CALCULUS Higher Concepts of MathematicsCALCULUSMany practical problems can be solved using arithmetic and algebra; however,many other practical problems involve quantities that cannot be adequatelydescribed using numbers which have fixed values.EO 4.1 STATE the graphical definition of a derivative.EO 4.2 STATE the graphical definition of an integral.DynamicSystemsArithmetic involves numbers that have fixed values. Algebra involves both literal and arithmeticnumbers. Although the literal numbers in algebraic problems can change value from onecalculation to the next, they also have fixed values in a given calculation. When a weight isdropped and allowed to fall freely, its velocity changes continually. The electric current in analternating current circuit changes continually. Both of these quantities have a different valueat successive instants of time. Physical systems that involve quantities that change continuallyare called dynamic systems. The solution of problems involving dynamic systems often involvesmathematical techniques different from those described in arithmetic and algebra. Calculusinvolves all the same mathematical techniques involved in arithmetic and algebra, such asaddition, subtraction, multiplication, division, equations, and functions, but it also involves severalother techniques. These techniques are not difficult to understand because they can be developedusing familiar physical systems, but they do involve new ideas and terminology.There are many dynamic systems encountered in nuclear facility work. The decay of radioactivematerials, the startup of a reactor, and a power change on a turbine generator all involvequantities which change continually. An analysis of these dynamic systems involves calculus.Although the operation of a nuclear facility does not require a detailed understanding of calculus,it is most helpful to understand certain of the basic ideas and terminology involved. These ideasand terminology are encountered frequently, and a brief introduction to the basic ideas andterminology of the mathematics of dynamic systems is discussed in this chapter.DifferentialsandDerivativesOne of the most commonly encountered applications of the mathematics of dynamic systemsinvolves the relationship between position and time for a moving object. Figure 2 represents anobject moving in a straight line from position P_{1} to position P_{2}. The distance to P_{1} from a fixedreference point, point 0, along the line of travel is represented by S_{1}; the distance to P_{2}frompoint 0 by S_{2}.MA-05 Page 30 Rev. 0