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Chapter 3: Problem 88
Which functions are exponential? (a) \(3 x\) (b) \(3 x^{2}\) (c) \(3^{x}\) (d) \(2^{-x}\)
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If $$\$ 1$$ is invested in an account over a 10-year period, the amount in the account, where \(t\) represents the time in years, is given by \(A=1+0.075 \llbracket t \rrbracket\) or \(A=e^{0.07 t}\) depending on whether the account pays simple interest at \(7 \frac{1}{2} \%\) or continuous compound interest at \(7 \%\). Graph each function on the same set of axes. Which grows at a higher rate? (Remember that \(\llbracket t \rrbracket\) is the greatest integer function discussed in Section 1.6.)
Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 3 \\ \hline y & 1 & \frac{1}{4} \\ \hline \end{array} $$
The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 4 x-\log (12+\sqrt{x})=2$$
Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.
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