Mon Prof Augmenté
Secant lines close in on the tangent
Age 16 · Maths
Draw a line through a fixed point on a curve and a second point on the curve, then slide the second point closer and closer to the first. The lines settle on a single line, the tangent, and its slope is the derivative at that point.
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Age 16: open this week
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In the picture
- The fixed point (a, f(a)): where we measure the derivative
- A secant line through a second point: its slope is the average rate of change
- The tangent line: the limit of the secant lines as the second point gets closer
- Rise over run along the tangent: this slope equals f′(a)
Examples
- A car’s speedometer shows your speed at one instant, which is the limit of average speeds over shorter and shorter time intervals.
- On a hiking trail, the steepness right under your feet is the slope of the tangent to the elevation profile at that spot.
Key takeaways
- f′(a) is the limit of [f(a+h) − f(a)] / h as h approaches 0.
- The tangent at a point has slope f′(a): y = f(a) + f′(a)(x − a).
Quiz: Secant lines close in on the tangent (Age 16)
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Which expression defines the derivative f′(a)?
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What does the slope of a secant line through two points on a curve tell you?
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Which equation gives the tangent line to y = f(x) at the point (a, f(a))?
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