Chapter 3: Problem 35
Find y^{\prime \prime}\( if \)y=\csc x
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Finding Profit The monthly profit (in thousands of dollars) of a software company is given by \(P(x)=\frac{10}{1+50 \cdot 2^{5-0.1 x}}\) where x is the number of software packages sold. (a) Graph \(P(x)\) (b) What values of \(x\) make sense in the problem situation? (c) Use NDER to graph \(P^{\prime}(x) .\) For what values of \(x\) is \(P\) relatively sensitive to changes in \(x\) ? (d) What is the profit when the marginal profit is greatest? (e) What is the marginal profit when 50 units are sold 100 units, 125 units, 150 units, 175 units, and 300 units? (f) What is \(\lim _{x \rightarrow \infty} P(x) ?\) What is the maximum profit possible? (g) Writing to Learn Is there a practical explanation to the maximum profit answer?
Which is Bigger, \(\pi^{e}\) or \(e^{\pi} ?\) Calculators have taken some of the
mystery out of this once-challenging question. (Go ahead and check; you will
see that it is a surprisingly close call.) You can answer the question without
a calculator, though, by using he result from Example 3 of this section.
Recall from that example that the line through the origin tangent to the graph
of \(y=\ln x\) has slope 1\(/ e\) .
(a) Find an equation for this tangent line.
(b) Give an argument based on the graphs of \(y=\ln x\) and the tangent line to
explain why \(\ln x
Marginal Revenue Suppose the weekly revenue in dollars from selling x custom- made office desks is \(r(x)=2000\left(1-\frac{1}{x+1}\right)\) (a) Draw the graph of \(r .\) What values of \(x\) make sense in this problem situation? (b) Find the marginal revenue when \(x\) desks are sold. (c) Use the function \(r^{\prime}(x)\) to estimate the increase in revenue that will result from increasing sales from 5 desks a week to 6 desks a week. (d) Writing to Learn Find the limit of \(r^{\prime}(x)\) as \(x \rightarrow \infty\) How would you interpret this number?
In Exercises \(37-42,\) find \(f^{\prime}(x)\) and state the domain of \(f^{\prime}\) $$f(x)=\ln (2 x+2)$$
In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=\ln 2 \cdot \log _{2} x$$
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