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The fundamental theorem of algebra states that in the eld of complex numbers every polynomial equation has at least one root As a consequence of this theorem, it can be proved that every nth degree polynomial has n roots in the complex eld When complex numbers are admitted, the polynomial theoretically may be expressed as the product of n linear factors; with our restriction to real numbers, it is possible that 2k of the roots may be complex In this case, the k factors generating them will be quadratic (The corresponding roots are in complex conjugate pairs) The polynomial x3 5x2 11x 15 x 3 x2 2x 5 illustrates this thought 2 Algebraic functions are functions y f x satisfying an equation of the form p0 x yn p1 x yn 1 pn 1 x y pn x 0 2 where p0 x ; .

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determines a number which depends on x. Thus, the expression de nes a function, called the derivative of f . De nition: The derivative f of f is the function de ned by the formula f (x) = lim f (x + h) f (x) h

; pn x are polynomials in x If the function can be expressed as the quotient of two polynomials, ie, P x =Q x where P x and Q x are polynomials, it is called a rational algebraic function; otherwise it is an irrational algebraic function 3 Transcendental functions are functions which are not algebraic, ie, they do not satisfy equations of the form (2)..

notation There are other notations traditionally used for the derivative: Dx f (x) and dy dx dy . We shall use whichever notation is most convenient dx

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If you want to create an elliptical arc, set the StartAngle and EndAngle properties after you create the full Ellipse object, as shown in Figure 8-3.

Note the analogy with real numbers, polynomials corresponding to integers, rational functions to rational numbers, and so on.

When a variable y represents f (x), the derivative is denoted by y , Dx y, or or customary in a given case.

TRANSCENDENTAL FUNCTIONS The following are sometimes called elementary transcendental functions. 1. 2. Exponential function: f x ax , a 6 0; 1. For properties, see Page 3.

The derivative is so important in all parts of pure and applied mathematics that we must devote a great deal of effort to nding formulas for the derivatives of various kinds of functions. If the limit in the above de nition exists, the function f is said to be differentiable at x, and the process of calculating f is called differentiation of f . EXAMPLES

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Logarithmic function: f x loga x, a 6 0; 1. This and the exponential function are inverse functions. If a e 2:71828 . . . ; called the natural base of logarithms, we write f x loge x ln x, called the natural logarithm of x. For properties, see Page 4. Trigonometric functions (Also called circular functions because of their geometric interpretation with respect to the unit circle): sin x; cos x; tan x sin x 1 1 1 cos x ; csc x ; sec x ; cot x cos x sin x cos x tan x sin x

(a) Let f (x) = 3x + 5 for all x. Then, f (x + h) = 3(x + h) + 5 = 3x + 3h + 5 f (x + h) f (x) = (3x + 3h + 5) (3x + 5) = 3x + 3h + 5 3x 5 = 3h 3h f (x + h) f (x) = =3 h h Hence, f (x) = lim

The following example creates a new Ellipse object by first setting the center axis, major axis, and radius ratio. It then gets a start and an end angle from the user to convert the closed Ellipse into an elliptical arc. Public Sub TestAddEllipse() Dim dblCenter(0 To 2) As Double Dim dblMajor(0 To 2) As Double Dim dblRatio As Double Dim dblStart As Double Dim dblEnd As Double Dim objEnt As AcadEllipse On Error Resume Next '' setup the ellipse parameters dblCenter(0) = 0: dblCenter(1) = 0: dblCenter(2) = 0 dblMajor(0) = 10: dblMajor(1) = 0: dblMajor(2) = 0 dblRatio = 0.5 '' draw the ellipse If ThisDrawing.ActiveSpace = acModelSpace Then Set objEnt = ThisDrawing.ModelSpace.AddEllipse(dblCenter, dblMajor, dblRatio) Else Set objEnt = ThisDrawing.PaperSpace.AddEllipse(dblCenter, dblMajor, dblRatio) End If objEnt.Update

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