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Chapter 8: Problem 104
Evaluate \(\int_{2}^{4} \frac{\sqrt{\ln (9-x)} d x}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}\)
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Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b]\), then $$ f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t $$
Find the area of the region bounded by the graphs of the equations. \(y=\sin ^{2} \pi x, \quad y=0, \quad x=0, \quad x=1\)
The Gamma Function \(\Gamma(n)\) is defined in terms of the integral of the function given by \(f(x)=x^{n-1} e^{-x}, \quad n>0\). Show that for any fixed value of \(n\) the limit of \(f(x)\) as \(x\) approaches infinity is zero.
The mean height of American men between 18 and 24 years old is 70 inches, and the standard deviation is 3 inches. An 18 - to 24 -year-old man is chosen at random from the population. The probability that he is 6 feet tall or taller is $$ P(72 \leq x<\infty)=\int_{72}^{\infty} \frac{1}{3 \sqrt{2 \pi}} e^{-(x-70)^{2} / 18} d x $$ (Source: National Center for Health Statistics) (a) Use a graphing utility to graph the integrand. Use the graphing utility to convince yourself that the area between the \(x\) -axis and the integrand is 1 . (b) Use a graphing utility to approximate \(P(72 \leq x<\infty)\). (c) Approximate \(0.5-P(70 \leq x \leq 72)\) using a graphing utility. Use the graph in part (a) to explain why this result is the same as the answer in part (b).
Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0\), and \(\lim _{x \rightarrow a} g(x)=-\infty\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty\).
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