AP Calculus- AB College Board
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68 topics in 7 modules
☑️ Analytical Applicatiions of Differentiation 12 topics
- Connecting to a Function, its first derivative, and its second derivative
- Determining Concavity of functions over their Domains
- Determining Intervals on Which a Function is Increasing or Decreasing
- Exploring Behaviors of Implicit Relations
- Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- Introduction to Optimization Problems
- Sketching Graphs of Functions and Their Derivatives
- Solving Optimization Problems
- Using the Candidates Test to Determine Absolute (Global) Extrema
- Using the First Derivative test to determine relative (local) Extrema
- Using the Mean Value Theorem
- Using the Second Derivative Test to Determine Extrema
☑️ Contextual Applications of Differentiation 7 topics
- Approximating values of a function using local linearity and linearization
- Interpreting the Meaning of the Derivative in Context
- Introduction to Related Rates
- Rates of Change in applied contexts other than motion
- Solving Related Rates Problems
- Straight Line Motion: Connecting Position, Velocity, and Acceleration
- Using L'hopital's rule for determining limits of indeterminite forms
☑️ Differential Equations 7 topics
- Exponential Models with Differential Equations
- FInding Particular Solutions Using Initial Conditions and Seperation of Variables
- Finding General Solutions Using Separation of Variables
- Modeling Situations with Differential Equations
- Reasoning Using Slope Fields
- Sketching Slope Fields
- Verifying Solutions for Differential Equations
☑️ Differentiation: Composite, Implicit, and Inverse Functions 6 topics
- Calculating Higher Order Derivatives
- Differentiating Inferse Trig Functions
- Differentiating Inverse Functions
- Implicit Differentiation
- Selecting Procedures for Calculating Derivatives
- The Chain Rule
☑️ Differentiation: Definition and Basic Derivative Rules 10 topics
- Applying the Power Rule
- Connecting Differentiability and Continuity: Determining when Derivative Do and Do Not Exist
- Defining Average and Instantaneous Rates of Change at a Point
- Defining the Derivative of a Function and Using Derivative Notation
- Definition and Basic Derivative Rules
- Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- Derivatives of cos x, sin x, ex LIM , and ln x
- Estimating Derivatives of a Function at a Point
- The Product Rule
- The Quotient Rule
☑️ Integration and Accumulation of Change 11 topics
- Applying properties of Definite Integrals
- Approximating Areas with Riemann Sums
- Exploring Accumulations of Change
- Finding antiderivatives and indefinite integrals: Basic Rules and Notation
- Integrating Functions using Long Division and Completing the Square
- Integrating Using Subsitition
- Interpreting the Behavior of Accumulation Functions involving Area
- Riemann Sums, Summation Notation, and Definite Integral Notation
- Selecting Techniques for Antidifferentiation
- The Fundamental Theorem of Calculus and Accumulation Functions
- The Fundamental theorem of calculus and definite integrals
☑️ Limits and Continuity 15 topics
- Confirming Continuity over an Interval
- Connectcing limits at infinity and horizontal asymptotes
- Connecting infinite limits and vertival asymptotesC
- Connecting multiple representations of limits
- Defining Continuity at a point
- Defining Limits and Using Limit Notation
- Determining Limits Using Algebraic Properties of Limits
- Determining Limits using Algebraig Manipulation
- Determining Limits using the squeeze theorem
- Estimating Limit Values from Graphs
- Estimating Limit Values from Tables
- Exploring Types of Discontinuities
- Introducing Calculus: Can Change Occur at an Instant
- Removing Discontinuities
- Working with the Intermediate Value Theorem
AP Calculus- AB Revision Content
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AP Calculus- AB - Analytical Applicatiions of Differentiation - Connecting to a Function, its first derivative, and its second derivative Content Preview
Analytical Applicatiions of Differentiation
Connecting to a Function, its first derivative, and its second derivative
The Relationship Between a Function, Its First Derivative, and Its Second Derivative
Basic Definitions
- A function describes a relationship where every input is associated with exactly one output.
- The first derivative of a function represents the function's rate of change i.e., the rate at which the value of the function is changing at each point.
- The second derivative of a function represents the rate at which the first derivative is changing, providing information about the function's concavity.
Understanding the Connection
- If a function, f(x), is increasing, the first derivative, f'(x) > 0.
- If a function f(x) is decreasing, the first derivative, f'(x) < 0.
- If the first derivative, f'(x), is increasing then the function is said to be concave up and the second derivative, f''(x) > 0.
- If the first derivative, f'(x), is decreasing then the function is said to be concave down and the second derivative, f''(x) < 0.
Critical Points
- Critical points of a function occur where the first derivative is zero or undefined.
- Local maxima and minima can only occur at critical points; they represent the highest or lowest points in a certain interval.
- The second derivative test can be used to determine whether a critical point is a local maximum, local minimum, or point of inflection.
Inflection Points and Concavity
- An inflection point is a point where the concavity of a function changes. It occurs where the second derivative is zero or undefined.
- If the function changes from concave up to concave down or vice versa, then it has an inflection point.
Real-World Applications
- In physics, a function can represent an object's distance over time; the first derivative gives the speed of the object and the second derivative gives the object's acceleration.
- In economics, the second derivative can be used to find points of diminishing returns, where an increase in production gives a lesser increase in output.
Question: Consider a function which is defined and differentiable everywhere. The function's first derivative is positive for x < 2, zero at x = 2, and negative for x > 2. The function's second derivative is negative for all x. At x = 2, is there a local maximum, local minimum, or point of inflection, and is the function concave up or concave down?
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