Unit 5 Review (5.1-5.9)

Front

What is the relationship between a function, its first derivative, and its second derivative at a point?

Back

For a twice differentiable function f at a point x=a, the values of f(a), f'(a), and f''(a) can indicate the function's behavior: f'(a) indicates the slope (increasing or decreasing), and f''(a) indicates concavity (concave up or down).

Front

What does it mean for a function to be continuous on a closed interval?

Back

A function is continuous on a closed interval [a,b] if it is defined at every point in the interval and the limit of the function as it approaches any point in the interval equals the function's value at that point.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which can be determined by analyzing the second derivative.

Front

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

Back

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

Front

What does it mean if the second derivative of a function is less than zero?

Back

If f''(x) < 0 for all x in an interval, the function is concave down on that interval.

Front

What is the significance of a critical point in a function?

Back

A critical point occurs where f'(x) = 0 or f'(x) does not exist, and it may indicate a local maximum, local minimum, or an inflection point.

Front

How can you find the number of inflection points in a polynomial function?

Back

To find inflection points, set the second derivative equal to zero and solve for x. The number of solutions indicates the potential inflection points.

Front

What does it mean for a function to be differentiable at a point?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning the function is smooth and has no sharp corners or discontinuities.

Front

What is the relationship between the graph of a function and the graph of its derivative?

Back

The graph of the derivative f'(x) indicates the slope of the original function f(x). Where f'(x) is positive, f(x) is increasing; where f'(x) is negative, f(x) is decreasing.

Front

What is the first derivative test?

Back

The first derivative test is a method to determine local extrema by analyzing the sign of f'(x) around critical points.