Graphs of functions and their 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

How do you determine if a function is increasing or decreasing at a point?

Back

A function is increasing at a point if its derivative is positive at that point, and it is decreasing if its derivative is negative.

Front

What does a positive slope indicate about a function's graph?

Back

A positive slope indicates that the function is increasing at that interval.

Front

What does a negative slope indicate about a function's graph?

Back

A negative slope indicates that the function is decreasing at that interval.

Front

What is the relationship between a function and its derivative?

Back

The derivative of a function provides information about the function's rate of change and can indicate where the function is increasing or decreasing.

Front

How can you find intervals where a function is increasing?

Back

To find intervals where a function is increasing, identify where its derivative is greater than zero (f'(x) > 0).

Front

What does it mean if the derivative of a function is zero at a point?

Back

If the derivative is zero at a point, it indicates a potential local maximum, local minimum, or a point of inflection.

Front

What is the significance of the second derivative?

Back

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

Front

How do you find the derivative of a polynomial function?

Back

To find the derivative of a polynomial function, apply the power rule: for each term ax^n, the derivative is n*ax^(n-1).

Front

What is a critical point?

Back

A critical point occurs where the derivative is zero or undefined, indicating potential locations for local maxima or minima.