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Chapter 6: Problem 30
Verify the integration formula. $$ \int(\ln u)^{n} d u=u(\ln u)^{n}-n \int(\ln u)^{n-1} d u $$
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Consider the function \(h(x)=\frac{x+\sin x}{x}\). (a) Use a graphing utility to graph the function. Then use the zoom and trace features to investigate \(\lim _{x \rightarrow \infty} h(x)\). (b) Find \(\lim _{x \rightarrow \infty} h(x)\) analytically by writing \(h(x)=\frac{x}{x}+\frac{\sin x}{x}\) (c) Can you use L'H么pital's Rule to find \(\lim _{x \rightarrow \infty} h(x) ?\) Explain your reasoning.
Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\).
Think About It In Exercises 55-58, L'Hopital's Rule is used incorrectly. Describe the error.\(\begin{aligned} \lim _{x \rightarrow \infty} \operatorname{xec} \operatorname{sen} \frac{1}{x} &=\lim _{x \rightarrow \infty} \frac{\cos (1 / x)}{1 / x} \\ &=\lim _{x \rightarrow \infty} \frac{-\sin (1 / x)]\left(1 / x^{2}\right)}{-1 \times x^{2}} \\ &=0 \end{aligned}\)
A nonnegative function \(f\) is called a probability density function if \(\int_{-\infty}^{\infty} f(t) d t=1 .\) The probability that \(x\) lies between \(a\) and \(b\) is given by \(P(a \leq x \leq b)=\int_{a}^{b} f(t) d t\) The expected value of \(x\) is given by \(E(x)=\int_{-\infty}^{\infty} t f(t) d t\). Show that the nonnegative function is a probability density function, (b) find \(P(0 \leq x \leq 4),\) and (c) find \(E(x)\). $$ f(t)=\left\\{\begin{array}{ll} \frac{1}{7} e^{-t / 7}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$
Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. (c) Use a graphing utility to verify the result of part (b). FOR FURTHER INFORMATION For a geometric approach to this exercise, see the article "A Geometric Proof of \(\lim _{l \rightarrow 0^{+}}(-d \ln d)=0\) " by John H. Mathews in the College Mathematics Journal. To view this article, go to the website www.matharticles.com.
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