Higher Concepts of MathematicsCALCULUSThe ideas of differentials and derivatives are fundamental to the application of mathematics todynamic systems. They are used not only to express relationships among distance traveled,elapsed time and velocity, but also to express relationships among many different physicalquantities. One of the most important parts of understanding these ideas is having a physicalinterpretation of their meaning. For example, when a relationship is written using a differentialor a derivative, the physical meaning in terms of incremental changes or rates of change shouldbe readily understood.When expressions are written using deltas, they can be understood in terms of changes. Thus,the expressionDT, where T is the symbol for temperature, represents a change in temperature.As previously discussed, a lower case delta, d, is used to represent very small changes. Thus,dTrepresents a very small change in temperature. The fractional change in a physical quantityis the change divided by the value of the quantity. Thus, dT is an incremental change intemperature, and dT/T is a fractional change in temperature. When expressions are written asderivatives, they can be understood in terms of rates of change. Thus, dT/dt is the rate ofchange of temperature with respect to time.Examples:1. Interpret the expressionDV /V , and write it in terms of adifferential. DV /V expresses the fractional change of velocity.It is the change in velocity divided by the velocity. It can bewritten as a differential when DV is taken as an incrementalchange.may be written as D VVdVV2.Give the physical interpretation of the following equation relatingthe work W done when a force Fmoves a body through adistance x.dW = FdxThis equation includes the differentials dW and dx which can beinterpreted in terms of incremental changes. The incrementalamount of work done equals the force applied multiplied by theincremental distance moved.Rev. 0 Page 33MA-05

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