Chapter 3: Applications of Derivatives

AP Exam Weight: 25-35% | Multiple Choice: 10-14 questions | Free Response: Major focus in several questions

📚 Table of Contents

1. Mean Value Theorem
2. Extrema & Optimization
3. Related Rates
4. Curve Analysis
5. L'Hôpital's Rule

1. Mean Value Theorem 📊

Understanding MVT

The Mean Value Theorem connects average and instantaneous rates of change. Think of it as:

• A car's average speed vs. instantaneous speed
• The exact moment you're going the average speed
• A mathematical guarantee about derivatives
• A tool for proving other theorems

Statement

If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) where: $f'(c) = \frac{f(b)-f(a)}{b-a}$

Intuitive Understanding

• Left side: instantaneous rate at some point
• Right side: average rate over interval
• Must match at least once
• Like hitting your average speed during a trip

Verification Process

1. Check continuity on [a,b]
2. Check differentiability on (a,b)
3. Calculate average rate
4. Find where derivative equals average rate

Example Walkthrough

Show MVT applies to f(x) = x² on [1,4]

1. Check conditions:
   • Continuous: polynomials are continuous
   • Differentiable: f'(x) = 2x exists everywhere
2. Calculate average rate:
   • $\frac{f(4)-f(1)}{4-1} = \frac{16-1}{3} = 5$
3. Find c:
   • f'(c) = 2c = 5
   • c = 2.5
4. Verify:
   • c is in (1,4)
   • f'(2.5) = 5 matches average rate

Rolle's Theorem

A special case of MVT where f(a) = f(b)

Key Insights

• Guarantees a horizontal tangent
• Like finding a peak or valley
• Used in proof of MVT
• Important for existence proofs

Applications

1. Proving Zero Existence

   • Finding roots of derivatives
   • Showing turning points exist
   • Locating horizontal tangents
   • Maximum/minimum locations

2. Real-World Examples

   • Return to starting height
   • Periodic motion
   • Temperature cycles
   • Financial breakeven points

Common Applications

1. Average vs. Instantaneous Rate

• Speed at a moment vs. average speed
• Growth rates in biology
• Reaction rates in chemistry
• Economic rates of change

2. Existence Proofs

• Showing solutions exist
• Finding intervals containing solutions
• Proving inequalities
• Demonstrating properties

3. Estimation

• Error bounds
• Approximation guarantees
• Interval calculations
• Numerical analysis

Common Mistakes

1. Condition Checking

   • Forgetting to verify continuity
   • Missing discontinuities
   • Not checking differentiability
   • Ignoring endpoint behavior

2. Interpretation

   • Confusing average and instantaneous rates
   • Misidentifying c value
   • Wrong interval notation
   • Incomplete conclusions

3. Application

   • Wrong formula use
   • Incorrect interval selection
   • Missing critical assumptions
   • Incomplete verification

Practice Strategies

1. Understanding the Theorem

   • Draw diagrams
   • Use real-world analogies
   • Connect to other concepts
   • Verify conditions carefully

2. Problem Solving

   • List given information
   • Check conditions systematically
   • Show clear work
   • Verify conclusions

3. Common Scenarios

   • Motion problems
   • Rate comparisons
   • Existence proofs
   • Inequality verifications

2. Extrema & Optimization 🎯

Understanding Extrema

Finding maximum and minimum values is crucial for optimization. Think of it as:

• Finding the highest/lowest points
• Locating best/worst scenarios
• Optimizing real-world situations
• Analyzing function behavior

First Derivative Test

Understanding the Test

• f'(x) > 0: Function increasing
• f'(x) < 0: Function decreasing
• f'(x) = 0 or DNE: Critical point

Process

1. Find critical points
2. Create number line
3. Test intervals
4. Identify extrema

Example Walkthrough

Find extrema of f(x) = x³ - 3x

1. Find f'(x):
   • f'(x) = 3x² - 3
   • = 3(x² - 1)
   • = 3(x+1)(x-1)
2. Critical points:
   • x = -1 and x = 1
3. Test intervals:
   • x < -1: negative
   • -1 < x < 1: positive
   • x > 1: negative
4. Conclude:
   • Local max at x = 1
   • Local min at x = -1

Second Derivative Test

Understanding the Test

At critical point x = c:

• f”(c) > 0: Local minimum
• f”(c) < 0: Local maximum
• f”(c) = 0: Inconclusive

When to Use

• Faster than first derivative test
• Only works at critical points
• Requires second derivative
• Must be differentiable twice

Optimization Process

Step-by-Step Approach

1. Define Variables

   • Identify what varies
   • Choose appropriate variables
   • Label diagram
   • Note relationships

2. Write Objective Function

   • Express quantity to optimize
   • Use given constraints
   • Simplify if possible
   • Check units

3. Find Domain Restrictions

   • Physical constraints
   • Geometric limitations
   • Practical bounds
   • Mathematical restrictions

4. Locate Critical Points

   • Take derivative
   • Set equal to zero
   • Solve equation
   • Check endpoints

5. Test Points

   • Use second derivative test
   • Compare values
   • Check endpoints
   • Verify answer makes sense

Common Applications

1. Geometric

   • Maximize area/volume
   • Minimize perimeter
   • Optimize dimensions
   • Find best shapes

2. Business

   • Maximize profit
   • Minimize cost
   • Optimize production
   • Best pricing strategies

3. Physics

   • Minimize time
   • Maximize distance
   • Optimize angles
   • Energy efficiency

3. Related Rates 🔄

Understanding Related Rates

Problems where multiple quantities change with respect to time. Think of it as:

• Connected rates of change
• Chain rule with time
• Implicit differentiation
• Real-world applications

Problem-Solving Process

1. Draw and Label Diagram

• Show all variables
• Mark known values
• Indicate what's changing
• Show relationships

2. Write Equation

• Connect variables
• Use geometric formulas
• Consider relationships
• Keep time dependence

3. Differentiate

• Use chain rule
• Remember implicit differentiation
• Keep track of units
• Watch for constant terms

4. Solve

• Substitute known values
• Solve for unknown rate
• Check units
• Verify reasonableness

Common Scenarios

1. Geometric Shapes

• Growing circles
• Changing triangles
• Expanding volumes
• Moving points

2. Real-World Applications

• Water filling tanks
• Shadow problems
• Distance/velocity
• Cost relationships

4. Curve Analysis 📈

Understanding Curve Analysis

A systematic way to understand function behavior. Think of it as:

• Creating a function's complete profile
• Mapping out all key features
• Understanding overall behavior
• Predicting function patterns

First Derivative Information

Understanding f'(x)

• Sign tells increasing/decreasing
• Zeros give critical points
• DNE points are critical
• Connects to motion/rates

Key Features

1. Increasing/Decreasing

   • f'(x) > 0: Increasing
   • f'(x) < 0: Decreasing
   • Create sign chart
   • Mark transitions

2. Critical Points

   • Where f'(x) = 0
   • Where f'(x) DNE
   • Potential extrema
   • Important features

3. Local Extrema

   • Use first derivative test
   • Compare neighboring intervals
   • Consider domain edges
   • Verify with second derivative

4. Tangent Lines

   • Slope = f'(x)
   • Point-slope form
   • Normal lines
   • Approximations

Second Derivative Information

Understanding f”(x)

• Sign tells concavity
• Zeros give inflection points
• Confirms extrema type
• Rate of change of slope

Key Features

1. Concavity

   • f”(x) > 0: Concave up
   • f”(x) < 0: Concave down
   • Like shape of smile/frown
   • Changes at inflection points

2. Inflection Points

   • Where f”(x) = 0 or DNE
   • Concavity changes
   • Important for shape
   • Often symmetric

3. Maximum/Minimum Verification

   • Second derivative test
   • Confirms extrema type
   • More efficient than first derivative
   • Not always conclusive

4. Linear Approximation

   • Tangent line approximation
   • Error bounds
   • Local behavior
   • Differentials

Curve Sketching Process

Step-by-Step Approach

1. Find Domain

   • Check denominators
   • Check radicals
   • Consider context
   • Note restrictions

2. Find Intercepts

   • x-intercepts: f(x) = 0
   • y-intercept: f(0)
   • Important points
   • Cross-check with domain

3. Find Asymptotes

   • Horizontal: $\lim_{x \to \pm\infty} f(x)$
   • Vertical: Where denominator = 0
   • Slant: Divide by highest power
   • Guide overall shape

4. Analyze First Derivative

   • Find f'(x)
   • Solve f'(x) = 0
   • Create sign chart
   • Mark critical points

5. Analyze Second Derivative

   • Find f”(x)
   • Solve f”(x) = 0
   • Determine concavity
   • Mark inflection points

6. Sketch the Curve

   • Plot key points
   • Follow sign charts
   • Respect asymptotes
   • Connect smoothly

Common Mistakes

1. Process Errors

   • Skipping steps
   • Missing key points
   • Wrong asymptotes
   • Inconsistent signs

2. Graphing Errors

   • Wrong concavity
   • Missing features
   • Incorrect crossings
   • Scale issues

5. L'Hôpital's Rule 🔄

Understanding L'Hôpital's Rule

A powerful tool for evaluating indeterminate limits. Think of it as:

• Converting "bad" limits to "good" ones
• Using derivatives to simplify
• Finding true limiting behavior
• Resolving uncertainty

Indeterminate Forms

Types and Recognition

1. $\frac{0}{0}$ Form

   • Both numerator and denominator → 0
   • Like $\lim_{x \to 0} \frac{\sin x}{x}$
   • Most common type
   • Often from cancellation

2. $\frac{\infty}{\infty}$ Form

   • Both parts grow without bound
   • Like $\lim_{x \to \infty} \frac{x^2}{x}$
   • Check growth rates
   • Consider simplification first

3. Other Forms

   • $0 \cdot \infty$: Rewrite as fraction
   • $\infty - \infty$: Common denominator
   • $0^0$, $1^\infty$, $\infty^0$: Use ln

Application Process

Step-by-Step Approach

1. Verify Indeterminate Form

   • Check limit of each part
   • Confirm indeterminate
   • Classify form
   • Consider alternatives

2. Apply Rule

   • Take derivatives of top and bottom
   • Keep limit statement
   • Simplify if possible
   • Check if still indeterminate

3. Evaluate Result

   • Compute final limit
   • Verify answer makes sense
   • Check domain issues
   • Consider context

Example Walkthrough

Find $\lim_{x \to 0} \frac{e^x - 1}{x}$

1. Check form:
   • Top → 0
   • Bottom → 0
   • $\frac{0}{0}$ form
2. Apply L'Hôpital's:
   • $\lim_{x \to 0} \frac{e^x}{1} = 1$
3. Verify:
   • Result is finite
   • Makes sense geometrically
   • Matches calculator

Common Mistakes

1. Application Errors

   • Using when not indeterminate
   • Wrong derivatives
   • Missing steps
   • Domain issues

2. Process Errors

   • Not checking form
   • Applying too soon
   • Missing simpler methods
   • Algebraic mistakes

Practice Strategies

1. Form Recognition

   • Make table of forms
   • Practice identification
   • Know alternatives
   • Check thoroughly

2. Technique Selection

   • Try algebra first
   • Consider substitution
   • Look for patterns
   • Use L'Hôpital's last

3. Verification

   • Use calculator
   • Check reasonableness
   • Consider graph
   • Confirm work

📝 AP-Style Examples

Example 1: Optimization

A box with square base and open top must have volume 1000 cubic inches. Find dimensions that minimize surface area.

Solution:

1. Let x = side length of base
2. Let h = height
3. Volume: $x^2h = 1000$
4. Surface area: $S = x^2 + 4xh$
5. Solve: $h = \frac{1000}{x^2}$
6. $S = x^2 + \frac{4000}{x}$
7. $S' = 2x - \frac{4000}{x^2}$
8. Solve $S' = 0$: $x = 10$
9. Therefore: x = 10, h = 10

Example 2: Related Rates

A conical water tank with radius 6 ft and height 10 ft is being filled at 3 cubic feet per minute. How fast is the height increasing when the water is 4 ft deep?

Solution:

1. Volume formula: $V = \frac{1}{3}\pi r^2h$
2. Similar triangles: $\frac{r}{h} = \frac{6}{10}$
3. Differentiate: $\frac{dV}{dt} = \pi r^2\frac{dh}{dt}$
4. Solve for $\frac{dh}{dt}$

💡 Success Strategies

1. Optimization Problems

• Draw diagram
• Define variables
• Write constraint equation
• Consider domain
• Check endpoints

2. Related Rates

• Draw picture
• Label variables
• Write equation
• Differentiate
• Substitute values

3. Common Mistakes

• Wrong units
• Missing critical points
• Incomplete testing
• Domain errors

🔍 AP Exam Focus

Free Response Tips

1. Show work clearly

   • Label steps
   • Include units
   • Justify conclusions

2. Common Questions

   • Optimization
   • Related rates
   • Curve sketching
   • Rate comparisons

Multiple Choice Strategy

1. Consider

   • Multiple approaches
   • Graphical insights
   • Sign analysis

2. Check

   • Units
   • Reasonableness
   • Domain restrictions

💡 Pro Tip: Applications of derivatives form the core of many real-world problems. Practice interpreting word problems carefully!