Graphs and Derivatives

Front

What is the derivative of a function?

Back

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the graph of the function at a given point.

Front

What does it mean if the derivative f'(x) is positive?

Back

If f'(x) is positive, it means that the function f(x) is increasing at that point.

Front

What does it mean if the derivative f'(x) is negative?

Back

If f'(x) is negative, it means that the function f(x) is decreasing at that point.

Front

What is a relative maximum?

Back

A relative maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.

Front

What is a relative minimum?

Back

A relative minimum is a point on the graph of a function where the function value is lower than the values of the function at nearby points.

Front

What does it mean if the derivative f'(x) is zero?

Back

If f'(x) is zero, it indicates a potential relative maximum, minimum, or a point of inflection.

Front

What is the relationship between increasing functions and their derivatives?

Back

When a function is increasing, its derivative is positive.

Front

What is the relationship between decreasing functions and their derivatives?

Back

When a function is decreasing, its derivative is negative.

Front

What is a tangent line?

Back

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

Front

What is the significance of the second derivative?

Back

The second derivative indicates the concavity of the function. If the second derivative is positive, the function is concave up; if negative, it is concave down.