The Tangent Line Shows the Slope

In the picture

1. 1The blue curve is the graph of a function f(x).
2. 2The red dot is the chosen point (a, f(a)) on the curve.
3. 3The orange tangent line touches the curve at the red dot and follows its direction there.
4. 4The green triangle measures rise over run along the tangent, and that ratio is the derivative f'(a).

Examples

• A car’s speedometer shows the derivative of distance travelled with respect to time at that exact instant.
• How steep a mountain road is right under your bike’s wheel is the slope of the tangent to the road’s profile at that spot.

Quiz: The Tangent Line Shows the Slope (Age 16)

1. 1

   What does the derivative of a function at a point tell you?

   • How steep the curve is exactly at that point
   • How steep the curve is on average across a whole interval
   • How high the curve is at that point
2. 2

   To find the derivative at a point on a curve, which straight line do you use?

   • A line through that point and a second point further along the curve
   • The line that just touches the curve at that point and follows its direction
   • The line that crosses the curve at a right angle at that point
3. 3

   At a certain point, the tangent line goes up 6 units for every 2 units it goes across. What is the derivative at that point?

   • 1/3
   • 12
   • 3

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.