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 $c in (a, b)$ such that $f(c) = 5$."

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

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

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:

f'(c) = rac{f(b) - f(a)}{b - a}

Geometric Meaning

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

Worked Example

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

Average rate: rac{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 $(3,9)$.

MVT 🎯

Key Takeaways — Part 2

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

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 $[a, b]$.

Finding Absolute Extrema (Closed Interval Method)

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

EVT & Closed Interval Method 🎯

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

Key Takeaways — Part 3

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

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 $c in (a, b)$ such that $f'(c) = 0$.

MVT for Speed

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

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

Theorem Applications

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

Theorem Selection Guide

Which Theorem? 🎯

Key Takeaways — Part 5

1. IVT: proving function values exist
2. MVT: proving derivative values exist
3. EVT: proving extrema exist

Theorem Applications

Part 6 of 7 — Practice Workshop

Mixed Theorem Practice 🎯

A table: $f(0) = 1$, $f(2) = 5$, $f(5) = 3$, $f(7) = 8$. $f$ is continuous and differentiable.

Workshop Complete!

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