As indicated above, the converse of this theorem is not true. in .NET framework

Generating QR-Code in .NET framework As indicated above, the converse of this theorem is not true.

As indicated above, the converse of this theorem is not true.
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DERIVATIVES
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RIGHT- AND LEFT-HAND DERIVATIVES The status of the derivative at end points of the domain of f , and in other special circumstances, is clari ed by the following de nitions. The right-hand derivative of f x at x x0 is de ned as
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0 f x0 lim h!0
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f x0 h f x0 h
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if this limit exists. Note that in this case h x is restricted only to positive values as it approaches zero. Similarly, the left-hand derivative of f x at x x0 is de ned as
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if this limit exists. In this case h is restricted to negative values as it approaches zero. 0 0 A function f has a derivative at x x0 if and only if f x0 f x0 .
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DIFFERENTIABILITY IN AN INTERVAL If a function has a derivative at all points of an interval, it is said to be di erentiable in the interval. In particular if f is de ned in the closed interval a @ x @ b, i.e. a; b , then f is di erentiable in the 0 0 interval if and only if f 0 x0 exists for each x0 such that a < x0 < b and if f a and f b both exist. If a function has a continuous derivative, it is sometimes called continuously di erentiable.
PIECEWISE DIFFERENTIABILITY A function is called piecewise di erentiable or piecewise smooth in an interval a @ x @ b if f 0 x is piecewise continuous. An example of a piecewise continuous function is shown graphically on Page 48. An equation for the tangent line to the curve y f x at the point where x x0 is given by y f x0 f 0 x0 x x0 7
The fact that a function can be continuous at a point and yet not be di erentiable there is shown graphically in Fig. 4-3. In this case there are two tangent lines at P represented by PM and PN. The 0 0 slopes of these tangent lines are f x0 and f x0 respectively.
DIFFERENTIALS Let x dx be an increment given to x. Then y f x x f x is called the increment in y f x . interval, then 8 If f x is continuous and has a continuous rst derivative in an 9
y f 0 x x  x f 0 x dx dx where  ! 0 as x ! 0. The expression dy f 0 x dx
10
is called the di erential of y or f(x) or the principal part of y. Note that y 6 dy in general. However if x dx is small, then dy is a close approximation of y (see Problem 11). The quantity dx, called the di erential of x, and dy need not be small.
DERIVATIVES
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Because of the de nitions (8) and (10), we often write dy f x x f x y f 0 x lim lim x!0 x!0 x dx x 11
It is emphasized that dx and dy are not the limits of x and y as x ! 0, since these limits are zero whereas dx and dy are not necessarily zero. Instead, given dx we determine dy from (10), i.e., dy is a dependent variable determined from the independent variable dx for a given x. Geometrically, dy is represented in Fig. 4-1, for the particular value x x0 , by the line segment SR, whereas y is represented by QR. The geometric interpretation of the derivative as the slope of the tangent line to a curve at one of its points is fundamental to its application. Also of importance is its use as representative of instantaneous velocity in the construction of physical models. In particular, this physical viewpoint may be used to introduce the notion of di erentials. Newton s Second and First Laws of Motion imply that the path of an object is determined by the forces acting on it, and that if those forces suddenly disappear, the object takes on the tangential direction of the path at the point of release. Thus, the nature of the path in a small neighborhood of the point of release becomes of interest. With this thought in mind, consider the following idea. Suppose the graph of a function f is represented by y f x . Let x x0 be a domain value at which f 0 exists (i.e., the function is di erentiable at that value). Construct a new linear function dy f 0 x0 dx with dx as the (independent) domain variable and dy the range variable generated by this rule. linear function has the graphical interpretation illustrated in Fig. 4-4. This
Fig. 4-4
That is, a coordinate system may be constructed with its origin at P0 and the dx and dy axes parallel to the x and y axes, respectively. In this system our linear equation is the equation of the tangent line to the graph at P0 . It is representative of the path in a small neighborhood of the point; and if the path is that of an object, the linear equation represents its new path when all forces are released. dx and dy are called di erentials of x and y, respectively. Because the above linear equation is valid at every point in the domain of f at which the function has a derivative, the subscript may be dropped and we can write dy f 0 x dx The following important observations should be made. y dy y , thus is not the same thing as . lim x!0 x dx x dy f x x f x f 0 x lim x!0 dx x
CHAP. 4]
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