1. Identify the Touch: A Tangent Line touches the curve at one local point.
2. Identify the Direction: The line must go the same way the curve is going at that moment.
3. Estimate the Slope: Use Rise/Run on the grid to estimate how steep the tangent is.
4. The Sign: Up = Positive. Down = Negative. Flat = Zero.
For each description, sketch the curve and the tangent line at the specified point.
The Peak: Draw a hill (parabola facing down). Draw the tangent line at the very top.
The Climb: Draw a curve that is going up steeply. Draw the tangent.
Using the function $y = x^2$. We want to find the slope at $x=2$.
The Wide Secant: Find the slope between $x=2$ and $x=3$.
The Narrow Secant: Find the slope between $x=2$ and $x=2.1$.
The Prediction: Based on the Narrow Secant, what whole number do you think the slope is approaching?
Why can't we just use the slope formula on the point $(2, 4)$ by itself? What happens if we try $\frac{4-4}{2-2}$?
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The Radius Rule: Draw a circle. Draw a Radius from the center to the edge. Draw a Tangent line at that edge point.
What is the angle between the Radius and the Tangent?
Take the function $y = \sin(x)$ (a wave).
If you zoom in very close to $x=0$, the curve looks like a straight line.
Task: Calculate $\sin(0.1)$ and $\sin(0.01)$ on your calculator. Compare them to $x$ (0.1 and 0.01).
Does $\sin(x) \approx x$ when $x$ is small?
Objective: Teach a younger student that "Straight" is hidden inside "Curved."
The Activity:
1. Draw a circle on the ground with chalk.
2. Have them walk the circle.
3. Tell them to "Freeze."
4. Lay a stick down next to their feet pointing the way they were facing. The stick is straight.
The Lesson: "Even when you walk in circles, each step is straight. God guides our steps (straight) to make our path (curve)."
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