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5.10 Introduction to Optimization Problems

sumi vora

⏱️ June 8, 2020

📅

🎥Watch: AP Calculus AB/BC - Optimization

Now that we understand how derivatives can determine the maximum and minimum values on a graph, we can also apply this knowledge to word problems. Optimization problems can look a little bit scary at first, but the most important thing to remember is that you need to take the derivative of the function you are trying to optimize! (the function you’re trying to find a maximum or minimum value for)

Optimization is different from the problems that we’ve seen before because it usually adds one more variable that we have to account for. This, however, doesn’t mean that we will have a function with three variables; instead, you need to find a relationship between two of the variables so that you can get rid of one.

Examples in the next section!

ap calc study guides

✍️ Free Response Questions (FRQ)

Calculus Free Response Questions

♾️ 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?

Limits

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

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 📌

2.5Applying the Power Rule

The Power Rule ✊

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 📝

Answers 📝

3.2Implicit Differentiation

What is Implicit Differentiation?

How to Differentiate Implicitly 🕵

Implicit Differentiation Practice 📝

Answers 📝

3.3Differentiating Inverse Functions

What are Inverse Functions? 🙃

How to Differentiate Inverse Functions 🕵

Differentiating Inverse Functions Practice 📝

Answers 📝

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 📝

Answers 📝

3.6Calculating Higher-Order Derivatives

What is a Higher-Order Derivative? 😵

How to Calculate Higher-Order Derivatives 🕵

Calculating Higher-Order Derivatives Practice 📝

Answers 📝

👀 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

Acceleration

4.3Rates of Change in Applied Contexts other than Motion

🔍 Examples

📓 Practice

4.4Intro to Related Rates

What are Related Rates?

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

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

Optimization

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

Practice FRQ

🔥 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

Problems

10.2Working with Geometric Series

What is a Geometric Series?

The Geometric Series Test

Problems

10.3The nth Term Test for Divergence

What is the nth Term Test?

Problems

10.4Integral Test for Convergence

What is the Integral Test?

Problems

10.5Harmonic Series and p-Series

P-Series Test

10.6Comparison Tests for Convergence

Intro to the Comparison Tests

What is the Direct Comparison Test?

What is the Limit Comparison Test?

Problems

10.7Alternating Series Test for Convergence

What are Alternating Sequences?

What is the Alternating Series Test?

Problems

10.8Ratio Test for Convergence

What is the Ratio Test?

Review Flowchart

Problems

10.9Determining Absolute or Conditional Convergence

Problems

10.10Alternating Series Error Bound

How do we find the error bound?

Problems

10.11Finding 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?

Problems

10.13Radius and Interval of Convergence of Power Series

Problems

10.14Finding Taylor or Maclaurin Series for a Function

Some Important Maclaurin Series to Remember

Problems

10.15Representing Functions as Power Series

What is a power series?

How can we manipulate Power Series?

Conclusion and What’s Next?

Sources

🧐 Multiple Choice Questions (MCQ)

AP Calculus AB/BC Multiple Choice Help (MCQ)

Multiple Choice Help (MCQ)

Calculus Multiple Choice Questions

Table of Contents 📊

🧠 Exam Format and Logistical Details

❓ Types of Problems

⭕️ How to Approach MC Questions

continue learning

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*ap® and advanced placement® are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.

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5.10 Introduction to Optimization Problems

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