MA Chapter 5 Review

Front

What does it mean for a function to have a horizontal tangent line?

Back

A function has a horizontal tangent line at a point if the derivative at that point is zero, indicating that the slope of the tangent line is flat.

Front

What is the Mean Value Theorem?

Back

The Mean Value Theorem states that if a function is continuous on a 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 does it indicate if the first derivative of a function is negative at a point?

Back

If the first derivative f'(x) is negative at a point, it indicates that the function f(x) is decreasing at that point.

Front

How do you find critical points of a function?

Back

Critical points occur where the first derivative f'(x) is zero or undefined. To find them, set f'(x) = 0 and solve for x.

Front

What is the significance of critical points in calculus?

Back

Critical points are important because they can indicate local maxima, local minima, or points of inflection in the graph of a function.

Front

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

Back

If f'(x) < 0 on an interval, it means that the function f(x) is decreasing throughout that interval.

Front

What is a zero of a function?

Back

A zero of a function is a value of x for which f(x) = 0. It represents the x-intercepts of the graph of the function.

Front

What is the relationship between the first derivative and the concavity of a function?

Back

The first derivative indicates the slope of the function (increasing or decreasing), while the second derivative indicates concavity (concave up or concave down).

Front

How can you determine if a function is increasing or decreasing?

Back

A function is increasing on an interval if its first derivative f'(x) > 0 for all x in that interval, and decreasing if f'(x) < 0.

Front

What is the second derivative test for concavity?

Back

The second derivative test states that if f''(x) > 0, the function is concave up at that point; if f''(x) < 0, the function is concave down.