From secant lines to the tangent

In the picture

1. 1The fixed point: the place where we want the slope of the curve
2. 2A far secant: a line through two distant points that gives only a rough average slope
3. 3A close secant: the second point is near, so its slope is almost the true one
4. 4The tangent: the limit of the secants, and its slope is the derivative at the point

Examples

• A car’s speedometer shows its speed at one instant, which is the limit of the average speed 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

Quiz: From secant lines to the tangent (Age 16)

1. 1

   How is the tangent to a curve at a point obtained from secant lines?

   • It is the secant through the two points of the curve that are farthest apart
   • It is the limit of the secants as the second point approaches the fixed point
   • It is the average of all the secants that pass through the fixed point
2. 2

   What does the slope of a secant through two distant points of a curve tell you?

   • The exact derivative at the fixed point
   • The slope of the tangent at the second point
   • Only a rough average slope between the two points
3. 3

   In f'(a) = lim as h→0 of [f(a+h) − f(a)] / h, what does f'(a) measure?

   • The slope of the tangent to the curve at a
   • The value of the function at the point a
   • The slope of the secant between a and a+1

The answers, each with its explanation, are in the app. This lesson has left the free week, so its quiz is part of the subscription.