By Derek F. Lawden

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The roots are found by factorisation, completing the square, or by the quadratic formula:  b  b 2  4ac 2a The quantity b2 – 4ac has special significance and is called the discriminant: b 2  4ac  0 crosses x axis at two points, 2 roots. b 2  4ac  0 touches the x axis at a single point. b 2  4ac  0 lies completely above or below the x axis. Later we shall see that complex roots can be defined for the case of b2-4ac < 0. A quadratic equation can have no roots (lies completely above or below the x axis), one (touches the x axis) or two roots but no more than two since it can only cross the x axis (y = 0) no more than twice.

The tangent to some points may be maxima or minima but the derivative is undefined. maximum y maximum inflection minimum inflection minimum x The tangent to some points of inflection is not horizontal and these points are not stationary points. The tangent at some points of inflection is horizontal. Maxima and minima in a function are called turning points or extrema. The points at which the first derivative is zero or is undefined are called critical points. points Extrema of the function may occur at the end points of the function, in which case they are referred to as end point extrema.

Such processes are the rate of cooling of a body from a high temperature, the number of atoms that disintegrate per second in a radioactive material, and the growth of bacteria in a food culture. Mathematically, we express this as: dN  N dt where  is a constant which is positive for growth and negative for decay processes. We can separate the variables in this type of equation to express the relationship in terms of differentials: 1 dN  dt N Both sides of the equation are now integrated: Nt  No t 1 dN   dt N  0 ln N t  t  ln N o N t  N o e t In this equation, No is the value of the variable at t = 0, and Nt is the value of the variable at time t.

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