Inverse, Exponential, and Logarithmic Functions
Preview of Inverse, Exponential, and Logarithmic Functions
This final unit in the study of calculus AB begins with a discussion of inverse functions and the algebraic and geometric relationship between a function f and its inverse f -1 . The geometric property of f -1 as a reflection of f across the line y = x is used to develop a formula for finding the derivative of f -1 from f .
Next is an introduction to the function
f (x) = e
x
and its inverse
f (x) = ln(x)
.
After a brief discussion of the properties of these functions, we see that the
derivative of
f (x) = e
x
is in fact
e
x
itself, and that the derivative of
f (x) = ln(x)
is the function
, which is the only power function
that could not be integrated by reversing the power rule. The derivatives of
e
x
and
ln(x)
are used to develop methods to differentiate functions where
x
is in the exponent. Finally, the general form of functions that exhibit
exponential growth or decay is presented.
