CHAPTER 3 APPLICATIONS OF DIFFERENTIATION One relatively new test available to physicians for diagnosing injuries and disease is the MRI. Magnetic resonance imaging (MRI) is used to visualize internal structures, such as torn cartilage in a knee. The ability to see the physical status of a knee or an internal organ without surgery is an invaluable aid to physicians and their patients. However, it still takes an experienced physician to distinguish the important features of an MRI from insignificant ones. If you have ever looked at an MRI or even a conventional x-ray, you have probably been amazed at the details that your physician could quickly identify. In the MRI below, can you identify any damage to the knee? Of course, it always helps to know what you are looking for. The ability to accurately read graphs is one of the primary goals of this chapter. By the end of section 3.6, you should have a good idea of what the significant features of a graph are. Although we will be looking only at two-dimensional graphs of functions, the language and skills that you acquire here will transfer to plots of seismic readings, sonar mappings of the ocean floor and other graphical displays of information that you may encounter. Most people do not recognize the vast amount of mathematical computation required to produce a viewable image from an MRI. In an MRI, magnetic fields and pulses of radio waves are used to determine the distribution of hydrogen atoms in the body (see Visualization by R. Friedhoff and W. Benzon for more details). The presence of hydrogen atoms, in turn, is deduced from the release of energy during the magnetization process. (This is a long way from a standard x-ray image!) By solving countless equations and performing lengthy calculations, a computer transforms the energy data into an accurate image of the interior of a human body. MRI of a knee. (FONAR Corporation) Likewise, it may surprise you how many calculations we must perform to draw an accurate graph of a function. At each stage of the graphing process, we must solve equations to identify significant features of the graph. Because of the central role that equation solving plays in this chapter, we devote the second section to a discussion of a powerful method that you can use to approximate solutions of difficult equations. But, you may ask, if a computer or calculator can do all of the calculations, why do you need to know what it's doing? One answer is that most computer algorithms are imperfect approximations that can occasionally result in significant errors on certain types of problems. By understanding how such algorithms work, you can anticipate and identify when a computer is in error. For example, the picture on the right below is a computer enhancement of the out-of-focus picture on the left. (Images courtesy of Vicom Systems.) Notice that the picture on the right shows a faint halo around the airplane. This is not a real ghost or aura or even sound waves, but instead a by-product of the computer algorithm used to sharpen the picture. If you understand the algorithm, you will not misinterpret the ghost. A ghost on an airplane is not serious, but the mathematics used to sharpen the picture is also used to produce MRIs, where a misinterpreted ghost could have serious consequences. (How would you feel if an MRI appeared to show a tumor that was not actually there?)