AP Calc Unit 5 Review

Front

What is the definition of a function being increasing?

Back

A function f(x) is said to be increasing on an interval if for any two points x1 and x2 in that interval, if x1 < x2, then f(x1) < f(x2).

Front

What does it mean for a function to be concave up?

Back

A function is concave up on an interval if its second derivative is positive on that interval, indicating that the slope of the function is increasing.

Front

How do you find the intervals where a function is increasing?

Back

To find the intervals where a function is increasing, determine where the first derivative f'(x) is greater than zero.

Front

What is the relationship between the first and second derivatives in determining concavity?

Back

The first derivative indicates the slope of the function, while the second derivative indicates the concavity. If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.

Front

What is the significance of critical points in calculus?

Back

Critical points occur where the first derivative is zero or undefined, and they are potential locations for local maxima, minima, or points of inflection.

Front

How do you determine the x-values where a function has a local maximum or minimum?

Back

To find local maxima or minima, analyze the critical points using the first derivative test or the second derivative test.

Front

What is the first derivative test?

Back

The first derivative test involves checking the sign of the first derivative before and after a critical point to determine if it is a local maximum or minimum.

Front

What is the second derivative test?

Back

The second derivative test uses the value of the second derivative at a critical point: if f''(x) > 0, it's a local minimum; if f''(x) < 0, it's a local maximum.

Front

What does it mean for a function to have an inflection point?

Back

An inflection point occurs where the function changes concavity, which can be found where the second derivative is zero or undefined.

Front

How do you find the intervals of concavity for a function?

Back

To find intervals of concavity, determine where the second derivative is positive (concave up) or negative (concave down).