A. The graph in the accompanying figure is made of line...

a. The graph in the accompanying figure is made of line segments
joined end to end. At which points of the interval $[-4,6]$ is
$f^{\prime}$ not defined? Give reasons for your answer.
b. Graph the derivative of $f$
The graph should show a step function.

Key Concepts

Step Function Representation of the Derivative

When a function is piecewise linear with constant slopes on each segment, its derivative will be constant on those intervals and will change values abruptly at the endpoints. This results in a derivative function that is a step function, exhibiting jumps at the points where the original function has corners.

Differentiability

Differentiability is a property of a function which indicates that at a given point, the function has a unique tangent line, meaning the limit of the difference quotient exists. When a function is made up of distinct segments, the function may not be differentiable at points where those segments join if the slopes (or directional changes) do not match, as the left-hand and right-hand derivatives differ.

Piecewise Linear Functions

Piecewise linear functions are constructed from multiple straight line segments, each defined on a part of the domain. Since the derivative of each individual segment is simply the constant slope of that segment, the overall derivative is also piecewise constant, except at the joining points where the slope might change abruptly.

Corner Points

Corner points occur where the graph of a function, especially a piecewise linear function, changes direction abruptly, resulting in a sharp turn or 'corner'. At these points, the derivative is not defined since the left-hand and right-hand slopes are different, meaning no single tangent line can be drawn.

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