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Chapter 3: Problem 106
Use the definitions of increasing and decreasing functions to prove that \(f(x)=x^{3}\) is increasing on \((-\infty, \infty)\).
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In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{2 x}{1-x} $$
Consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates \((4,1),(5,6),\) and \((10,3) .\) A trunk line will run from the distribution center along the line \(y=m x,\) and feeder lines will run to the three factories. The objective is to find \(m\) such that the lengths of the feeder lines are minimized. Minimize the sum of the absolute values of the lengths of vertical feeder lines given by \(S_{2}=|4 m-1|+|5 m-6|+|10 m-3|\) Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines.
Timber Yield The yield \(V\) (in millions of cubic feet per acre) for a stand of timber at age \(t\) (in years) is \(V=7.1 e^{(-48.1) / t}\) (a) Find the limiting volume of wood per acre as \(t\) approaches infinity. (b) Find the rates at which the yield is changing when \(t=20\) years and \(t=60\) years.
Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=3 f(x)-3 \quad g^{\prime}(-5) \quad 0 $$
Prove that if \(p(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0}\) and \(q(x)=b_{m}
x^{m}+\cdots+b_{1} x+b_{0}\left(a_{n} \neq 0, b_{m} \neq 0\right),\) then \(\lim
_{x \rightarrow \infty} \frac{p(x)}{q(x)}=\left\\{\begin{array}{ll}0, & n
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