Tangent Line and Slope

In the picture

1. 1A point on the curve: the location where the tangent and secant lines meet.
2. 2The tangent line: it touches the curve at that point and follows its direction there.
3. 3A secant line to a farther point: its slope is the average change over a wider interval.
4. 4A secant line to a closer point: its slope moves toward the tangent slope.
5. 5A secant line to a very close point: its slope nearly equals the derivative at the point.

Examples

• A speedometer shows instantaneous speed, the derivative of distance with respect to time.
• The steepness of a curved slide at one spot is the slope of its tangent line there.

Quiz: Tangent Line and Slope (Age 16)

1. 1

   What does the tangent line at a point on a curve give?

   • The location where the tangent and secant lines meet
   • The average change over a wider interval
   • The direction of the curve at that point
2. 2

   As the second point of a secant line moves closer to the first point, what happens to the secant line’s slope?

   • It stays exactly the same
   • It becomes the average change over a wider interval
   • It moves toward the tangent slope
3. 3

   What is the derivative at a point on a curve?

   • The average change over a wider interval
   • The limit of secant-line slopes as the second point approaches that point
   • The location where the tangent and secant lines meet

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.