AP Calculus Rates of Change Review

Front

What is the average rate of change?

Back

The average rate of change of a function over an interval is the change in the function's value divided by the change in the input value over that interval.

Front

How do you find the derivative of a function?

Back

To find the derivative of a function, apply the power rule: if f(x) = x^n, then f'(x) = n*x^(n-1).

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 tangent line exists and is unique.

Front

What is the difference quotient?

Back

Front

What is the significance of a zero slope on a tangent line?

Back

A zero slope on a tangent line indicates that the function has a local maximum or minimum at that point.

Front

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

Back

The average rate of change over an interval approaches the instantaneous rate of change (the derivative) as the interval shrinks to zero.

Front

What is the power rule for derivatives?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Front

What does it mean for a function to be continuous?

Back

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point.

Front

How do you determine if a function is differentiable at a point?

Back

A function is differentiable at a point if it is continuous at that point and the left-hand and right-hand limits of the derivative exist and are equal.

Front

What is a tangent line?

Back

A tangent line to a curve at a given point is the straight line that just touches the curve at that point and has the same slope as the curve at that point.