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 particular, two of the m
...
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 particular, 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.
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