AP Calculus BC · Unit 5
Analytical Applications of Differentiation
In short
Unit 5: Analytical Applications of Differentiation in AP Calculus BC covers 12 topics: Using the Mean Value Theorem, Extreme Value Theorem, Global Versus Local Extrema, and Critical Points, Determining Intervals on Which a Function Is Increasing or Decreasing, Using the First Derivative Test to Determine Relative (Local) Extrema and 8 more. Open each one for what you need to know, a self-check list and how it is tested on the exam.
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What is in Unit 5: Analytical Applications of Differentiation?
Unit 5: Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
- I can state and verify the conditions of the Mean Value Theorem (continuity on [a, b] and differentiability on (a, b)) to guarantee a point where the instantaneous rate of change equals the average rate of change.
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- I can apply the Extreme Value Theorem to guarantee a minimum and maximum value for a continuous function on a closed interval.
- I can find critical points of a function by determining where its first derivative is zero or undefined.
- I can explain that all local extrema occur at critical points, but not all critical points result in local extrema.
5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- I can use the first derivative of a function to determine intervals where the function is increasing or decreasing.
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- I can apply the first derivative test to determine and justify relative (local) extrema.
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- I can use the candidates test to find absolute (global) extrema on a closed interval by evaluating the function at critical points and endpoints.
5.6 Determining Concavity of Functions over Their Domains
- I can determine if the graph of a function is concave up or down on an open interval by checking if its derivative is increasing or decreasing.
- I can use the second derivative of a function to find intervals of upward or downward concavity.
5.7 Using the Second Derivative Test to Determine Extrema
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Analytical Applications of Differentiation: common questions
What is in Unit 5: Analytical Applications of Differentiation?
It covers 12 topics: 5.1 Using the Mean Value Theorem; 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points; 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing; 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema; 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema; 5.6 Determining Concavity of Functions over Their Domains; 5.7 Using the Second Derivative Test to Determine Extrema; 5.8 Sketching Graphs of Functions and Their Derivatives; 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative; 5.10 Introduction to Optimization Problems; 5.11 Solving Optimization Problems; 5.12 Exploring Behaviors of Implicit Relations.
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