MVT, EVT, IVT (SPHS)

Front

What is the Intermediate Value Theorem (IVT)?

Back

The IVT states that if a function is continuous on a closed interval [a, b], then for any value L between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = L.

Front

What is the Mean Value Theorem (MVT)?

Back

The MVT states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What is the Extreme Value Theorem (EVT)?

Back

The EVT states that if a function is continuous on a closed interval [a, b], then it must attain a maximum and a minimum value at least once in that interval.

Front

When does the IVT apply?

Back

The IVT applies when a function is continuous on a closed interval [a, b] and the values of the function at the endpoints f(a) and f(b) are different.

Front

What does it mean for a function to be continuous?

Back

A function is continuous if there are no breaks, jumps, or holes in its graph, meaning it can be drawn without lifting the pencil.

Front

Provide an example of a function that satisfies the conditions of the IVT.

Back

f(x) = x^2 is continuous on [1, 4]. Since f(1) = 1 and f(4) = 16, the IVT guarantees that there is a c in (1, 4) such that f(c) = 10.

Front

What is the significance of the derivative in the MVT?

Back

The derivative f'(c) represents the instantaneous rate of change of the function at point c, which equals the average rate of change over the interval [a, b].

Front

What is a critical point?

Back

A critical point of a function occurs where its derivative is zero or undefined, indicating potential local maxima or minima.

Front

How do you determine if a function has a zero in an interval using the IVT?

Back

If f(a) and f(b) have opposite signs (f(a) * f(b) < 0), then by the IVT, there is at least one c in (a, b) such that f(c) = 0.

Front

What is the relationship between the EVT and the endpoints of an interval?

Back

The EVT guarantees that the maximum and minimum values of a continuous function on [a, b] will occur either at critical points or at the endpoints a or b.