Mean Value Theorem The mean value theorem states that if f(x)=g(x) for all x in some interval containing a and b, thenthere exists at least one number c in this interval such that f(c)=g(c). Now let's talk about some definitions and equations.An average rate of change can be expressedalgebraically as the algebraic slope between two points: f(b) − f(a) divided by b − a If a function iscontinuous and differentiable, then it has an instantaneous rate of change expressed as thederivative .Thus, if f is continuous and differentiable on an interval [a, b], then f(a) must equal f(b), where f(a)is the average rate of change. So there's the average rate of change. That red line equals the average rate of change. Todetermine the limit of a function, we will trace its graph. When you feel that this line is tangent toor parallel with the curve, stop tracing. And right there, the line representing the derivative is parallel to the average rate of change.Because we're talking about slopes and when two slopes are equal to each other, they areparallel. And this happens again at another point. We'll follow the curve until the average rate of change equals zero, or until it feels like the tangentline is parallel to the curve at that point. As the slope decreases and we approach the point atwhich the curve levels off, the tangent line is parallel to the average rate of change. Thus f'(x) is defined by the formula f'(c) = (f(b) - f(a))/(b - a), where f(x) is the average rate ofchange of x.The mean value theorem states that if we find the average slope of a functionbetween two points on the graph, there must be some point on the graph where the derivativeequals that average.
Let us illustrate this point with a simple example. Consider the points (a,3) and (b,3). Is it possibleto connect these two points without using a horizontal tangent? If you have to do thiscontinuously and differentiably (without any breaks), it turns out it's not possible. There's no wayto connect these two points without having some sort of horizontal tangent somewhere inbetween.One method for finding the tangent slope of a curve at a given point is to draw a straight linethrough the point and the curve, as above. The slope of the chord connecting these two pointswill be exactly equal to the slope of the tangent at that point. In this case, the slope of the chordis 0, so we have a tangent line with a slope of 0.The mean value theorem states that the slope of a function is equal to the average of its twoendpoints. This must be true at some point in time because the slope between any two points ona curve must be 0.