Unit 5: Analytical Applications of Differentiation

📋 Unit Overview

Unit 5 focuses on advanced analytical techniques using derivatives. You'll learn how to:

Use L'Hôpital's rule to evaluate indeterminate forms
Analyze function behavior using derivatives
Sketch curves using calculus techniques
Apply the Extreme Value Theorem
Use Rolle's Theorem and the Mean Value Theorem
Solve advanced optimization problems

💡 Key Insight: This unit brings together all the differentiation techniques you've learned to create a powerful toolkit for analyzing any function. You'll be able to determine where functions increase, decrease, have extrema, and much more!

🔢 L'Hôpital's Rule

Understanding Indeterminate Forms

L'Hôpital's rule helps us evaluate limits that result in indeterminate forms:

If lim_x→a f(x)/g(x) = 0/0 or ∞/∞

Then lim_x→a f(x)/g(x) = lim_x→a f'(x)/g'(x)

(provided the limit on the right exists)

🎯 Key Concept

L'Hôpital's rule can be applied repeatedly until you get a determinate form. Always check that you have 0/0 or ∞/∞ before applying!

📝 Example 5.1: Basic L'Hôpital's Rule

Find lim_x→0 (sin(x))/x

Solution:

1) Direct substitution gives 0/0 (indeterminate)

2) Apply L'Hôpital's rule: lim_x→0 (sin(x))/x = lim_x→0 cos(x)/1

3) Evaluate: cos(0)/1 = 1/1 = 1

Other Indeterminate Forms

Sometimes you need to manipulate the expression first:

Form	Strategy	Example
0 · ∞	Rewrite as 0/(1/∞) or ∞/(1/0)	x ln(x) as x→0⁺
∞ - ∞	Factor or rationalize	√(x²+1) - x as x→-∞
0⁰, 1^∞, ∞⁰	Take natural log	x^x as x→0⁺

📊 Function Analysis

First and Second Derivative Tests

Use derivatives to analyze function behavior:

First Derivative Test

f'(x) > 0 → f(x) increasing
f'(x) < 0 → f(x) decreasing
f'(x) = 0 → possible extrema

Second Derivative Test

f''(x) > 0 → concave up
f''(x) < 0 → concave down
f''(x) = 0 → possible inflection point

📝 Example 5.2: Analyzing f(x) = x³ - 3x² + 2

Solution:

1) f'(x) = 3x² - 6x = 3x(x - 2)

2) Critical points: x = 0, x = 2

3) f''(x) = 6x - 6 = 6(x - 1)

4) Inflection point: x = 1

5) Analysis:

Increasing on (-∞, 0) and (2, ∞)
Decreasing on (0, 2)
Concave up on (1, ∞), concave down on (-∞, 1)

🎨 Curve Sketching

Steps for Curve Sketching

Follow these steps to sketch any function:

1. Find domain and intercepts

2. Find asymptotes (vertical, horizontal, oblique)

3. Find critical points and intervals of increase/decrease

4. Find inflection points and concavity

5. Sketch the curve

📝 Example 5.3: Sketch f(x) = (x²-1)/(x²-4)

Solution:

1) Domain: x ≠ ±2

2) Intercepts: (0, 1/4), (±1, 0)

3) Asymptotes: x = ±2 (vertical), y = 1 (horizontal)

4) f'(x) = 6x/(x²-4)², critical point at x = 0

5) f''(x) = -6(3x²+4)/(x²-4)³, always negative

6) Function is concave down everywhere

📐 Important Theorems

Extreme Value Theorem

If f is continuous on [a,b], then f has both an absolute maximum and absolute minimum on [a,b].

🎯 Key Concept

Always check the endpoints when finding absolute extrema on a closed interval!

Rolle's Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) such that f'(c) = 0.

📝 Example 5.4: Applying Rolle's Theorem

For f(x) = x² - 4x + 3 on [1, 3], find c guaranteed by Rolle's theorem.

Solution:

1) f(1) = 0 and f(3) = 0, so f(1) = f(3)

2) f'(x) = 2x - 4

3) Set f'(c) = 0: 2c - 4 = 0, so c = 2

4) Check: 2 is in (1, 3) ✓

🎯 Advanced Optimization

Optimization problems can involve multiple variables and constraints:

📝 Example 5.5: Advanced Optimization

Find the point on the parabola y = x² closest to the point (0, 1).

Solution:

1) Distance formula: D = √(x² + (x²-1)²)

2) Minimize D² = x² + (x²-1)² = x⁴ - x² + 1

3) d(D²)/dx = 4x³ - 2x = 2x(2x² - 1) = 0

4) Critical points: x = 0, x = ±1/√2

5) Check second derivative: d²(D²)/dx² = 12x² - 2

6) At x = ±1/√2: 12(1/2) - 2 = 4 > 0, so minimum

7) Closest points: (±1/√2, 1/2)

🧮 Practice Problems

Problem 1

Find lim_x→0 (e^x - 1 - x)/x² using L'Hôpital's rule.

Solution:

1) Direct substitution gives 0/0

2) Apply L'Hôpital's rule: lim_x→0 (e^x - 1)/(2x)

3) Still 0/0, apply again: lim_x→0 e^x/2

4) Evaluate: e^0/2 = 1/2

Problem 2

Find all critical points and inflection points of f(x) = x⁴ - 4x³ + 6x² - 4x + 1.

Solution:

1) f'(x) = 4x³ - 12x² + 12x - 4 = 4(x³ - 3x² + 3x - 1)

2) Factor: f'(x) = 4(x - 1)³

3) Critical point: x = 1

4) f''(x) = 12x² - 24x + 12 = 12(x² - 2x + 1) = 12(x - 1)²

5) f''(x) = 0 when x = 1, but f''(x) ≥ 0 everywhere

6) No inflection points (function is always concave up)

Problem 3

Find the absolute maximum and minimum of f(x) = x³ - 3x + 1 on [-2, 2].

Solution:

1) f'(x) = 3x² - 3 = 3(x² - 1) = 3(x - 1)(x + 1)

2) Critical points: x = ±1

3) Check endpoints and critical points:

- f(-2) = -8 + 6 + 1 = -1

- f(-1) = -1 + 3 + 1 = 3

- f(1) = 1 - 3 + 1 = -1

- f(2) = 8 - 6 + 1 = 3

4) Absolute maximum: 3 at x = ±1

5) Absolute minimum: -1 at x = ±2

💡 Exam Tips

🎯 Multiple Choice

Always check if L'Hôpital's rule applies
Remember to check endpoints for absolute extrema
Use the second derivative test when possible
Look for patterns in function behavior

✍️ Free Response

Show all steps in L'Hôpital's rule applications
Justify your analysis with derivatives
Check all conditions for theorems
Draw clear, labeled graphs

⚠️ Common Mistakes:

Applying L'Hôpital's rule to non-indeterminate forms
Forgetting to check endpoints in optimization
Not verifying theorem conditions
Mixing up concavity and increasing/decreasing

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➡️ What's Next?

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In Unit 6, you'll learn about integration and accumulation of change, building the foundation for integral calculus.

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