Slope and Concavity

Calculus, Slope and Concavity

Slope and Concavity

If f is differentiable at x, the slope of f at x is f′(x). This is a definition, but a convenient one, because it makes slope a continuous function as two points move together along the curve and merge into one. The difference quotient gives the slope of the chord, and as the chord becomes the tangent line, the difference quotient approaches the derivative.

The concavity of f at x is f′′(x), representing the change in slope. f is concave up at x if the concavity is positive, concave down if the concavity is negative. If f is concave up, the slope of the tangent line is increasing as we pass through x. In other words, the curve is bending upward. If f is concave down, the slope of the tangent line is decreasing as we pass through x. In other words, the curve is bending downward. If the concavity is 0, x is a point of inflection, or an inflection point. The curve is not bending downward or upward at that point. Perhaps it was bending up or down before or after x, but not at x.