Nick starts jogging and runs faster and faster for 3...

Nick starts jogging and runs faster and faster for 3 mintues, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes. (a) Sketch a possible graph of the distance $s$ Nick has covered after $t$ minutes. (b) Sketch a graph of $d s / d t$

Nick starts jogging and runs faster and faster for 3 mintues, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.
(a) Sketch a possible graph of the distance $s$ Nick has covered after $t$ minutes.
(b) Sketch a graph of $d s / d t$

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Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions in different intervals. In the context of motion, such functions can model segments where behavior changes, such as accelerating, decelerating, constant-speed movement, or stopping. Understanding how each segment translates into different parts of a graph is essential.

Interpretation of the Derivative as Instantaneous Rate of Change

The derivative of a function, when interpreted in the context of distance and time, represents the instantaneous speed or velocity. A positive derivative means increasing distance with time, a zero derivative indicates no movement (a stop), and changes in the derivative's value reflect acceleration or deceleration.

Relating Graphs of Functions and Their Derivatives

The graph of a function, like distance vs. time, is directly related to the graph of its derivative. The slope of the tangent line at any point on the distance graph corresponds to the instantaneous rate of change (or speed) at that time, which is depicted in the derivative's graph. Recognizing this relationship is key to understanding motion graphs.

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