Rates of change

Front

What is the definition of the average rate of change of a function?

Back

The average rate of change of a function over an interval [a, b] is defined as the change in the function's value divided by the change in the input value, calculated as \( \frac{f(b) - f(a)}{b - a} \).

Front

What is a continuous function?

Back

A function is continuous if there are no breaks, jumps, or holes in its graph. Formally, a function f(x) is continuous at a point c if \( \lim_{x \to c} f(x) = f(c) \).

Front

What is a removable discontinuity?

Back

A removable discontinuity occurs at a point where a function is not defined, but can be made continuous by defining or redefining the function at that point.

Front

What is a jump discontinuity?

Back

A jump discontinuity occurs when the left-hand limit and the right-hand limit of a function at a certain point exist but are not equal, causing a 'jump' in the graph.

Front

What is an infinite discontinuity?

Back

An infinite discontinuity occurs when the function approaches infinity or negative infinity as it approaches a certain point, resulting in a vertical asymptote.

Front

How do you find the slope of a function?

Back

The slope of a function at a point is found by calculating the derivative of the function at that point, which represents the instantaneous rate of change.

Front

What is the derivative of the function \( f(x) = 3x^2 + 2x \)?

Back

The derivative of the function is \( f'(x) = 6x + 2 \).

Front

What is the significance of the slope in the context of rates of change?

Back

The slope represents the rate of change of the function's output with respect to its input, indicating how steeply the function rises or falls.

Front

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

Back

The average rate of change over an interval approaches the derivative as the interval shrinks to a point.

Front

What does it mean for a function to be differentiable?

Back

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