Calculus (ii)—Integration and Differential Equations

The last chapter showed simple business problems being solved by differentiation, the calculation at particular points of the rate at which a function changed in relationship to a variable, where the rate itself was in continuous change. Each calculation involved an analytical or ‘narrowing down’ process, conveniently summarised in the relationship <math display='block'> <mrow> <munder> <mrow> <mi>F</mi><mi>U</mi><mi>N</mi><mi>C</mi><mi>T</mi><mi>I</mi><mi>O</mi><mi>N</mi><mo>&#x2192;</mo><mi>D</mi><mi>E</mi><mi>R</mi><mi>I</mi><mi>V</mi><mi>A</mi><mi>T</mi><mi>I</mi><mi>V</mi><mi>E</mi> </mrow> <mrow> <mo stretchy='false'>(</mo><mi>D</mi><mi>i</mi><mi>f</mi><mi>f</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy='false'>)</mo> </mrow> </munder> </mrow> </math>$$\mathop {FUNCTION \to DERIVATIVE}\limits_{(Differentiation)}$$ e.g. in Example 7.1, we have <math display='block'> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>&#x2192;</mo><mn>2</mn><mi>X</mi> </mrow> </math>$${X^2} \to 2X$$ This chapter considers the inverse relationship. We are given, for example 2X, and asked to state a function from which it was obtained by differentiation: and because we can recall or have recorded the process, we cab state X2 as the required value. It is the integral of 2X, the connotation of the term indicating a process of expansion or building up, synthetical, instead of analytical. Chapter 7 gives other example.