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Chapter 6: Problem 4
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} \ln \left(x^{2}\right) d x $$
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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 \(E(x)\).$$ f(t)=\left\\{\begin{array}{ll} \frac{2}{5} e^{-2 t / 5}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$
Graphical Analysis In Exercises \(\mathbf{6 1}\) and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'H么pital's Rule? \(f(x)=\sin 3 x, \quad g(x)=\sin 4 x\)
In Exercises 65 and 66, apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) \(\begin{array}{l} \underline{\text { Functions }} \\ f(x)=\ln x, \quad g(x)=x^{3} \end{array} \quad \frac{\text { Interval }}{\left[1,4\right]}\)
Graphical Analysis In Exercises 61 and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'H么pital's Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\)
The magnetic potential \(P\) at a point on the axis of a circular coil is given by \(P=\frac{2 \pi N I r}{k} \int_{c}^{\infty} \frac{1}{\left(r^{2}+x^{2}\right)^{3 / 2}} d x\) where \(N, I, r, k,\) and \(c\) are constants. Find \(P\)
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