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Dynamic Systems - h1014v2_99
Graphical Understanding of Derivatives - h1014v2_101

Mathematics Volume 2 of 2
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CALCULUS Higher Concepts of Mathematics 3. Give the physical interpretation of the following equation relating the  force,  F,  applied  to  an  object,  its  mass  m,  its  instantaneous velocity v and time t. F mdv dt This  equation  includes  the  derivative  dv/dt;  the  derivative  of  the velocity with respect to time.   It is the rate of change of velocity with  respect  to  time.   The  force  applied  to  an  object  equals  the mass of the object multiplied by the rate of change of velocity with respect to time. 4. Give the physical interpretation of the following equation relating the acceleration a, the velocity v, and the time t. a dv dt This  equation  includes  the  derivative  dv/dt;  the  derivative  of  the velocity  with  respect  to  time.     It  is  a  rate  of  change.     The acceleration  equals  the  rate  of  change  of  velocity  with  respect  to time. Graphical Understanding of Derivatives A function expresses a relationship between two or more variables.   For example, the distance traveled by a moving body is a function of the body’s velocity and the elapsed time.   When a functional relationship is presented in graphical form, an important understanding of the meaning of derivatives can be developed. Figure 4 is a graph of the distance traveled by an object as a function of the elapsed time.   The functional relationship shown is given by the following equation: s = 40t (5-4) The  instantaneous  velocity  v,  which  is  the  velocity  at  a  given  instant  of  time,  equals  the derivative of the distance traveled with respect to time, ds/dt.   It is the rate of change of s with respect to t. MA-05 Page 34 Rev. 0







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