Without this you can’t understand how neural networks learn.
A common example is the speed of a car. We’ll do the same.
What is speed?
It’s the ratio of distance to time.
A car traveling at an average speed of 60 km/h — if we increase time \( t \) by one unit, distance \( s \) increases by 60 units. If we multiply \( t \) by 10, \( s \) is multiplied by 10 as well. The growth is linear; on a graph it’s a straight line.
But over time the car moves unevenly — speed increases and decreases.
Impossible
Distance traveled cannot decrease. Such an s(t) graph would be wrong:
Self-check questions
1. When is the speed at its maximum?
Hint
Look at the v(t) graph. Where does it reach its highest value?
2. What does the slope of s(t) at a given point represent?
Hint
Velocity is the ratio of distance change to time. How does that relate to slope?
3. Why does v(t) first increase and then decrease?
Hint
v is the derivative of s. How does the slope of s(t) change from start to end? Recall the S-shape.