Chapter6_ Inverse Functions

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Inverse Functions: 6 Exponential, Logarithmic, and Inverse Trigonometric Functions The common theme that links the functions of this chapter is that they occur as pairs of inverse functions. In part... icular, two of the most important functions that occur in mathematics and its applications are the exponential function and its inverse function, the logarithmic function . In this chapter we investigate their properties, compute their derivatives, and use them to describe exponential growth and decay in biology, physics, chemistry, and other sciences. We also study the inverses of trigonometric and hyperbolic functions. Finally, we look at a method (l’Hospital’s Rule) for computing difficult limits and apply it to sketching curves. There are two possible ways of defining the exponential and logarithmic functions and developing their properties and derivatives. One is to start with the exponential function (defined as in algebra or precalculus courses) and then define the logarithm as its inverse. That is the approach taken in Sections 6.2, 6.3, and 6.4 and is probably the most intuitive method. The other way is to start by defining the logarithm as an integral and then define the exponential function as its inverse. This approach is followed in Sections 6.2*, 6.3*, and 6.4* and, although it is less intuitive, many instructors prefer it because it is more rigorous and the properties follow more easily. You need only read one of these two approaches (whichever your instructor recommends). fx a x tx loga x 383 © Pakmor / Shutterstock Exponential functions are used to describe the rapid growth of populations, including the bacteria pictured here. [Show More]

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