Chapter 5

Predicting Wind Speed Wind speed typically varies in the first 20 meters above the ground. Close to the ground, wind speed is often less than it is at 20 meters above the ground. For this reason, the National Weather Service usually measures wind speeds at heights between 5 and 10 meters. For a particular day, let the formula \(f(x)=1.2 \ln x+2.3\) compute the wind speed in meters per second at a height \(x\) meters above the ground for \(x \geq 1 .\) (IMAGE CANNOT COPY) (a) Find the wind speed at a height of 5 meters. (b) Graph \(f\) in the window \([0,20,5]\) by \([0,7,1]\) Interpret the graph. (c) Estimate the height where the wind speed is 5 meters per second.

Rise in Sea Level Because of the greenhouse effect, the global sea level could rise due to partial melting of the polar ice caps. The table represents a function \(R\) that models this expected rise in sea level in centimeters for the year \(t\). (This model assumes no changes in current trends.) $$ \begin{array}{|rccccc} \hline t(\mathrm{yr}) & 1990 & 2000 & 2030 & 2070 & 2100 \\ \hline R(t)(\mathrm{cm}) & 0 & 1 & 18 & 44 & 66 \end{array} $$ (a) Is \(R\) a one-to-one function? Explain. (b) Use \(R(t)\) to find a table for \(R^{-1}(t)\). Interpret \(R^{-1}\)

Cooling an Object \(A\) pot of boiling water with a temperature of \(100^{\circ} \mathrm{C}\) is set in a room with a temperature of \(20^{\circ} \mathrm{C}\). The temperature \(T\) of the water after \(x\) hours is given by \(T(x)=20+80 e^{-x}\) (a) Estimate the temperature of the water after 1 hour. (b) How long did it take the water to cool to \(60^{\circ} \mathrm{C} ?\)

Explain how to find verbal, numerical, graphical, and symbolic representations of an inverse function. Give examples.

Can a one-to-one function have more than one \(x\) -intercept or more than one \(y\) -intercept? Explain.

If the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\) intersect at a point \((a, b),\) what can be said about this point? Explain.

If \(f(x)=a x^{2}+b x+c\) with \(a \neq 0,\) does \(f^{-1}(x)\) exist? Explain.