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by visual and auditory illusions or hallucinations and paranoid delusions (a veritable psychosis) are variable features These psychologic disturbances may appear in many contexts Confusion, as de ned in this way, is an essential ingredient of the singular state called delirium (discussed further on), in which a agitation, hallucinations, and sometimes convulsions and tremor accompany the core confusional state Also, as pointed out in Chap 17, a confusional state may appear at any stage in the evolution and resolution of a number of diseases that lead to drowsiness, stupor, and coma typically in the metabolic encephalopathies but also in diseases affecting the regions of the brain that maintain normal arousal.

It should be pointed out that confusion is also a characteristic feature of the chronic syndrome of dementia, where it is the product of a progressive failure of language, memory, and other intellectual functions; here it is mainly the long-standing and progressive nature of the mental confusion that differentiates the process from the acute confusional and delirious states that carry quite different implications Intense emotional disturbances, of either manic or depressive type, may interfere with attentiveness and coherence of thinking and thereby produce an apparent confusional state Finally, special forms of confusion may appear in association with certain focal cerebral lesions, particularly of the frontal, parietal, and sometimes the temporal lobe association areas Then, instead of a global inattention and incoherence, there may be unilateral neglect of self or of the environment, an inability to identify persons or objects, and sensorimotor defects.

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Yet another unique form of confusion arises as a result of disordered language function, which apparently alters the stream of thought; this is usually a consequence of lesions in the specialized language areas of the left temporal lobe The many mental and behavioral aberrations that are seen in confused patients, and their occurrence in various combinations and clinical contexts, make it unlikely that they all derive from a single elementary psychologic abnormality such as a disturbance of attention While attention is near the core of confusion, phenomena as diverse as drowsiness and stupor, hallucinations and delusions, disorders of perception and registration, impersistence and perseveration, and so forth are not easily reduced to a disorder of one psychologic or physiologic mechanism More likely, a number of separable disorders of function are involved, all acute and usually reversible.

Indeed, one view of the confusional state that we nd attractive envisions confusion as a loss of the integrative functions among the elementary and localizable cerebral functions such as symbolic language, memory retrieval, and apperception All of these are often included under the rubric of the confusional states, for want of a better term We have reserved the term delirium, in distinction to the use of this term in psychiatric parlance, to denote a special type of confusional state In addition to many of the negative elements mentioned above, delirium is marked by a prominent disorder of perception; hallucinations and vivid dreams; a kaleidoscopic array of strange and absurd fantasies and delusions; inability to sleep; a.

ax 2 + bx + c = 0 If the discriminant is a positive real number, then the associated quadratic has two real roots. If the discriminant is equal to 0, then the quadratic has one real root with multiplicity 2. There are plenty of quadratic equations in which the discriminant is a negative real number. Here s an example: 4x 2 4x + 36 = 0 This is almost exactly the same equation as we solved in the second example above. The only difference is that the second coefficient is 4 rather than 24. The discriminant here is b 2 4ac = 42 4 4 36 = 16 576 = 560 When we apply the quadratic formula, we must take the square root of 560. That s an imaginary number! A negative discriminant gives us imaginary or complex roots. We ll explore situations like this in the next chapter. We ll also see what happens when one or more of the coefficients in a quadratic equation are imaginary or complex.

ln Chap. 15 (n - l)th-order polynomialswere used to interpolatebetweenn data points. polynomial. This For example, for eight points, we can derive a perfect seventh-order (at curve would captureall the meanderings leastup to and including seventhderivatives) suggested the points. However,thereare caseswhere thesefunctionscan lead to erroby neous results becauseof round-off error and oscillations.An alternativeapproachis to of apply lower-orderpolynomialsin a piecewisefashionto subsets datapoints. Such connectingpolynomialsare calledsplinefunctions. For example, third-order curves employed to connect each pair of data points are Thesefunctionscan be constructed that the connections so calledcubic spl.ines. between

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