AP Calculus BC - MCQs and Free response -Exam Style Practice question and Answer
AP Calculus BC Exam Style Questions – MCQs and FRQs – New syllabys
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AP Calculus BC - Multiple Choice Questions (MCQs) Questions
Unit 1: Limits and Continuity – MCQs
- 1.1 Introducing Calculus: Can Change Occur at an Instant?
- 1.2 Defining Limits and Using Limit Notation
- 1.3 Estimating Limit Values from Graphs
- 1.4 Estimating Limit Values from Tables
- 1.5 Determining Limits Using Algebraic Properties of Limits
- 1.6 Determining Limits Using Algebraic Manipulation
- 1.7 Selecting Procedures for Determining Limits
- 1.8 Determining Limits Using the Squeeze Theorem
- 1.9 Connecting Multiple Representations of Limits
- 1.10 Exploring Types of 3 Discontinuities
- 1.11 Defining Continuity at a Point
- 1.12 Confirming Continuity over an Interval
- 1.13 Removing Discontinuities
- 1.14 Connecting Infinite Limits and Vertical Asymptotes
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
Unit 2: Differentiation: Definition and Basic Derivative Rules – MCQs
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative of a Function and Using Derivative Notation
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5 Applying the Power Rule
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, ex, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions- MCQs
Unit 4: Contextual Applications of Differentiation- MCQs
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3 Rates of Change in Applied Contexts Other Than Motion
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
Unit 5: Analytical Applications of Differentiation- MCQs
- 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
Unit 6: Integration and Accumulation of Change- MCQs
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integrating Using Integration by Parts bc only
- 6.12 Using Linear Partial Fractions bc only
- 6.13 Evaluating Improper Integrals bc only
- 6.14 Selecting Techniques for Antidifferentiation
Unit 7: Differential Equations- MCQs
- 7.1 Modeling Situations with Differential Equations
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler’s Method bc only
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8 Exponential Models with Differential Equations
- 7.9 Logistic Models with Differential Equations bc only
Unit 8: Applications of Integration- MCQs
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.5 Finding the Area Between Curves Expressed as Functions of y
- 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.8 Volumes with Cross Sections: Triangles and Semicircles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10 Volume with Disc Method: Revolving Around Other Axes
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled bc only
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions bc only- MCQs
- 9.1 Defining and Differentiating Parametric Equations
- 9.2 Second Derivatives of Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector- Valued Functions
- 9.5 Integrating Vector- Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector- Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
Unit 10 : Infinite Sequences and Series bc only- MCQs
- 10.1 Defining Convergent and Divergent Infinite Series
- 10.2 Working with Geometric Series
- 10.3 The nth Term Test for Divergence
- 10.4 Integral Test for Convergence
- 10.5 Harmonic Series and p-Series
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
AP Calculus BC -Free-Response Questions(FRQs) Questions
Unit 1: Limits and Continuity – FRQs
- 1.1 Introducing Calculus: Can Change Occur at an Instant?
- 1.2 Defining Limits and Using Limit Notation
- 1.3 Estimating Limit Values from Graphs
- 1.4 Estimating Limit Values from Tables
- 1.5 Determining Limits Using Algebraic Properties of Limits
- 1.6 Determining Limits Using Algebraic Manipulation
- 1.7 Selecting Procedures for Determining Limits
- 1.8 Determining Limits Using the Squeeze Theorem
- 1.9 Connecting Multiple Representations of Limits
- 1.10 Exploring Types of 3 Discontinuities
- 1.11 Defining Continuity at a Point
- 1.12 Confirming Continuity over an Interval
- 1.13 Removing Discontinuities
- 1.14 Connecting Infinite Limits and Vertical Asymptotes
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
Unit 2: Differentiation: Definition and Basic Derivative Rules – FRQs
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative of a Function and Using Derivative Notation
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5 Applying the Power Rule
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, ex, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions – FRQs
Unit 4: Contextual Applications of Differentiation – FRQs
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3 Rates of Change in Applied Contexts Other Than Motion
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
Unit 5: Analytical Applications of Differentiation – FRQs
- 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
Unit 6: Integration and Accumulation of Change – FRQs
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integrating Using Integration by Parts bc only
- 6.12 Using Linear Partial Fractions bc only
- 6.13 Evaluating Improper Integrals bc only
- 6.14 Selecting Techniques for Antidifferentiation
Unit 7: Differential Equations – FRQs
- 7.1 Modeling Situations with Differential Equations
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler’s Method bc only
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8 Exponential Models with Differential Equations
- 7.9 Logistic Models with Differential Equations bc only
Unit 8: Applications of Integration – FRQs
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.5 Finding the Area Between Curves Expressed as Functions of y
- 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.8 Volumes with Cross Sections: Triangles and Semicircles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10 Volume with Disc Method: Revolving Around Other Axes
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled bc only
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions bc only – FRQs
- 9.1 Defining and Differentiating Parametric Equations
- 9.2 Second Derivatives of Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector- Valued Functions
- 9.5 Integrating Vector- Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector- Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
Unit 10 : Infinite Sequences and Series bc only – FRQs
- 10.1 Defining Convergent and Divergent Infinite Series
- 10.2 Working with Geometric Series
- 10.3 The nth Term Test for Divergence
- 10.4 Integral Test for Convergence
- 10.5 Harmonic Series and p-Series
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Exam Overview
The AP Calculus AB and BC Exams assess student understanding of the mathematical practices and learning objectives outlined in the course framework. The exams are both 3 hours and 15 minute long and include 45 multiple-choice questions and 6 free-response questions. The details of the exams, including exam weighting, timing, and calculator requirements, can be found below:
The AP Exams also assess each of the units of the course—eight units for AP Calculus AB and 10 for AP Calculus BC—with the following exam weighting on the multiple-choice section: