An object moving along a straight line has position given by {{MATH_241}}, where position is measured in meters and time in seconds. Find the (instantaneous) velocity of the object at time {{MATH_242}}. {{CMD_243}} Velocity is the instantaneous rate of change of position with respect to time. We are being asked to find {{MATH_244}}. We have to adjust the definition of the the derivative, namely, replace {{MATH_245}} with {{MATH_246}}, {{MATH_247}} Multiply by {{MATH_248}}, in order to rationalize the expression, {{MATH_249}} Now simplify the numerator, {{MATH_250}} Combine like-terms, {{MATH_251}} Take the limit as {{MATH_252}} goes to {{MATH_253}}, {{MATH_254}} The object has velocity {{MATH_255}} {{MATH_256}} at time {{MATH_257}}. {{CMD_258}} Below we can see the graph of {{MATH_259}} and the tangent line at {{MATH_260}}, with a slope of {{MATH_261}}. Notice, again, how the line fits the graph of the function {{MATH_262}} near the point {{MATH_263}}. {{CMD_264}}{{ENV_265}}{{CMD_266}} {{CMD_267}} {{CMD_268}} {{CMD_269}}