How does a tangent line show the steepness of a curved hill?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
In differential calculus, a tangent line represents the instantaneous rate of change of a function at a specific point, which geometrically corresponds to the steepness of the curve at that location.12
If you imagine a curved hill as the graph of a function, the tangent line acts as a straight line that perfectly matches the hill's inclination at a single point. The steepness of the hill at that exact spot is defined by the slope of this tangent line, calculated as the ratio of the vertical rise to the horizontal run.1
Mathematically, this slope is determined by the function's derivative. As you move along the hill:1
* If the derivative is positive, the hill is sloping upward.2
* If the derivative is negative, the hill is sloping downward.2
* If the derivative is zero, the hill has leveled out, such as at a peak or a valley.2
By analyzing the slope of the tangent line, you can determine the precise steepness and direction of the hill at any given point.2