Question

The graph of a function \(y=f(x)\) is given. Use this graph to sketch the graph of \(y=f^{\prime}(x)\).

Short answer

Sketch the derivative by analyzing slope changes and critical points.

Step-by-step solution

Understand the Relationship

The graph of a derivative, such as \(y=f'(x)\), shows the slope of the original function \(y=f(x)\) at any point \(x\). We need to identify how \(f(x)\) changes to determine \(f'(x)\).

Identify Critical Points

Find where the graph of \(y=f(x)\) has horizontal tangents (i.e., the slope is zero). These are points where \(f'(x) = 0\). These typically occur at peaks, valleys, or points of inflection.

Assess Increasing and Decreasing Intervals

Determine where \(f(x)\) is increasing (\(f'(x) > 0\)) and where \(f(x)\) is decreasing (\(f'(x) < 0\)). This involves identifying intervals on the graph where the slope is positive or negative.

Examine Concavity for Inflection Points

Inflection points on \(f(x)\) occur where the concavity changes. At these points, \(f'(x)\) will change from increasing to decreasing or vice versa, indicating the slope changes direction.

Sketch the Derivative Graph

Using the information from previous steps, sketch the graph of \(y=f'(x)\). When \(f(x)\) is rising, \(f'(x)\) should be above the x-axis, and when \(f(x)\) is falling, \(f'(x)\) should be below the x-axis. At critical points, mark where \(f'(x) = 0\).

Key concepts

Derivative

In calculus, a derivative represents the rate at which a function, like \( y = f(x) \), is changing at any particular point. If you imagine a function graph as a hill, the derivative at each point tells you how steep that hill is at that moment. To put it simply, it answers the question: "How fast is something changing?" This involves recognizing slopes across the graph.

• When the graph is rising, the derivative is positive, indicating an increase.
• When the graph is falling, the derivative is negative, showing a decrease.
• If the graph levels off (like at the top of a hill or bottom of a valley), the derivative is zero.

Recognizing these changes in slope helps sketch the derivative. Remember that what you're really drawing is the pattern of slopes, not the original function itself.

Function Graphs

Function graphs provide a visual picture of a function's behavior. They graphically represent how the output \( y \) changes in response to changes in the input \( x \). For any given function \( y = f(x) \), a graph is a helpful way to see all kinds of information at once.

• Peaks and valleys on the graph denote local maximums or minimums.
• Increasing and decreasing portions show places where the derivative is positive or negative.
• Flat areas reveal where the derivative equals zero, hinting at critical points.

By studying these graph features, one can begin to understand more complex ideas such as derivatives and inflection points, without any calculations.

Critical Points

Critical points on a function's graph are places where the derivative equals zero or is undefined. At these specific spots, the tangent line is horizontal, meaning no change in terms of slope direction.

• They can indicate local maxima (hilltops) or minima (valley bottoms) of the function.
• Not every critical point is a max or min; sometimes, they are inflection points where the curve bends but does not peak or dip.
• Finding these helps sketch the derivative graph by marking `where` it touches the x-axis.

By identifying critical points, one bridges the understanding between the original graph and its derivative representation.

Inflection Points

Inflection points are special spots on a graph where the curve changes concavity. This means where the shape of the graph shifts from curving upwards to downwards or vice versa. While these are not always critical points, they still play an important role in understanding the graph's behavior.

• At an inflection point, the second derivative changes sign (for those exploring deeper into calculus).
• This change often corresponds to a change in how \( f'(x) \) is increasing or decreasing.
• Inflection points help in sketching how \( f'(x) \) might shift from rising to falling.

Spotting inflection points contributes to a fuller picture of how the derivative graph relates to the original function graph.