Graphical Understanding of Derivatives

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