The slope of a curve at one point

In the picture

1. 1The fixed point on the curve, at x = a
2. 2A second point on the curve, far from the first, giving a secant line
3. 3A secant line whose second point is much closer, already almost like the tangent
4. 4The rise of the slope triangle: the change in y over a small run in x
5. 5The tangent line, y = f(a) + f'(a)(x − a), touching the curve at one point

Examples

• The speedometer of a car shows its speed at one instant, found from smaller and smaller distance-over-time intervals.
• The steepness of a ski slope at the exact spot where you stand, not the average over the whole run.

Quiz: The slope of a curve at one point (Age 16)

1. 1

   A secant line passes through the point at x = a and a second point at x = a + h on the curve. Which expression gives its slope?

   • (f(a+h) + f(a)) / h
   • (f(a+h) − f(a)) / h
   • (f(a+h) − f(a)) · h
2. 2

   As the second point slides closer and closer to the fixed point, what happens to the slopes of the secant lines?

   • They all shrink to zero
   • They grow larger and larger without limit
   • They settle on one value, the slope of the tangent
3. 3

   Which equation gives the tangent line to the curve y = f(x) at the point where x = a?

   • y = f(a) + f'(a)x
   • y = f(a) + f'(a)(x − a)
   • y = f'(a) + f(a)(x − a)

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.