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Chapter 1: Problem 61
Determine whether the function has an inverse function. If it does, then find the inverse function. $$p(x)=-4$$
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(a) Write the linear function \(f\) such that it has the indicated function values and (b) Sketch the graph of the function. $$f(1)=4, \quad f(0)=6$$
Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=4-2 \sqrt{x}$$
The cost per unit in the production of an MP3 player is 60 dollars. The manufacturer charges 90 dollars per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by 0.15 dollars per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of 87 dollars per MP3 player for an order size of 120 ). (a) The table shows the profits \(P\) (in dollars) for various numbers of units ordered, \(x .\) Use the table to estimate the maximum profit. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Units, } x & 130 & 140 & 150 & 160 & 170 \\\\\hline \text { Profit, } P & 3315 & 3360 & 3375 & 3360 & 3315 \\\\\hline\end{array}$$ (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x ?\) (c) Given that \(P\) is a function of \(x,\) write the function and determine its domain. (Note: \(P=R-C\) where \(R\) is revenue and \(C\) is cost.)
Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(P\) varies directly as \(x\) and inversely as the square of \(y .\) \(\left(P=\frac{28}{3} \text { when } x=42 \text { and } y=9 .\right)\)
Finding a Mathematical Model In Exercises \(41-50\), find a mathematical model for the verbal statement. Newton's Law of Cooling: The rate of change \(R\) of the temperature of an object is directly proportional to the difference between the temperature \(T\) of the object and the temperature \(T_{e}\) of the environment in which the object is placed.
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