📐 Concept diagram

04-07 - Applications of Derivatives

Phase: 4 | Subject: 04-07 Prerequisites: 04-06-implicit-differentiation.md Next subject: 04-08-optimization.md

Learning Objectives

By the end of this subject, you will be able to:

1. Find equations of tangent and normal lines
2. Identify critical points
3. Apply the first derivative test
4. Apply the second derivative test
5. Sketch curves using derivative information

Core Content

Tangent and Normal Lines

Tangent line: Has the same slope as the curve at the point. Slope = f'(a) at point (a, f(a)).

Normal line: Perpendicular to the tangent. Slope = -1/f'(a) (negative reciprocal).

Example: y = x² at x = 1. f'(x) = 2x, so f'(1) = 2. Tangent: y - 1 = 2(x - 1), so y = 2x - 1. Normal: slope = -1/2. y - 1 = -1/2(x - 1), so y = -x/2 + 3/2.

Critical Points

A critical point occurs where f'(x) = 0 or f'(x) is undefined (but f(x) is defined).

First Derivative Test

⚠️ THIS IS CRITICAL — the first derivative test is your primary tool for classifying critical points. The sign of f' tells you whether the function is rising or falling as you cross the critical point, revealing maxima, minima, or neither.

Examines the SIGN of f'(x) around a critical point.

• f' changes from + to -: local MAXIMUM
• f' changes from - to +: local MINIMUM
• f' doesn't change sign: neither (inflection point or saddle)

Example: f(x) = x³ - 3x f'(x) = 3x² - 3 = 3(x² - 1) = 3(x - 1)(x + 1) Critical points: x = -1, x = 1

At x = -1: f' changes from - to + → local minimum At x = 1: f' changes from + to - → local maximum

Second Derivative Test

If f'(c) = 0 and f''(c) exists:

• f''(c) > 0: local MINIMUM (concave up)
• f''(c) < 0: local MAXIMUM (concave down)
• f''(c) = 0: inconclusive, use first derivative test

Example: f(x) = x³ - 3x f''(x) = 6x At x = -1: f''(-1) = -6 < 0 → local maximum At x = 1: f''(1) = 6 > 0 → local minimum

Verification with the first derivative test: For x < -1: f'(-2) = 3(4) - 3 = 9 > 0 For -1 < x < 1: f'(0) = -3 < 0 For x > 1: f'(2) = 9 > 0

Curve Sketching Checklist

1. Domain and range
2. Intercepts (x and y)
3. Symmetry (even, odd, neither)
4. Asymptotes
5. Critical points and classification
6. Intervals of increase/decrease
7. Concavity and inflection points
8. Sketch!

Key Terms

• 04 07 Applications Of Derivatives
• Correct: B)
• Correct: C)
• Critical Points
• Curve Sketching Checklist
• Example 1: Tangent and normal
• Example 2: Curve sketching
• Example 3: Concavity and inflection
• First Derivative Test
• Second Derivative Test
• Tangent and Normal Lines
• critical point

Worked Examples

Example 1: Tangent and normal

y = √x at (4, 2) f'(x) = 1/(2√x), f'(4) = 1/4 Tangent: y - 2 = (1/4)(x - 4), so y = x/4 + 1 Normal: slope = -4. y - 2 = -4(x - 4), so y = -4x + 18

Example 2: Curve sketching

f(x) = x³ - 3x² + 2

1. f'(x) = 3x² - 6x = 3x(x - 2)
2. Critical: x = 0, x = 2
3. f''(x) = 6x - 6
4. f''(0) = -6 < 0 → local max at x = 0, f(0) = 2
5. f''(2) = 6 > 0 → local min at x = 2, f(2) = -2
6. f''' (not needed for classification)
7. Intervals: increasing on (-∞, 0), decreasing on (0, 2), increasing on (2, ∞)

Example 3: Concavity and inflection

f(x) = x⁴ - 6x²

1. f'(x) = 4x³ - 12x = 4x(x² - 3)
2. Critical: x = 0, ±√3
3. f''(x) = 12x² - 12 = 12(x² - 1)
4. f''(x) = 0 at x = ±1. These are inflection points (concavity changes).
5. Classification:
6. At x = 0: f''(0) = -12 < 0 → local max, f(0) = 0
7. At x = √3: f''(√3) = 24 > 0 → local min, f(√3) = -9
8. At x = -√3: same as √3 → local min, f(-√3) = -9

Quiz

Q1: What does the concept of Critical Points primarily refer to in this subject?

A) A historical anecdote about Critical Points B) The definition and application of Critical Points C) A visual representation of Critical Points D) A computational error related to Critical Points

Correct: B)

• If you chose A: This is incorrect. Critical Points is defined as: the definition and application of critical points. The other options describe different aspects that are not the primary focus.
• If you chose B: Critical Points is defined as: the definition and application of critical points. The other options describe different aspects that are not the primary focus. Correct!
• If you chose C: This is incorrect. Critical Points is defined as: the definition and application of critical points. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Critical Points is defined as: the definition and application of critical points. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Curve Sketching Checklist?

A) It is used only in advanced research contexts B) It replaces all other methods in this domain C) It is used to curve sketching checklist in mathematical analysis D) It is primarily a historical notation system

Correct: C)

• If you chose A: This is incorrect. Curve Sketching Checklist serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Curve Sketching Checklist serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Curve Sketching Checklist serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Curve Sketching Checklist serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about First Derivative Test is TRUE?

A) First Derivative Test is a fundamental concept covered in this subject B) First Derivative Test is mentioned only as a historical footnote C) First Derivative Test is not related to this subject D) First Derivative Test is an advanced topic beyond this subject's scope

Correct: A)

• If you chose A: First Derivative Test is a fundamental concept covered in this subject. This subject covers First Derivative Test as part of its core content. Correct!
• If you chose B: This is incorrect. First Derivative Test is a fundamental concept covered in this subject. This subject covers First Derivative Test as part of its core content.
• If you chose C: This is incorrect. First Derivative Test is a fundamental concept covered in this subject. This subject covers First Derivative Test as part of its core content.
• If you chose D: This is incorrect. First Derivative Test is a fundamental concept covered in this subject. This subject covers First Derivative Test as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) 2x - 1. C) The inverse of the correct answer D) A different result from a common mistake

Correct: B)

• If you chose A: This is incorrect. The worked examples show that the result is 2x - 1.. The other options represent common errors.
• If you chose B: The worked examples show that the result is 2x - 1.. The other options represent common errors. Correct!
• If you chose C: This is incorrect. The worked examples show that the result is 2x - 1.. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 2x - 1.. The other options represent common errors.

Q5: How are First Derivative Test and Second Derivative Test related?

A) First Derivative Test and Second Derivative Test are completely unrelated topics B) First Derivative Test and Second Derivative Test are closely related concepts C) First Derivative Test is a special case of Second Derivative Test D) First Derivative Test is the inverse of Second Derivative Test

Correct: B)

• If you chose A: This is incorrect. Both First Derivative Test and Second Derivative Test are covered in this subject as interconnected topics.
• If you chose B: Both First Derivative Test and Second Derivative Test are covered in this subject as interconnected topics. Correct!
• If you chose C: This is incorrect. Both First Derivative Test and Second Derivative Test are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both First Derivative Test and Second Derivative Test are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Tangent and Normal Lines?

A) Tangent and Normal Lines has no common misconceptions B) A common mistake is confusing Tangent and Normal Lines with a similar concept C) Tangent and Normal Lines is always computed the same way in all contexts D) The main error with Tangent and Normal Lines is using it when it is not needed

Correct: B)

• If you chose A: This is incorrect. Students often confuse Tangent and Normal Lines with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse Tangent and Normal Lines with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse Tangent and Normal Lines with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Tangent and Normal Lines with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Example 1: Tangent And Normal?

A) Avoid Example 1: Tangent And Normal unless explicitly instructed B) Apply Example 1: Tangent And Normal to solve problems in this subject's domain C) Example 1: Tangent And Normal is not practically useful D) Use Example 1: Tangent And Normal only in pure mathematics contexts

Correct: B)

• If you chose A: This is incorrect. Example 1: Tangent And Normal is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: Example 1: Tangent And Normal is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. Example 1: Tangent And Normal is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Example 1: Tangent And Normal is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Tangent to y = x² at (1, 1) Answer: y' = 2x, slope at x=1 is 2. y - 1 = 2(x - 1), y = 2x - 1.

2. Critical points of f(x) = x³ - 3x Answer: f'(x) = 3x² - 3 = 0 when x = ±1.

3. Classify the critical point at x = 0 for f(x) = x³ - 3x² + 1 Answer: f'(x) = 3x² - 6x. f'(0) = 0 ✓ critical. f''(x) = 6x - 6. f''(0) = -6 < 0 → local maximum. f(0) = 1.

4. Normal to y = 1/x at x = 2 Answer: y' = -1/x². At x=2: y' = -1/4. Normal slope = 4. Point: (2, 1/2). y - 1/2 = 4(x - 2), y = 4x - 15/2.

5. Find intervals of concavity for f(x) = x³ - 3x² + 1 Answer: f''(x) = 6x - 6. f''(x) = 0 at x = 1. Concave down on (-∞, 1), concave up on (1, ∞). Inflection point at x = 1.

Summary

Key takeaways:

• Tangent slope = f'(a); normal slope = -1/f'(a)
• Critical points: where f' = 0 or undefined
• First derivative test: sign change of f'
• Second derivative test: sign of f'' at critical point
• f'' > 0: concave up; f'' < 0: concave down

Pitfalls

• Confusing the first and second derivative tests. The first derivative test uses sign changes of f' around a critical point. The second derivative test uses the sign of f'' at the critical point. They answer the same question but with different data — mixing them up leads to wrong conclusions.
• Declaring an inflection point from f''(c) = 0 alone. f''(c) = 0 is necessary but not sufficient — you must verify that f'' actually changes sign at c. For f(x) = x⁴, f''(0) = 0 but there's no inflection (concave up on both sides).
• Using the wrong slope for the normal line. The normal slope is -1/f'(a), NOT 1/f'(a) or f'(a). The negative reciprocal is easy to forget.
• Calling a point "critical" when f is undefined there. A critical point requires f to be defined. f'(x) = 1/x has no critical point at x = 0 because f is undefined there.
• Forgetting to test intervals between ALL critical points. When determining intervals of increase/decrease, partition the number line at every critical point. Skipping one can flip your sign analysis.

Next Steps

Next up: 04-08-optimization.md