Someone in your group walked in front of the motion sensor today — forward, pause, drift back — and the class watched a position–time graph draw itself in real time. Then came the request that turns walking into calculus: walk so the graph comes out straight; now walk so it curves upward. Position, how position changes, and how that changes: the whole page is those three layers.
If is position at time , then:
- Velocity is the derivative of position: . Its sign is direction; its size is speed.
- Acceleration is the derivative of velocity: — the rate of change of the rate of change, and your first meeting with a second derivative.
That double-prime is a genuinely new idea wearing familiar clothes. The Which One Doesn’t Belong set of motion graphs turned on it: two graphs can climb equally fast on average while one of them is easing off and the other is winding up.
Speeding up or slowing down?
Here is the trap the sensor walk exposed: acceleration is not “speeding up”. A ball thrown upward with has and constant — yet it slows on the way up and speeds up on the way down, with the same acceleration throughout. What matters is the agreement between velocity and acceleration:
| The object is | ||
|---|---|---|
| moving in the positive direction, speeding up | ||
| moving in the positive direction, slowing down | ||
| moving in the negative direction, slowing down | ||
| moving in the negative direction, speeding up |
Same signs, speeding up; opposite signs, slowing down. The ball turns around at , when — twenty metres up, momentarily still, accelerating the whole time.
The pattern behind motion
Motion is the curriculum’s favourite costume for derivatives, but the same two-layer reading works on any quantity that changes: population and growth rate, prices and inflation, a tank’s volume and its rate of flow. A question about “the moment inflation peaked” is a question about the derivative of a derivative, whoever is asking. The Smooth Landing task lives here — height, velocity, and acceleration all behaving at once — and the motion questions in Curve Sketching Practice rehearse the sign analysis above. The same first-and-second-derivative reading, applied to graphs instead of walkers, is Curve Sketching — it is the next page, and it is the same page.
Curriculum connection
B1.2
recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function
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B2.1
make connections between the concept of motion (i.e., displacement, velocity, acceleration) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically) Sample problem: Generate a displacement–time graph by walking in front of a motion sensor connected to a graphing calculator. Use your knowledge of derivatives to sketch the velocity–time and acceleration–time graphs. Verify the sketches by displaying the graphs on the graphing calculator.
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B2.2
make connections between the graphical or algebraic representations of derivatives and real-world applications (e.g., population and rates of population change, prices and inflation rates, volume and rates of flow, height and growth rates) Sample problem: Given a graph of prices over time, identify the periods of inflation and deflation, and the time at which the maximum rate of inflation occurred. Explain how derivatives helped solve the problem.
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