🎯⭐ INTERACTIVE LESSON

Theorem Applications

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Theorem Applications - Complete Interactive Lesson

Part 1: The Intermediate Value Theorem (IVT)

Theorem Applications

Part 1 of 7 — The Intermediate Value Theorem (IVT)

Statement

If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$ , then there exists $c in (a, b)$ such that $f(c) = N$ .

What It Means

A continuous function takes on every value between $f(a)$ and $f(b)$ .

AP Usage

"Since $f$ is continuous on $[a, b]$ , $f(a) = 2$ , and $f(b) = 7$ , by the IVT there exists such that ."

Important: IVT Does NOT Tell You

Where $c$ is
How many such $c$ values exist
Only that at least one exists

IVT 🎯

Key Takeaways — Part 1

IVT requires continuity
Guarantees existence of a value, not location
Always cite continuity when using IVT on the AP exam

Part 2: The Mean Value Theorem (MVT)

Theorem Applications

Part 2 of 7 — The Mean Value Theorem (MVT)

Statement

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there exists $c in (a, b)$ such that:

Part 3: The Extreme Value Theorem (EVT)

Theorem Applications

Part 3 of 7 — The Extreme Value Theorem (EVT)

Statement

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

Part 4: Rolle's Theorem & MVT Applications

Theorem Applications

Part 4 of 7 — Rolle's Theorem & MVT Applications

Rolle's Theorem (Special Case of MVT)

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

Part 5: FTC and When to Use Each Theorem

Theorem Applications

Part 5 of 7 — FTC and When to Use Each Theorem

Theorem Selection Guide

Scenario	Theorem
Show $f(c) = N$ for some $c$	IVT
Show $f'(c) = m$ for some

Part 6: Practice Workshop

Theorem Applications

Part 6 of 7 — Practice Workshop

Mixed Theorem Practice 🎯

A table: $f(0) = 1$ , $f(2) = 5$ , $f(5) = 3$ , . is continuous and differentiable.

Part 7: Final Assessment

Theorem Applications — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Theorem Applications — Complete! ✅

You have mastered:

✅ Intermediate Value Theorem (IVT)
✅ Mean Value Theorem (MVT)
✅ Extreme Value Theorem (EVT)
✅ Rolle's Theorem
✅ Choosing the right theorem

$f'(c) = \frac{f(b) - f(a)}{b - a}$

There's a point where the tangent line is parallel to the secant line.

$f(x) = x^2$ on $[1, 3]$ .

Average rate: $\frac{9-1}{3-1} = 4$ .

$f'(c) = 2c = 4 implies c = 2$ .

The tangent at $x = 2$ is parallel to the secant from $(1,1)$ to $(3,9)$ .

Key Takeaways — Part 2

MVT: instantaneous rate = average rate at some point
Requires continuity AND differentiability
On the AP exam, always verify both conditions

Finding Absolute Extrema (Closed Interval Method)

Find all critical points in $(a, b)$ where $f' = 0$ or $f'$ DNE
Evaluate $f$ at critical points AND at endpoints
Largest value = absolute max, smallest = absolute min

EVT & Closed Interval Method 🎯

$f(x) = x^3 - 3x + 1$ on $[-2, 2]$ .

Key Takeaways — Part 3

EVT guarantees max and min exist on closed intervals
Check critical points AND endpoints
Only requires continuity

If a car travels 120 miles in 2 hours, then at some moment the speedometer reads exactly 60 mph.

This is MVT applied: average speed = instantaneous speed at some point!

Rolle's Theorem & MVT Apps 🎯

Key Takeaways — Part 4

Rolle's Theorem: same endpoints → horizontal tangent somewhere
MVT has real-world applications (speed, rates)

Which Theorem? 🎯

Key Takeaways — Part 5

IVT: proving function values exist
MVT: proving derivative values exist
EVT: proving extrema exist

Theorem Applications

Geometric Meaning

Worked Example

Key Takeaways — Part 2

Finding Absolute Extrema (Closed Interval Method)

Key Takeaways — Part 3

MVT for Speed

Key Takeaways — Part 4

Key Takeaways — Part 5

Workshop Complete!