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ap calc study guides

✍️ Free Response Questions (FRQ)

Calculus Free Response Questions

✋ Justification

♾️ Derivatives

♾️ Integrals

♾️ New Representation

♾️ Appropriate Units

Part A - Calculator Section

📖 Worked Example

📊 Past Trends

📱 A Note on Calculators

Part B - Non-Calculator Section

📖 Worked Example

📊 Past Trends

FAQs About FRQs

👑 Unit 1: Limits & Continuity

1.0Unit 1 Overview: Limits and Continuity

What will you learn? 🤔

1.1Introducing Calculus: Can Change Occur at An Instant?

1.2Defining Limits and Using Limit Notation

Defining Limits and Using Limit Notation

1.3Estimating Limit Values from Graphs

Estimating a Limit Value: Step by Step

1.4Estimating Limit Values from Tables

Example Problem ❓

1.5Determining Limits Using Algebraic Properties of Limits

Example Problem: Using the Sum Rule to Determine a Limit ➕

Example Problem: Using the Constant Multiple Rule to Determine a Limit ➡

Example Problem: Using the Product Rule to Determine a Limit ✖️

Example Problem: Using the Exponent/Power Rule to Determine a Limit 💥

1.6Determining Limits Using Algebraic Manipulation

Example Problem: Determining a Limit Using Factoring

Example Problem: Determining a Limit Using Alternate Forms of Trigonometric Functions

1.7Selecting Procedures for Determining Limits

Selecting Procedures for Determining Limits

1.8Determining Limits Using the Squeeze Theorem

The Squeeze Theorem 🥪

Example Problem 💡

1.10Exploring Types of Discontinuities

Discontinuities

Jump Discontinuity ↔

Point Discontinuity 🕳

Essential Discontinuity ♾️

Removable Discontinuity 🚫

1.11Defining Continuity at a Point

Conditions for Continuity

Example Problem: Using Conditions to Determine Continuity

1.12Confirming Continuity over an Interval

Confirming Continuity Over an Open Interval 🔓

Confirming Continuity Over a Closed Interval 🔐

1.13Removing Discontinuities

Removing Discontinuities Using Factoring

Removing Discontinuities Using Rationalization

1.14Connecting Infinite Limits and Vertical Asymptotes

1.15Connecting Limits at Infinity and Horizontal Asymptotes

Asymptotes, Continued

1.16Working with the Intermediate Value Theorem (IVT)

The Intermediate Value Theorem (IVT)

🤓 Unit 2: Differentiation: Definition & Fundamental Properties

2.0Unit 2 Overview: Differentiation

An Introduction to Differentiation ⤴

2.1Defining Average and Instantaneous Rates of Change at a Point

What is a Derivative? 🤔

2.2Defining the Derivative of a Function and Using Derivative Notation

Derivatives Defined

2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Differentiability Rules 📌

2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Properties of Derivatives 🚩

2.7Derivatives of cos x, sinx, e^x, and ln x

Derivatives of Special Functions 💭

2.8The Product Rule

The Product Rule ✖️

2.9The Quotient Rule

The Quotient Rule➗

2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Derivatives of Special Functions, Continued!💬

🤙🏽 Unit 3: Differentiation: Composite, Implicit & Inverse Functions

3.0Unit 3 Overview: Differentiation: Composite, Implicit, and Inverse Functions

Exam Weighting ⚖

Developing Understanding 🧠

3.1The Chain Rule

What is the Chain Rule? 🔗

When to Use the Chain Rule 📆

How to Apply the Chain Rule 🤔

Chain Rule Practice 📝

3.2Implicit Differentiation

What is Implicit Differentiation?

How to Differentiate Implicitly 🕵

Implicit Differentiation Practice 📝

3.3Differentiating Inverse Functions

What are Inverse Functions? 🙃

How to Differentiate Inverse Functions 🕵

Differentiating Inverse Functions Practice 📝

3.4Differentiating Inverse Trigonometric Functions

What is an Inverse Trigonometric Function? 🤠

How to Differentiate Inverse Trigonometric Functions 🕵

3.5Selecting Procedures for Calculating Derivatives

Calculating Derivatives Practice 📝

3.6Calculating Higher-Order Derivatives

What is a Higher-Order Derivative? 😵

How to Calculate Higher-Order Derivatives 🕵

Calculating Higher-Order Derivatives Practice 📝

👀 Unit 4: Contextual Applications of the Differentiation

4.0Unit 4 Overview: Contextual Applications of Differentiation

🔍 Prerequisite Information

4.1Interpreting the Meaning of the Derivative in Context

Review: Implicit Differentiation ✔️

4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration

Position and Velocity

4.3Rates of Change in Applied Contexts other than Motion

4.4Intro to Related Rates

What are Related Rates?

The Process 📠

Example Problem:

Equation to Relate

Plugging In And Solving

4.5Solving Related Rates Problems

Special Example: Volume of a Cone 🍦

4.6Approximating Values of a Function Using Local Linearity and Linearization

Under and Overestimates

Linearization and Point-Slope

4.7Using L'Hopitals Rule for Determining Limits in Indeterminate Forms

Indeterminate Forms

L'Hopital's Rule

When You Can Use L'Hopitals

✨ Unit 5: Analytical Applications of Differentiation

5.0Unit 5 Overview: Analytical Applications of Differentiation

Graphical Analysis

Mean Value Theorem and Extreme Value Theorem

5.1Using the Mean Value Theorem

Okay, let’s break this down: 🔍

Rolle's Theorem

5.2Extreme Value Theorem, Global vs Local Extrema, and Critical Points

The Extreme Value Theorem

Finding Absolute Extrema

5.3Determining Intervals on Which a Function is Increasing or Decreasing

Derivatives and Direction

Critical Values and Relative/Local Extrema

5.4Using the First Derivative Test to Determine Relative (Local) Extrema

What is the First Derivative Test?

Finding Relative Extrema Using the First Derivative Test

5.6Determining Concavity

The Second Derivative and Concavity

Analytically Determining Concavity

Points of Inflection

5.7Using the Second Derivative Test to Determine Extrema

Using the Second Derivative Test

Example Problems

5.11Solving Optimization Problems

Example Problem

🔥 Unit 6: Integration and Accumulation of Change

6.11Integrating Using Integration by Parts

AP Description & Expectations 🎯

What is Integration by Parts?

Another Form of the Equation ❓

How and Why Would I Use This Formula? 🤔

Steps to Take for Integration by Parts

Worked Examples 📔

6.12Using Linear Partial Fractions

AP Description & Expectations 🎯

Integrating Rational Functions Using Partial Fractions 💡

Steps to Using Partial Fraction Decomposition 🏠

Worked Examples 📔

🐶 Unit 8: Applications of Integration

8.0Unit 8 Overview: Applications of Integration

What will you learn? 🤔

8.1Finding the Average Value of a Function on an Interval

What is the Average Value of a Function? 🤔

How do you find the Average Value? 🔍

8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals

How do we use Integrals to Connect Position, Velocity, and Acceleration? 🧩

Integration is Going Backwards

The College Board stresses the importance of this topic in integral calculus: ❗️

8.3Using Accumulation Functions and Definite Integrals in Applied Contexts

What should you know about this topic? 🧠

What will this look like on the AP Exam? ✍

8.4Finding the Area Between Curves Expressed as Functions of x

The Area Between Two Curves

🔍 Example Problem: Finding the Area Between Two Curves

🔍 Example Problem: Finding the Area Between Two Curves #2

8.5Finding the Area Between Curves Expressed as Functions of y

Using the y-axis to Find Area Between Two Curves

🔍 Example Problem: Finding the Area Between Two Curves Using Horizontal Slices

8.6Finding the Area Between Curves That Intersect at More Than Two Points

When Curves Intersect More Than Once

🔍 Example Problem: Finding the Area Between Two Curves That Intersect at More Than Two Points

8.7Volumes with Cross Sections: Squares and Rectangles

Finding Volume Using Square Cross Sections

🔍 Example Problem: Finding the Volumes of a Shape with Square Cross Sections

Using Rectangular Cross Sections

🔍 Example Problem: Finding the Volumes of a Shape with Rectangular Cross Sections

8.8Volumes with Cross Sections: Triangles and Semicircles

Using Triangles as Cross Sections

🔍 Example Problem: Finding the Volumes of a Shape with Triangle Cross Sections

8.9Volume with Disc Method: Revolving Around the x- or y-Axis

🔍 Example Problem: Finding the Volume Using the Disc Method

🔍 Example Problem: Finding the Volume Using the Disc Method #2

8.10Volume with Disc Method: Revolving Around Other Axes

🔍 Example Problem: Finding the Volume Using the Disc Method Around Other Axes

8.11Volume with Washer Method: Revolving Around the x- or y-Axis

What is the Disk Method?

🔍 Example Problem: Finding the Volume Using the Washer Method

🔍 Example Problem: Finding the Volume Using the Washer Method #2

8.12Volume with Washer Method: Revolving Around Other Axes

🔍 Example Problem: Finding the Volume Using the Washer Method Around Other Axes

🦖 Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)

9.0Unit 9 Overview: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

What To Expect 🤓

9.1Defining and Differentiating Parametric Equations

What is a Parametric Function?

Derivatives of Parametric Functions

9.4Defining and Differentiating Vector-Valued Functions

Parametrics and Motion

9.7Defining Polar Coordinates and Differentiating in Polar Form

Derivatives of Polar Functions

9.9Finding the Area of the Region Bounded by Two Polar Curves

Polar Arc Length

♾ Unit 10: Infinite Sequences and Series (BC Only)

10.0Unit 10 Overview: Infinite Series and Sequences

Exam Weighting ⚖

Developing Understanding 🧠

Mathematical Practices ➗

Main Ideas for this Unit 💬

10.1Defining Convergent and Divergent Infinite Series

What is a Sequence?

The Limit of A Sequence

Some Notation and Terminology on Sequences

What is a Series?

Convergence and Divergence of Series

10.2Working with Geometric Series

What is a Geometric Series?

The Geometric Series Test

10.3The nth Term Test for Divergence

What is the nth Term Test?

10.4Integral Test for Convergence

What is the Integral Test?

10.5Harmonic Series and p-Series

10.6Comparison Tests for Convergence

Intro to the Comparison Tests

What is the Direct Comparison Test?

What is the Limit Comparison Test?

10.7Alternating Series Test for Convergence

What are Alternating Sequences?

What is the Alternating Series Test?

10.8Ratio Test for Convergence

What is the Ratio Test?

Review Flowchart

10.9Determining Absolute or Conditional Convergence

10.10Alternating Series Error Bound

How do we find the error bound?

10.1110.11 Finding Taylor Polynomial Approximations of Functions

What are Taylor Series, and how do you find them?

10.12Lagrange Error Bound

What is the Lagrange Error Bound and how do we find it?

10.13Radius and Interval of Convergence of Power Series

10.14Finding Taylor or Maclaurin Series for a Function

Some Important Maclaurin Series to Remember

10.15Representing Functions as Power Series

What is a power series?

How can we manipulate Power Series?

Conclusion and What’s Next?

🧐 Multiple Choice Questions (MCQ)

AP Calculus AB/BC Multiple Choice Help (MCQ)

Multiple Choice Help (MCQ)

Overview ♾️

Tips and Tricks ✏️

Where to Practice 📍

Closing Thoughts 💭

Calculus Multiple Choice Questions

Table of Contents 📊

🧠 Exam Format and Logistical Details

❓ Types of Problems

⭕️ How to Approach MC Questions

♾️ ap calc

💎 jump to Unit 7 library

7.4 Reasoning Using Slope Fields

#slopefields

⏱️ 1 min read

written by

Jacob Jeffries

jacob jeffries

June 8, 2020

available on hyper typer

Continued from 7.3 Sketching Slope Fields.

The actual solution (which can actually be manipulated to be separable*) to the differential equation in Eq. 39 is the following:

By going to https://www.desmos.com/calculator/fjli4efhcj and clicking the play button on equation 18, you can see that this curve does indeed fit the given slope field for any constant C. Clicking the play button will show different curves for different values of C.

The most intuitive way to think of a slope field is to picture a fluid flowing and then placing an object on the fluid that will trace out a path. This path is approximate to the solution to the curve that represents the differential equation. 😀

Review

Fill in the table below for different values of y’ at different coordinate points. Use a calculator to find the values to two decimal places. Create a slope field and then solve the differential equation and confirm that your slope field matches the solution to the differential equation. ✍

Answer

Varying values of C plotted over the slope field are shown here:

https://www.desmos.com/calculator/ab0swk21ao

Footnotes

*One can make this separable by doing a substitution:

From here, one can solve the latter differential equation (which will give a solution that is only a function of x) and substitute this into u = x + y, which will give the aforementioned solution to the original differential equation.

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