Mon Prof Augmenté
Tangent from Secants
A curve is shown with one fixed point and a nearby moving point. As the nearby point slides toward the fixed point, the secant line through them turns toward the tangent, and the derivative is the slope of that tangent.
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Age 17: open this week
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In the picture
- A red point on the curve: the fixed location where the change is measured.
- A red straight line touching the curve at the red point: the tangent direction at that point.
- A yellow point on the curve near the red point: a second location used to form a secant.
- A dashed line through the red and yellow points: the secant slope over that interval.
- A faint dashed line close to the red tangent: a secant that has moved nearer the fixed point.
Examples
- A car’s speedometer shows instantaneous speed, like the slope of a distance-time curve at one moment.
- The steepness of a hill at one spot, read from a tangent to its profile.
Key takeaways
- A secant slope averages change between two points.
- The tangent slope at a point is the limit of secant slopes as the points come together.
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Three questions on this lesson. It has left the free week, so its quiz is for subscribers.
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