Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 24. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Interpreting the Derivative Meaning and Notation 211 EXAMPLE Burning Calories While Bicycling Suppose C s gives the number of calories used per mile of bicycle riding measured in calories mile as a function of speed s measured in miles hour .13 Then C s also written dC is the function that gives the instantaneous rate of change of calories mile with respect to speed. lim so the units for are cal ries mi. ds As o As ds mi hr C 12 7 tells us that when the speed of the cyclist is 12 mph the number of calories used per mile is increasing at a rate of 7 calories mile per mph. Practically speaking this means that if you currently ride at a pace of 12 mph increasing your speed by 1 mph will result in your burning approximately 7 more calories for every mile you Notice that C 12 7 does not give us any information about the calories being burned by riding 1 mile at 12 mph. C 12 would tell us that. EXERCISE Suppose we know not only that C 12 7 but also that C 12 22. Then we can estimate that C 13 29. Explain the reasoning behind this. A Dash of History. During the decade from 1665 to 1675 about a century before the American War of Independence both Isaac Newton in England and Gottfried Leibniz in Continental Europe developed the ideas of calculus. Newton began his work during years of turmoil. The years 1665-1666 were the years of the Great Plague which wiped out about one quarter of the population of London. The Great Fire of London erupted in 1666 destroying almost half of Cloistered in his small hometown Newton developed calculus in order to understand physical phenomena. The language of fluxions he used to explain his ideas reflected his scientific perspective his terminology is no longer in common usage. Newton used the notation y to denote a derivative. Although we will not adopt this notation it is still used by some physicists and engineers. Around the same time that Newton did his work Leibniz was also developing calculus. The language
