AP Calculus AB: Unit 5 Quest

Front

What does it mean for a function f'(x) to be positive and increasing?

Back

It means that the slope of the tangent line to the graph of f is positive and that the rate of change of f'(x) is increasing, indicating that f is becoming steeper.

Front

What is the definition of concavity in relation to a function?

Back

A function is concave up on an interval if its second derivative is positive on that interval, indicating that the graph of the function is curving upwards.

Front

What is the First Derivative Test?

Back

The First Derivative Test is a method used to determine relative extrema of a function by analyzing the sign changes of its first derivative.

Front

What is the Extreme Value Theorem (EVT)?

Back

The EVT states that if a function is continuous on a closed interval [a, b], then it must have both an absolute maximum and an absolute minimum on that interval.

Front

What does it indicate if f has a relative minimum at x = c?

Back

It indicates that f(c) is less than the values of f at points immediately to the left and right of c.

Front

What is a point of inflection?

Back

A point of inflection is where the concavity of the function changes, which occurs when the second derivative is zero or undefined.

Front

How do you determine intervals of concavity using the second derivative?

Back

By finding where the second derivative is positive (concave up) or negative (concave down) and identifying the intervals accordingly.

Front

What is the significance of critical points in calculus?

Back

Critical points are where the first derivative is zero or undefined, and they are potential locations for relative extrema.

Front

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

Back

A function is 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 is the relationship between the first and second derivatives?

Back

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