Mean Value Theorem
The mean value theorem states that if f(x)=g(x) for all x in some interval containing a and b, then there 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 expressed algebraically as the algebraic slope between two points: f(b) – f(a) divided by b – a If a function is continuous and differentiable, then it has an instantaneous rate of change expressed as the derivative Thus, if f is continuous and differentiable on an interval [a, b], then f(a) must equal fb), 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. To determine the limit of a function, we will trace its graph. When you feel that this line is tangent to or 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 are parallel. 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 tangent line is parallel to the curve at that point. As the slope decreases and we approach the point at which 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 of change of .The mean value theorem states that if we find the average slope of a function between two points on the graph, there must be some point on the graph where the derivative equals that average. Average rate of change=instantaneous rate of change [Graph] (f(b)-f(a))/(b-a)=f^1(x) must be true if f is continuous and differentiable [a, b] mean value theorem. [Graph] Let us illustrate this point with a simple example. Consider the points (a,3) and b,3). Is it possible to connect these two points without using a horizontal tangent? If you have to do this continuously and differentiably (without any breaks), it turns out it’s not possible. There’s no way to connect these two points without having some sort of horizontal tangent somewhere in between. One method for finding the tangent slope of a curve at a given point is to draw a straight line through the point and the curve, as above. The slope of the chord connecting these two points will be exactly equal to the slope of the tangent at that point. In this case, the slope of the chord is 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 two endpoints. This must be true at some point in time because the slope between any two points on a curve must be 0.
