91Ó°ÊÓ

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative. If \(f(x)=x^{2}+1,\) find \(\frac{d}{d x}[(f \circ f)(x)]\).

The circumference \(C\), in centimeters, of a healing wound is approximated by \(C(r)=6.28 r\) where \(r\) is the wound's radius, in centimeters. a) Find the rate of change of the circumference with respect to the radius. b) Explain the meaning of your answer to part (a).

Differentiate. $$y=\sqrt{(2 x-3)^{2}+1}$$

The population of a city grows from an initial size of 100,000 to a size \(P\) given by \(P(t)=100,000+2000 t^{2}\) where \(t\) is in years. a) Find the growth rate, \(d P / d t\) b) Find the population after 10 yr. c) Find the growth rate at \(t=10\) d) Explain the meaning of your answer to part (c).

Tongue-Tied Sauces, Inc., finds that the cost, in dollars, of producing \(x\) bottles of barbecue sauce is given by \(C(x)=375+0.75 x^{3 / 4} .\) Find the rate at which the average cost is changing when 81 bottles of barbecue sauce have been produced.

Graph each of the following. Then estimate the x-values at which tangent lines are horizontal. $$f(x)=10.2 x^{4}-6.9 x^{3}$$

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=x^{4}-3 x^{2}+1$$