Unit 5 Review

Front

What is the Mean Value Theorem?

Back

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

Front

What is a required condition for the Mean Value Theorem?

Back

f(x) is continuous on [a, b].

Front

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

Back

A function f is decreasing on an interval if for any two points x1 and x2 in the interval, if x1 < x2, then f(x1) > f(x2).

Front

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

Back

A function is decreasing where its derivative f'(x) is less than 0.

Front

What is concavity in relation to a function?

Back

Concavity refers to the direction of the curvature of the graph of a function. A function is concave up if its graph opens upwards and concave down if it opens downwards.

Front

How do you find intervals of concavity?

Back

To find intervals of concavity, analyze the second derivative f''(x). If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Front

What is the significance of critical points in calculus?

Back

Critical points are where the derivative f'(x) is zero or undefined. They are potential locations for local maxima, minima, or points of inflection.

Front

How do you find the maximum value of a function on a closed interval?

Back

To find the maximum value of a function on a closed interval, evaluate the function at the endpoints and at any critical points within the interval.

Front

What is the relationship between average rate of change and instantaneous rate of change?

Back

The average rate of change of a function over an interval is the slope of the secant line, while the instantaneous rate of change at a point is the slope of the tangent line at that point.

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.