Session 05-05 - Graphical Calculus Mastery

Section 05: Differential Calculus

Entry Quiz - 10 Minutes

Quick Review from Session 05-04

Test your understanding of chain rule and implicit differentiation

1. Differentiate \(f(x) = (2x^3 - 5)^4\) using the chain rule.

2. Find \(\frac{dy}{dx}\) if \(x^2 + xy = 10\).

3. If \(f'(3) = 0\), what does this tell about the graph of \(f(x)\) at \(x = 3\)?

4. If \(f''(x) > 0\) for all \(x\) in an interval, what does this tell you about the shape of \(f(x)\)?

Homework Discussion - 15 Minutes

Your questions from Session 05-04

What questions do you have regarding the previous session?

Learning Objectives

What You’ll Master Today

• Sketch \(f'(x)\) from the graph of \(f(x)\) by analyzing slopes
• Determine properties of \(f(x)\) from the graph of \(f'(x)\)
• Understand concavity through second derivatives \(f''(x)\)
• Identify critical points and their classification
• Find inflection points where concavity changes
• Master visual analysis - a heavily tested exam skill!

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NoteKey Insight

Being able to translate between graphs of \(f\), \(f'\), and \(f''\) is one of the most important visual skills in calculus and frequently tested!

Part A: From Function to Derivative

Understanding the Derivative Graph

The graph of \(f'(x)\) shows the slope of \(f(x)\) at each point.

. . .

What to look for in \(f(x)\):

• Where \(f\) is increasing → \(f'(x) > 0\) (derivative is positive)
• Where \(f\) is decreasing → \(f'(x) < 0\) (derivative is negative)
• Where \(f\) has a horizontal tangent → \(f'(x) = 0\) (crosses x-axis)
• Where \(f\) is steep → \(|f'(x)|\) is large
• Where \(f\) is flat → \(|f'(x)|\) is small

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This shouldn’t be too complicated, right?

Sketching \(f'\) from \(f\)

A parabola and its derivative: \(f(x) = x^2\)

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\(f\) has minimum at \(x = 0\) → \(f'\) crosses zero

Step-by-Step: Sketching the Derivative

The process is rather straightforward:

1. Identify critical points: Where does \(f\) have horizontal tangents? Mark these as zeros of \(f'\)
2. Determine sign: Where is \(f\) increasing/decreasing? Make \(f'\) positive/negative accordingly
3. Consider steepness:
   • Where is \(f\) very steep? Make \(|f'|\) large.
   • Where is \(f\) nearly flat? Make \(|f'|\) small.
4. Check concavity: Is \(f'\) increasing or decreasing? This tells you about the concavity of \(f\)

Complex Example

A cubic function and its derivative.

. . .

\(f\) increasing, then decreasing, then increasing → \(f'\) crosses zero twice

Critical Points

A point \(x = c\) where either:

• \(f'(c) = 0\) (horizontal tangent), or
• \(f'(c)\) does not exist (corner, cusp, vertical tangent)

. . .

TipClassification of Critical Points

• Local maximum: \(f'\) changes from positive to negative
• Local minimum: \(f'\) changes from negative to positive
  
• Neither: \(f'\) doesn’t change sign (e.g., \(f(x) = x^3\) at \(x = 0\))

Part B: From Derivative to Function

Reading Information from \(f'(x)\)

Given the graph of \(f'(x)\), we can determine:

1. Where \(f\) is increasing/decreasing:

   • \(f'(x) > 0\) → \(f\) is increasing
   • \(f'(x) < 0\) → \(f\) is decreasing

2. Where \(f\) has local extrema: \(f'\) crosses zero

   • Sign change determines type!

3. Where \(f\) is steepest: Where \(|f'(x)|\) is largest

4. Relative heights: Cannot determine absolute \(y\)-values!

   • Can determine relative changes!

Example: Reading from \(f'(x)\)

. . .

• \(f' > 0\) on \((-2, 2)\) → \(f\) increasing, \(f' < 0\) for \(|x| > 2\) → \(f\) decreasing
• \(f'\) changes at \(x = -2, 2\) → \(f\) has local extrema

Quick Practice - 10 Minutes

Individual Exercise I

Sketch the derivative of this function!

Individual Exercise II

The Questions:

1. Where is \(f'(x) > 0\), \(f'(x) < 0\), and \(f'(x) = 0\)?

2. Sketch the graph of \(f'(x)\).

3. At what points does \(f'(x)\) not exist?

Break - 10 Minutes

Part C: Second Derivatives and Concavity

Understanding Concavity

Describes the curving behavior of a function.

. . .

• Concave up (⌣): Curves upward like a smile
  • \(f''(x) > 0\)
  • The slope \(f'(x)\) is increasing
  • “Holds water”
• Concave down (⌢): Curves downward like a frown
  • \(f''(x) < 0\)
  • The slope \(f'(x)\) is decreasing
  • “Spills water”

Visualizing Concavity

Concave up vs. concave down

. . .

\(f''(x)\) tells us how \(f'(x)\) is changing, which determines the shape of \(f(x)\).

Inflection Points

A point where concavity changes!

. . .

• \(f''(x) = 0\) or \(f''(x)\) does not exist
• Concavity changes (from ⌣ to ⌢ or vice versa)

. . .

Example: \(f(x) = x^3\)

• \(f'(x) = 3x^2\)
• \(f''(x) = 6x\)
• \(f''(0) = 0\) and concavity changes at \(x = 0\)
• So \((0, 0)\) is an inflection point

Relationships Between \(f\), \(f'\), and \(f''\)

. . .

• Where \(f'' < 0\): \(f\) concave down, \(f'\) decreasing, where \(f'' > 0\): \(f\) concave up, \(f'\) increasing
• Where \(f'' = 0\): Inflection point in \(f\), extremum in \(f'\)

Part D: Complete Analysis

The Complete Picture

Given a function \(f(x)\), complete analysis involves:

1. Critical points: Solve \(f'(x) = 0\)
2. First derivative test: Check sign changes of \(f'\) to classify extrema
3. Inflection points: Solve \(f''(x) = 0\) and check for concavity change
4. Intervals: Determine where \(f\) is increasing/decreasing and where concave up/down
5. Key points: Evaluate \(f\) at critical points and inflection points

Example: Complete Analysis

Analyze \(f(x) = x^4 - 4x^3\)

. . .

First derivative:

. . .

\[f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)\]

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Critical points: \(x = 0, 3\)

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Second derivative:

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\[f''(x) = 12x^2 - 24x = 12x(x - 2)\]

. . .

Inflection points: \(x = 0, 2\)

Analysis Visualization

Complete analysis of f(x) = x⁴ - 4x³

Guided Practice

Practice Set A: Sketching Derivatives

For each function graphed below, sketch \(f'(x)\) and identify:

• Where \(f'(x) > 0, < 0, = 0\) and any points where \(f'\) does not exist

Practice B: From Derivative I

Continue working for 10 minutes

Given the graph of \(f'(x)\), answer:

1. Where is \(f(x)\) increasing/decreasing?
2. Where does \(f(x)\) have local extrema? Classify them.
3. Sketch a possible graph of \(f(x)\).

Practice B: From Derivative II

Quick Practice: Derivative Sketching

Work individually for 5 minutes

For each function described, sketch both \(f(x)\) and \(f'(x)\):

1. \(f(x) = x^3 - 3x\) (cubic with local max and min)

2. \(f(x) = |x - 2|\) (V-shape shifted right)

3. \(f(x)\) is constant for \(x < 0\), then increases linearly for \(x \geq 0\)

4. \(f(x)\) has \(f'(x) > 0\) everywhere but \(f'(x) \to 0\) as \(x \to \infty\)

Coffee Break - 15 Minutes

Business Applications

Profit Function Analysis

Business Context: A company’s monthly profit (in thousands €) is modeled by: \[P(t) = -t^3 + 12t^2 - 36t + 50\] where \(t\) is months since product launch.

. . .

Questions:

1. When is profit increasing/decreasing?
2. When does profit reach local extrema?
3. When is the rate of profit change accelerating/decelerating?

Profit Analysis Solution

First derivative (profit rate of change): \[P'(t) = -3t^2 + 24t - 36 = -3(t^2 - 8t + 12) = -3(t - 2)(t - 6)\]

. . .

Critical points: \(t = 2, 6\) months

. . .

Second derivative (acceleration of profit change): \[P''(t) = -6t + 24 = -6(t - 4)\]

. . .

Inflection point: \(t = 4\) months

Profit Visualization

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Question: How would you describe the behaviour?

Profit Interpretation

TipReading Business Graphs: Quick Rules

• \(P'(t) > 0\) → Profit is growing (good news!)
• \(P'(t) < 0\) → Profit is shrinking (warning sign)
• \(P'(t) = 0\) → Profit has reached a turning point (decision time)

The sign of the derivative tells you the direction of change!

. . .

NoteInflection Points in Business

The inflection point (\(P''(t) = 0\) at month 4) marks where:

• Before: Profit is accelerating (growth speeding up)
• After: Profit is decelerating (growth slowing down)

Business insight: Month 4 is when the company should start planning for the eventual peak. Growth is still positive but momentum is fading!

Collaborative Problem-Solving

Challenge: Complete Function Analysis

Scenario: An economist models consumer demand response using:

\[D(p) = \frac{100}{p + 1} - 2\]

where \(D\) is demand (thousands of units) and \(p\) is price (€).

Tasks

Work in groups of 3-4

1. Find \(D'(p)\) and interpret its sign. What does this tell you economically?

2. Find \(D''(p)\). What does its sign tell you about how demand sensitivity changes with price?

3. Sketch the graphs of \(D(p)\), \(D'(p)\), and \(D''(p)\) on \(p \in [0, 10]\).

4. At what price is the rate of demand decrease (i.e., \(|D'(p)|\)) exactly 1 unit per €?

5. Business question: If you’re a monopolist who can set price, explain using calculus concepts why you wouldn’t set price arbitrarily high even though higher prices mean more revenue per unit.

Think-Pair-Share - 7 Minutes

Discussion Question

Think individually, then discuss with class

Question: Consider these scenarios:

• Company A: Stock price \(S_A(t)\) is \(S'_A > 0\) but \(S''_A < 0\)
• Company B: Stock price \(S_B(t)\) is \(S'_B < 0\) but \(S''_B > 0\)

1. Which company is in a better position right now?
2. Which company shows more promising momentum for the future?
3. In financial terms, what do \(S'\) and \(S''\) represent?
4. Can you think of real-world examples of each scenario?

Wrap-Up & Key Takeaways

The Derivative Overview

If you know…	You can determine…
\(f' > 0\)	\(f\) is increasing
\(f' < 0\)	\(f\) is decreasing
\(f' = 0\)	Possible local extremum
\(f'' > 0\)	\(f\) concave up, \(f'\) increasing
\(f'' < 0\)	\(f\) concave down, \(f'\) decreasing
\(f'' = 0\)	Possible inflection point

. . .

These are heavily Tested Skills in FSP!!!

Final Assessment - 5 Minutes

Quick Check I

Consider this function:

Quick Check II

Work individually, then we compare

1. Where is \(f'(x) > 0\)?

2. At \(x = 1\), is this a local max, local min, or neither?

3. Where is \(f(x)\) concave up?

4. True or False: If \(f'(c) = 0\), then \(f\) must have a local extremum at \(x = c\).

Next Session Preview

Session 05-06

Optimization & Curve Sketching

• First and second derivative tests for extrema classification
• Global maxima/minima on closed intervals
• Complete curve sketching algorithm (6 steps)
• Business optimization: profit maximization, cost minimization
• Interpreting results in real-world context

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Complete Tasks 05-05!