AP Calculus- BC College Board
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109 topics in 10 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
☑️ Applications of Integration 13 topics
- Connecting Position, Velocity, and Acceleration of Functions using Integrals
- Finding the Areas between curves as expressed as Functions of x
- Finding the Average Value of a Function on an Interval
- Finding the area between two curves that intersect at more than two points
- Finding the areas between curves as expressed as functions of y
- The arc length of a smooth, planar curve and distance travelled
- Using Accumulation Functions and Definite Integrals in Applied Contexts
- Volume with Disc Method: Revolving around other axes
- Volume with Disc Method: Revolving around the x- or y-axis
- Volume with Washer Method: Revolving around other axes
- Volume with Washer Method: Revolving around the x- or y-axis
- Volumes with cross sections: squares and rectangles
- Volumes with cross sections: triangles and semicircles
☑️ 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 9 topics
- Approximating Solutions Using Euler's Method
- Exponential Models with Differential Equations
- FInding Particular Solutions Using Initial Conditions and Seperation of Variables
- Finding General Solutions Using Separation of Variables
- Logistic Models with Differential Equations
- 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
☑️ Infinite Sequences and Series 15 topics
- Alternating Series Error Bound
- Alternating Series Test for Convergence
- Comparison Test for Convergence
- Defining Convergent and Divergent Infinite Series
- Determining Absolute or Conditional Convergence
- Finding Taylor Polynomial Approximations of Functions
- Finding Taylor or Maclaurin Series for a Function
- Harmonic Series and p-Series
- Integral test for convergence
- Lagrange Error Bound
- Radius and Interval of Convergence of Power Series
- Ratio Test for Convergence
- Representing Functions as Power Series
- The nth Term Test for Divergence
- Working with Geometric Series
☑️ Integration and Accumulation of Change 14 topics
- Applying properties of Definite Integrals
- Approximating Areas with Riemann Sums
- Evaluating Improper Integrals
- 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
- Integrating using integration by parts
- 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
- Using Linear Partial Fractions
☑️ 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
☑️ Parametric Equations, Polar Coordinates, and Vector Valued Functions 8 topics
- Defining and Differentiating Parametric Equations
- Defining and Differentiating Vector Valued Functions
- Defining polar coordinates and differentiating in polar form
- Finding Arc Lengths of Curves given by Parametric Equations
- Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- Finding the Area of the Region Bounded by Two Polar Curves
- Second Derivative of Parametric Equations
- Solving Motion Problems using parametric and vector valued functions
AP Calculus- BC Revision Content
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AP Calculus- BC - 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
Understanding Differentiation
- Differentiation provides a method to compute rates of change and determine the slope of a function at any point.
- The first derivative of a function, denoted as f'(x) or df/dx, represents the instantaneous rate of change at a point, or the slope of the tangent line to the curve at that point.
- A function will be increasing where its first derivative is positive, and decreasing where its first derivative is negative.
- Local maxima and minima of a function can be found where its first derivative equals zero.
Connecting a Function and Its First Derivative
- If f'(x) > 0, then the function f(x) is rising at x.
- If f'(x) = 0, then the function f(x) has a horizontal tangent at x.
- If f'(x) < 0, then the function f(x) is falling at x.
- Points of inflection are points at which the function changes concavity (i.e., switches between curving up and curving down). You can locate these by finding where the first derivative changes sign.
Understanding Second Derivative
- The second derivative, denoted as f''(x), provides information about the concavity or curvature of the graph.
- Specifically, if the second derivative is positive at a point, the function is concave up at that point. Conversely, if the second derivative is negative, the function is concave down.
- More intuitively, when a function is concave up, it curves upwards like the shape of a bowl. When a function is concave down, it curves downwards.
Connecting a Function, its First and Second Derivatives
- A function itself gives us information about the shape, position, and values of particular points.
- The first derivative gives us information about the rate of change of the function, identifying where the function has local extrema and inflection points.
- The second derivative confers information about the concavity or curvature of the function. This helps further classify local extrema and also identify inflection points.
Question: Given a function f(x) whose first derivative f'(x) is positive and second derivative f''(x) is negative for certain value of x, what can be inferred about the behaviour of the function f(x) at that point?
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