1.16 Working with the Intermediate Value Theorem (IVT)

Keywords

Existence Theorem 存在性定理 • Closed Interval 闭区间

Continuity Requirement 连续性要求 • Guaranteed Value 保证值

The Intermediate Value Theorem

The IVT states that if a continuous function moves from one $y$-value to another, it must pass through every possible $y$-value in between.

核心要点：The Intermediate Value Theorem (IVT)

If $f$ is a continuous function on the closed interval $[a, b]$ and $d$ is a number between $f(a)$ and $f(b)$, then the IVT guarantees that there is at least one number $c$ between $a$ and $b$ such that:

\[ f(c) = d \]

The Checklist for IVT

To apply the IVT on the AP Exam, you must verify these two conditions:

Continuity: The function must be continuous on the closed interval $[a, b]$. If there is a hole, jump, or asymptote in the interval, the IVT does not apply.
Interval: The target value $d$ must actually lie between the function's values at the endpoints ($f(a) < d < f(b)$ or $f(b) < d < f(a)$).

Example: Finding a Root

Does the function $f(x) = x^2 - 2$ have a zero in the interval $[0, 2]$?

$f(0) = -2$
$f(2) = 2$
Since $f(x)$ is a polynomial (continuous) and $0$ is between $-2$ and $2$, the IVT guarantees there is at least one $c$ in $(0, 2)$ such that $f(c) = 0$.

Why Continuity Matters

If a function is not continuous, it can "jump" over the target value $d$.

Important Note

The IVT only tells you that a value exists. It does not tell you:

How many such values exist (there could be 1, 3, or 100).
The exact $x$-coordinate of $c$.
The formula for the function.

Extension: Real-World IVT

Think of the IVT like a person walking up a staircase versus walking up a ramp.

If you are on a staircase (discontinuous), you can move from the 1st floor to the 2nd floor without ever having your feet exactly 1.5 meters above the ground.

If you are on a ramp (continuous), you cannot get to the 2nd floor without passing through every single height in between.

Deep Application: Proving Intersections

In many AP problems, you aren't just looking for where a function equals a number; you are asked to prove that two functions, $f(x)$ and $g(x)$, cross each other.

The Strategy: Creating a Difference Function

To prove $f(x) = g(x)$ at some point, we define a new function: $$ h(x) = f(x) - g(x) $$ If we can prove that $h(c) = 0$ using the IVT, then $f(c) - g(c) = 0$, which means $f(c) = g(c)$.

Example:

Prove that $f(x) = x$ and $g(x) = \cos(x)$ intersect at least once in the interval $[0, \pi/2]$.

Define $h(x)$: Let $h(x) = x - \cos(x)$.
Check Continuity: $h(x)$ is continuous because it is the difference of a linear and a trig function.
Check Endpoints:
- $h(0) = 0 - \cos(0) = -1$ (Negative)
- $h(\pi/2) = \pi/2 - \cos(\pi/2) = \pi/2 \approx 1.57$ (Positive)
Conclusion: Since $h(x)$ is continuous and $0$ lies between $-1$ and $1.57$, by the IVT, there must be at least one $c$ in $(0, \pi/2)$ such that $h(c) = 0$, meaning $x = \cos(x)$.

Extension: The 'Wobbly Table' Theorem

Did you know the IVT can be used to fix a wobbly table? If you have a four-legged table on uneven ground, the IVT guarantees that if you rotate the table, there is at least one orientation (less than 90 degrees of rotation) where all four legs will touch the ground simultaneously.

The "wobble" is a continuous function of the angle of rotation. If the wobble is positive in one orientation and negative after turning it 90 degrees, it must be zero at some point in between!