AP Calculus AB - Unit 3

What does the First Derivative Test help determine?

It uses zeroes, discontinuities, and sign behavior of a function's derivative to determine overall function behavior.

What indicates that a function is increasing on an interval?

The derivative f'(x) is greater than zero (f'(x) > 0).

What indicates that a function is decreasing on an interval?

The derivative f'(x) is less than zero (f'(x) < 0).

What does a horizontal tangent line indicate in the context of the First Derivative Test?

It indicates a potential maximum or minimum where f'(x) equals zero (f'(x) = 0).

What does a change from positive to negative in the derivative indicate?

It indicates a local maximum.

What does a change from negative to positive in the derivative indicate?

It indicates a local minimum.

What is a critical point?

A point where f'(x) = 0 or f'(x) does not exist (f'(x) DNE).

What must be true for candidates for maximums and minimums?

They must be proved by the sign chart.

What does the Increasing & Decreasing Functions Theorem state?

If f'(x) > 0 on an interval, then f(x) is increasing; if f'(x) < 0, then f(x) is decreasing.

What does the Local Extreme Value Theorem state?

If a function has a local maximum or minimum at point c in the interior domain and f'(c) exists, then f(c) is either a local maximum or minimum.

What is concavity in relation to a function?

A function is concave up if its first derivative f'(x) is increasing, and concave down if f'(x) is decreasing.

What is a point of inflection?

A point where the concavity of the function changes, occurring when f''(x) = 0 or f''(x) DNE.

What does the Second Derivative Test indicate about concavity?

If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

What does the Extreme Value Existence Theorem guarantee?

If f is continuous on a closed interval [a, b], then f has both a maximum and minimum within that interval.

What is the Intermediate Value Existence Theorem?

If a function f is continuous on a closed interval [a, b], it takes on every value between f(a) and f(b).

What does the Mean Value Existence Theorem state?

If f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

What is Rolle's Existence Theorem?

If f is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there is at least one c in (a, b) such that f'(c) = 0.