1. Average vs. Instant: Average requires two points ($y_2 - y_1$). Instant requires one point and a "Rule."
2. The Tangent Line: Touches the curve at exactly one point. Its slope is the derivative.
3. The Power Rule: If $f(x) = x^n$, then the derivative $f'(x) = nx^{(n-1)}$.
4. Constant Rule: The derivative of a constant number (like 5) is zero. (Still things have no speed!).
A traveler walks according to the function $d(t) = t^2$ (where $d$ is meters and $t$ is seconds).
The Interval:
A) Where is the traveler at $t = 1$ second?
B) Where is the traveler at $t = 4$ seconds?
C) What is the "Average Speed" from $t=1$ to $t=4$?
Use the Power Rule: $f(x) = x^n f'(x) = nx^{(n-1)}$.
The Speed Rule: If the position is $d(t) = t^2$, what is the "Speed Formula" ($d'$)?
The Speedometer: Using your new formula ($2t$), how fast is the traveler going at exactly $t = 2$ seconds? How about $t = 10$ seconds?
Look back at Part I. The average speed was 5 m/s. Is the instantaneous speed always 5 m/s? Or is it sometimes faster and sometimes slower? Why?
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The Slope of the Line: Draw a curve. Choose a point where the curve is going UP. Draw a tangent line at that point. Is the slope positive or negative?
The Power Rule works for each term in a sequence!
Example: $f(x) = x^3 + 5x f'(x) = 3x^2 + 5$.
Task: Find the speed formula ($H'$) for a plant that grows at $H(t) = 4t^2 + 10t + 5$. Then calculate the speed at $t = 3$.
Objective: Explain "Instantaneous" to a younger student using a camera.
The Activity:
1. Have them jump in the air.
2. Take a photo of them at the highest point.
3. Ask: "In the photo, are you moving? No. But were you moving in real life? Yes!"
The Lesson: "Math can see the movement even inside the 'still' photo. It can tell us exactly how fast you were going at the moment I clicked the button."
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