Front

What does the Mean Value Theorem state?

Back

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

Front

Define a continuous function.

Back

A continuous function is a function that does not have any breaks, jumps, or holes in its graph. Formally, a function f(x) is continuous at a point x = a if the limit of f(x) as x approaches a equals f(a).

Front

What is the significance of a derivative being equal to zero?

Back

A derivative equal to zero at a point indicates that the function has a horizontal tangent line at that point, which may correspond to a local maximum, local minimum, or a point of inflection.

Tags

CCSS.HSF.IF.A.2

Front

How do you determine where a function is increasing or decreasing using its derivative?

Back

A function f(x) is increasing on intervals where f'(x) > 0 and decreasing where f'(x) < 0.

Front

What does it mean if f''(x) < 0?

Back

If f''(x) < 0, it indicates that the function f(x) is concave down at that point, suggesting that the slope of the tangent line is decreasing.

Tags

CCSS.HSF.LE.B.5

Front

What is the relationship between the first and second derivatives?

Back

The first derivative f'(x) provides information about the slope and increasing/decreasing behavior of the function, while the second derivative f''(x) provides information about the concavity of the function.

Front

What is a critical point?

Back

A critical point of a function occurs where the derivative is either zero or undefined. Critical points are potential locations for local maxima and minima.

Front

Explain the concept of differentiability.

Back

A function is differentiable at a point if it has a derivative at that point. This means the function must be continuous at that point and have a defined slope.

Front

What does it mean for a function to be not differentiable at a point?

Back

A function is not differentiable at a point if it has a discontinuity, a sharp corner, or a vertical tangent line at that point.

Front

How can you use the first derivative test to identify local extrema?

Back

The first derivative test involves analyzing the sign of f'(x) around critical points. If f' changes from positive to negative, there is a local maximum; if it changes from negative to positive, there is a local minimum.