AP Calculus AB Flashcards: Working with the Intermediate Value Theorem (IVT)

Question 1

A function $f$ is continuous on $[0, 5]$, with $f(0) = -2$ and $f(5) = 4$. Can the IVT be used to show that $f(c) = 6$ for some $c$ in $(0, 5)$?

Accepted Answer

No, the IVT cannot be used for this case. The value $d=6$ is not between the function's endpoint values of $f(0)=-2$ and $f(5)=4$.

Question 2

What is the primary use of the Intermediate Value Theorem in analyzing a function?

Accepted Answer

The IVT is used to explain the behavior of a continuous function on an interval by guaranteeing it takes on every intermediate value between its starting and ending points.

Question 3

What does the Intermediate Value Theorem guarantee about the number $c$?

Accepted Answer

The IVT guarantees the *existence* of at least one number $c$ between $a$ and $b$ where the function's value is $d$. It does not provide a method for finding the exact value of $c$.

Question 4

If the IVT guarantees a value $c$ such that $f(c)=d$, where must the value $c$ be located?

Accepted Answer

The theorem guarantees that the number $c$ is located between $a$ and $b$, meaning it is within the open interval $(a, b)$.

Question 5

A function $g$ has values $g(1)=-3$ and $g(4)=3$. Why can't we use the IVT to guarantee a root (where $g(c)=0$) exists between 1 and 4?

Accepted Answer

We cannot apply the IVT because we do not know if the function $g$ is continuous on the closed interval $[1, 4]$, which is a mandatory condition for the theorem.

Question 6

What are the two essential conditions that must be met to apply the Intermediate Value Theorem?

Accepted Answer

First, the function $f$ must be continuous on the closed interval $[a, b]$. Second, the value $d$ must be a number between the function's endpoint values, $f(a)$ and $f(b)$.

Question 7

Does the Intermediate Value Theorem guarantee that there is *only one* number $c$ for which $f(c)=d$?

Accepted Answer

No, the theorem does not guarantee uniqueness. It only guarantees that there is *at least one* such number $c$ on the interval.

Question 8

What is the Intermediate Value Theorem (IVT)?

Accepted Answer

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

Question 9

A function $f$ is continuous on $[2, 8]$, with $f(2) = 10$ and $f(8) = 20$. Does the IVT guarantee a value $c$ such that $f(c) = 15$?

Accepted Answer

Yes, it does. Since the function is continuous on the closed interval and $d=15$ is between $f(2)=10$ and $f(8)=20$, the IVT guarantees at least one $c$ between 2 and 8 exists.

Question 10

In the statement of the IVT, what is the required property of the function $f$ on the interval $[a, b]$?

Accepted Answer

The function $f$ must be a continuous function on the closed interval $[a, b]$.

AP Calculus AB Flashcards: Working with the Intermediate Value Theorem (IVT)

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