Understanding Derivatives and Graphs

What does the graph of the first derivative tell us about the original function?

• It represents the area under the curve of the original function.

• It provides the average value of the original function.

• It indicates the maximum value of the original function.

• It shows the rate of change of the original function.

How is the second derivative graphically represented when the first derivative is a linear function?

• As a sine wave

• As a parabola

• As a horizontal line

• As a vertical line

When sketching a derivative without knowing the parent function, what is the first step?

• Identify the maximum value of the function

• Find the average rate of change

• Determine where the slope is zero

• Calculate the area under the curve

What happens to the derivative graph at points of discontinuity?

• It becomes a smooth curve

• It remains constant

• It shows a jump discontinuity

• It forms a closed loop

How can you determine when a function is increasing using its first derivative graph?

• When the first derivative is constant

• When the first derivative is above the x-axis

• When the first derivative is zero

• When the first derivative is below the x-axis

What indicates a relative maximum on the graph of the first derivative?

• The slope is zero

• The slope changes from negative to positive

• The slope changes from positive to negative

• The slope remains constant

In the context of derivatives, what does concavity refer to?

• The width of the graph

• The direction of the slope

• The curvature of the graph

• The height of the graph

What is an inflection point in terms of concavity?

• A point where the function reaches its maximum

• A point where the concavity changes

• A point where the function reaches its minimum

• A point where the slope is zero

How can you identify a minimum on the graph of the first derivative?

• The slope changes from positive to negative

• The slope changes from negative to positive

• The slope remains constant

• The slope is zero

What does a horizontal tangent line on the graph of a function indicate?

• A point of inflection

• A critical point

• A point of discontinuity

• A point of maximum curvature