Mean-value theoremFor any sufficiently “smooth” continuous curve (one without corners), the average (mean) slope between two of its points (here, a and b) must be the same as the slope at some intermediate point (c).
mean-value theorem, theorem in mathematical analysis dealing with a type of average useful for approximations and for establishing other theorems, such as the fundamental theorem of calculus.
The theorem states that the slope of a line connecting any two points on a “smooth” curve is the same as the slope of some line tangent to the curve at a point between the two points. In other words, at some point the slope of the curve must equal its average slope (seefigure). In symbols, if the functionf(x) represents the curve, a and b the two endpoints, and c the point between, then [f(b) − f(a)]/(b − a) = f′(c), in which f′(c) represents the slope of the tangent line at c, as given by the derivative.
Although the mean-value theorem seemed obvious geometrically, proving the result without appeal to diagrams involved a deep examination of the properties of real numbers and continuous functions. Other mean-value theorems can be obtained from this basic one by letting f(x) be some special function.
derivative, in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems (dynamical systems) to obtain the rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.
Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more precisely, as the slope of the tangentline at a point. Its calculation, in fact, derives from the slope formula for a straight line, except that a limiting process must be used for curves. The slope is often expressed as the “rise” over the “run,” or, in Cartesian terms, the ratio of the change in y to the change in x. For the straight line shown in the figure, the formula for the slope is (y1 − y0)/(x1 − x0). Another way to express this formula is [f(x0 + h) − f(x0)]/h, if h is used for x1 − x0 and f(x) for y. This change in notation is useful for advancing from the idea of the slope of a line to the more general concept of the derivative of a function.
slope of a curveThe slope, or instantaneous rate of change, for a curve at a particular point (x0, f(x0)) can be determined by observing the limit of the average rate of change as a second point (x0 + h, f(x0 + h)) approaches the original point.
For a curve, this ratio depends on where the points are chosen, reflecting the fact that curves do not have a constant slope. To find the slope at a desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, in general, the ratio will represent only an average slope between the points, rather than the actual slope at either point (seefigure). To get around this difficulty, a limiting process is used whereby the second point is not fixed but specified by a variable, as h in the ratio for the straight line above. Finding the limit in this case is a process of finding a number that the ratio approaches as h approaches 0, so that the limiting ratio will represent the actual slope at the given point. Some manipulations must be done on the quotient [f(x0 + h) − f(x0)]/h so that it can be rewritten in a form in which the limit as h approaches 0 can be seen more directly. Consider, for example, the parabola given by x2. In finding the derivative of x2 when x is 2, the quotient is [(2 + h)2 − 22]/h. By expanding the numerator, the quotient becomes (4 + 4h + h2 − 4)/h = (4h + h2)/h. Both numerator and denominator still approach 0, but if h is not actually zero but only very close to it, then h can be divided out, giving 4 + h, which is easily seen to approach 4 as h approaches 0.
To sum up, the derivative of f(x) at x0, written as f′(x0), (df/dx)(x0), or Df(x0), is defined as if this limit exists.
Differentiation—i.e., calculating the derivative—seldom requires the use of the basic definition but can instead be accomplished through a knowledge of the three basic derivatives, the use of four rules of operation, and a knowledge of how to manipulate functions.
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