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Chapter 2: Problem 36

Sketch the graph of the derivative \(f^{\prime}\) of the function \(f\) whose graph is given.

Short Answer

Expert verified
To sketch the graph of the derivative \(f'(x)\) of a given function \(f(x)\), first identify intervals where the function is increasing or decreasing. Next, find points of local maxima or minima, where \(f'(x)=0\), and any points where the graph is not differentiable. Finally, using this information, sketch a plausible graph of the derivative, with positive slopes for increasing intervals, negative slopes for decreasing intervals, and breaks in continuity at non-differentiable points.

Step by step solution

01

Identify intervals where the function is increasing or decreasing.

To find the slopes at different points on the graph, first observe the graph of the function \(f\). If the function is increasing, the slope of the tangent line will be positive, so \(f'(x) > 0\). If the function is decreasing, the slope of the tangent line will be negative, so \(f'(x) < 0\). Based on the graph of \(f\), identify the intervals where \(f\) is increasing or decreasing.
02

Identify points where the function has local maxima or minima.

Local maxima and minima will appear as "peaks" and "valleys" on the graph of the function. At these points, the slope of the tangent line changes from positive to negative (for maxima) or from negative to positive (for minima). The slopes of the tangent line (hence, the derivative) at these points will be zero, so \(f'(x) = 0\).
03

Identify points where the graph is not differentiable.

There might be points on the graph of the function where the graph takes a sharp turn or has a cusp. These points represent places where the derivative does not exist or is not continuous. Mark these points on the graph and remember that the graph of the derivative will either not be defined at these points or will have a break in continuity.
04

Sketch the graph of the derivative.

Using the information gathered in Steps 1-3, you can now sketch the graph of the derivative, \(f'(x)\). 1. Place points on the graph where \(f'(x)=0\) (local maxima and minima from Step 2). 2. Draw positive slopes for intervals where the function is increasing, and negative slopes for intervals where it is decreasing (from Step 1). 3. Mark the points where the graph is not differentiable or has breaks in continuity (from Step 3). 4. Based on these markings, sketch a plausible graph of the function's derivative. Consider the general shape of the function's graph and transitions between intervals of increasing and decreasing.

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