - Interpreting, Estimating and Using the Derivative
- 2.14 Let Me Take You Higher - The Second Derivative
- 2.15 MakingConnections - Limits, Continuity and Differentiability
- 2.16 On a Tangent - The Tangent Line Approximation

Unit 2: The Basics of Differentiation
- 2.21 Power Up! - Computing Basic Differentiation
- 2.22 Periodic Behavior - Differentiating Basic Trig Functions
- 2.23 Combining Forces - The Product and Quotient Rules of Differentiation
- 2.24 Tan-z and the Reciprocals - Derivatives of Other Trigonometric Functions
- 2.25 The Structure of Compositions - The Chain Rule
- 2.26 Anything You Can Do… I Can Un-Do - Derivatives of Inverse Functions/li>
- 2.27 Buried Treasures - Derivatives of Functions Given Implicitly
- 2.2H L’Hôpital to the Rescue - Using Derivatives to Evaluate Limits

Unit 3: Applying derivatives to analyze functions
- 2.31 Mind Your Behavior - Using Derivatives to Identify Extreme Values
- 2.32 To Be or Not To Be - Mean Value Theorem, and the other Existence Theorems
- 2.33 All in the Family 􁈺BC􁈻 - Using Parameters to Describe Families of Functions
- 2.34 Max Effort with the EVT - Global Optimization
- 2.35 Get Real - Applied Optimization Problems
- 2.36 Related Rates - complete lesson on Related Rates
- 2.37 Parametric - Defining and differentiating parametric equations
- 2.38 Second Derivative - Parametric - Second derivatives of parametric equations
- 2.39 Vectors - Defining and differentiating vector-valued functions
- 2.39 Polar Curves - Defining polar coordinates and differentiating in polar form

Assessment Tasks

FRQ - Related Rates - from AP Central
Project - Cooperative Task - Related Rates
Summative Assessment - Group or Individual Task

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