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Chapter 5: Problem 22
In Exercises \(5-28\) , find the indefinite integral. $$\int \frac{x^{3}-4 x^{2}-4 x+20}{x^{2}-5} d x$$
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Population Growth A lake is stocked with 500 fish, and the population \(p\) is growing according to the logistic curve \(p(t)=\frac{10,000}{1+19 e^{-t / 5}}\) where \(t\) is measured in months. (a) Use a graphing utility to graph the function. (b) Find the fish populations after 6 months, 12 months, 24 months, 36 months, and 48 months. What is the limiting size of the fish population? (c) Find the rates at which the fish population is changing after 1 month and after 10 months. (d) After how many months is the population increasing most rapidly?
Using Properties of Exponents In Exercises \(107-110\) , find the exact value of the expression. \(5^{1 / \ln 5}\)
Analyzing a Graph Consider \(f(x)=x e^{-k x}\) for \(k>0 .\) Find the relative extrema and the points of inflection of the function.
(a) Show that \(\left(2^{3}\right)^{2} \neq 2^{(3)}\) . (b Are \(f(x)=\left(x^{x}\right)^{x}\) and \(g(x)=x^{\left(x^{\prime}\right)}\) the same function? Why or why not? (c) Find \(f^{\prime}(x)\) and \(g^{\prime}(x)\)
Analyzing a Function Let \(f(x)=\frac{\ln x}{x}\) .
(a) Graph \(f\) on \((0, \infty)\) and show that \(f\) is strictly decreasing on
\((e, \infty) .\)
(b) Show that if \(e \leq A
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