1.16 Intermediate Value Theorem (IVT)

Front

What is the Intermediate Value Theorem (IVT)?

Back

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and takes on different values at the endpoints, then it must take on every value between f(a) and f(b) at least once within that interval.

Front

What does it mean for a function to be continuous on an interval?

Back

A function is continuous on an interval if there are no breaks, jumps, or holes in the graph of the function over that interval.

Front

If g is continuous on [-1, 4] and g(-1) = -4, g(4) = 1, what can we conclude using IVT?

Back

By the IVT, there exists at least one c in [-1, 4] such that g(c) = -3.

Front

Given f(x) is continuous on [1, 5] with f(1) = 1 and f(5) = -3, what value must f(c) take on?

Back

By the IVT, f(c) must equal -2 for at least one c in [1, 5].

Front

What type of values does the IVT primarily concern?

Back

The IVT primarily concerns y-values.

Front

If g(x) is continuous on [-2, 4], which intervals must contain a solution to g(x) = -1?

Back

All of the intervals [-2, 2], [0, 2], and [2, 4] must contain a solution to g(x) = -1.

Front

What does the IVT guarantee about continuous functions?

Back

The IVT guarantees that a continuous function will take on every value between its minimum and maximum on a closed interval.

Front

If f is continuous and f(-2) = 1, f(5) = -3, what does IVT guarantee?

Back

IVT guarantees that there exists at least one c in [-2, 5] such that f(c) = 0.

Front

What is an example of a function that satisfies the conditions of the IVT?

Back

An example is f(x) = x^2 - 4, which is continuous on any interval and takes on all values between f(a) and f(b) for any a and b.

Front

How can you visually represent the IVT?

Back

You can represent the IVT by drawing a continuous curve between two points on a graph, showing that the curve crosses every y-value between the two endpoints.