Chapter 7

Let \(p\) be a positive constant. Suppose that \(X\) is a random variable with probability density function \(f(x)=(p+1)\) \(x^{p}\) for \(0 \leq x \leq 1\) a. Show that \(g(u)=1+u-2^{u} \quad\) satisfies \(\quad 0

A solid has as its base the ellipse \(x^{2}+4 y^{2}=16\). The vertical slices parallel to the line \(y=2 x\) are equilateral triangles. Find the volume.

Du Nouy's equation for the decreasing surface tension \(T\) of blood serum is $$ \frac{d T}{d t}=-k \frac{T}{\sqrt{t}} $$ for some constant \(k\). Solve this equation if \(T=\tau\) at \(t=0\).

Suppose that \(f\) is a continuous positive function on the unbounded interval \([a, \infty)\). Is it appropriate to make the definition $$ f_{\text {avg }}=\lim _{N \rightarrow \infty} \frac{1}{N-a} \int_{a}^{N} f(x) d x ? $$ Discuss why or why not.

The base of a solid \(S\) is the disk \(x^{2}+y^{2} \leq 25\). For each \(k \in[-5,5]\), the plane through the line \(x=k\) and perpendicular to the \(x y\) -plane intersects \(S\) in a square. Find the volume of \(S\).

Let \(x\) and \(y\) be the measures of two body parts with relative growth rates that are proportional to a common factor \(\Phi(t)\) $$ \frac{1}{x} \cdot \frac{d x}{d t}=\alpha \cdot \Phi(t) \quad \text { and } \quad \frac{1}{y} \cdot \frac{d y}{d t}=\beta \cdot \Phi(t) $$ Show that \(x\) and \(y\) satisfy the Huxley Allometry Equation \(y=k x^{p}\) for suitable constants \(k\) and \(p\)

Let \(\lambda\) be a positive constant. Show that \(f(x)=\) \(\lambda \exp (-\lambda x), 0 \leq x<\infty\) is a probability density function. Show that the mean of a random variable with probability density function \(f\) is \(1 / \lambda\).

A solid has as its base the region bounded by the parabola \(x-y^{2}=-8\) and the left branch of the hyperbola \(x^{2}-y^{2}-4=0 .\) The vertical slices perpendicular to the \(x\) -axis are squares. Find the volume of the solid.

The shape of a Bessel horn is determined by the equation $$ \frac{d a}{d x}=-\gamma \cdot \frac{a}{x+x_{0}} $$ where \(a(x)\) is the bore radius at distance \(x\) from the edge of the bell of the horn, and \(x_{0}\) and \(\gamma\) are positive constants. Solve for a as a function of \(x\). Determine \(a(x)\) for a trumpet with \(\gamma=0.7, a(0)=10,\) and \(a(30)=2 .\) Sketch the graph of \(y=a(x)\) for \(0 \leq x \leq 30\).

Old Boniface, he took his cheer, Then he drilled a hole in a solid sphere, Clear through the center straight and strong And the hole was just 10 inches long. Now tell us when the end was gained What volume in the sphere remained. Sounds like you've not been told enough. But that's all you need, it's not too tough.

Verify that \(y(x)=\exp \left(-x^{2}\right) \int_{0}^{x} \exp \left(t^{2}\right) d t,\) which is known as Dawson's Integral, is the solution of the initial value problem \(\frac{d y}{d x}=1-2 x y, y(0)=0\)

If \(\mu\) is any real number, and \(\sigma\) is any positive real number, then $$ f_{\mu, \sigma}(x)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right),-\infty

An open cylindrical beaker with circular base has height \(L\) and radius \(r\). It is partially filled with a volume \(V\) of a fluid. Consider the parameters \(L, r,\) and \(V\) to be constant. The axis of symmetry of the beaker is along the positive \(y\) -axis and one diameter of its base is along the \(x\) -axis. When the tank is revolved about the \(y\) -axis with angular speed \(\omega\), the surface of the fluid assumes a shape that is the paraboloid of revolution that results when the curve $$ y=h+\omega^{2} x^{2} /(2 g), \quad 0 \leq x \leq r $$ is revolved about the \(y\) -axis. This formula is valid for angular speeds at which the surface of the fluid has not yet touched the base or the mouth of the beaker. The number \(h=h(\omega)\) is in the interval \(\left[0, V /\left(\pi r^{2}\right)\right]\) and depends on \(\omega\). (When \(\omega=0,\) then \(h=V /\left(\pi r^{2}\right)\). As \(\omega\) increases, \(h\) decreases.) a. Find a formula for \(h(\omega)\). b. At what value \(\omega_{S}\) of \(\omega\) does spilling begin, assuming $$ \text { that } h(\omega)>0 \text { for } \omega>\omega_{S} ? $$ c. At what value \(\omega_{B}\) of \(\omega\) does the surface touch the bottom of the beaker, assuming that spilling does not $$ \text { occur for } \omega<\omega_{B} ? $$ d. As \(\omega\) increases, does the surface of the fluid touch the bottom of the beaker or the mouth of the beaker first?

For each \(x>0,\) there is a unique \(y>0\) such that \(y e^{y}=x\) Denote this value of \(y\) by \(W(x)\). Show that the function \(W\) (called Lambert's \(W\) function \()\) is a solution of the differential equation $$ \frac{d y}{d x}=\frac{y}{x(1+y)} $$

In each of Exercises \(85-88\), a function \(f\) is given. Let \(A_{f}(c)\) denote the average value of \(f\) over the interval \([c-1 / 4, c+1 / 4]\) Plot \(y=f(x)\) and \(y=A_{f}(x)\) for \(-1 \leq x \leq 1 .\) The resulting plot will illustrate the gain in smoothness that results from averaging. $$ f(x)=|x| $$

Suppose an object of mass \(m\) is propelled upwards from the surface of the earth with initial velocity \(v_{0}\). Suppose that the (downward) force of air resistance \(R(v)\) is proportional to the square of the speed: \(R(v)=-k \cdot v^{2},\) where \(k\) is a positive constant that carries the units of mass/ length. (This is the quadratic drag law.) Solve the initial value problem for motion: $$ m \frac{d v}{d t}=-k v^{2}-m g, \quad v(0)=v_{0} $$

In each of Exercises \(85-88\), a function \(f\) is given. Let \(A_{f}(c)\) denote the average value of \(f\) over the interval \([c-1 / 4, c+1 / 4]\) Plot \(y=f(x)\) and \(y=A_{f}(x)\) for \(-1 \leq x \leq 1 .\) The resulting plot will illustrate the gain in smoothness that results from averaging. $$ f(x)=\sqrt{|x|} $$

The region below the graph of \(y=\exp \left(-x^{2}\right),-1 \leq x \leq 1\) is rotated about the \(x\) -axis. Use Simpson's Rule to calculate the resulting volume to four decimal places.

Suppose that the force \(R\) of air resistance on a \(0.2 \mathrm{~kg}\) object thrown straight up with initial velocity \(28 \mathrm{~m} / \mathrm{s}\) is given by the quadratic drag law, $$ R(v)=-k v^{2} $$ where \(k=4 \times 10^{-6} \mathrm{~kg} / \mathrm{m}\). Find an equation for the time \(\tau\) at which the object begins to fall.

In each of Exercises \(85-88\), a function \(f\) is given. Let \(A_{f}(c)\) denote the average value of \(f\) over the interval \([c-1 / 4, c+1 / 4]\) Plot \(y=f(x)\) and \(y=A_{f}(x)\) for \(-1 \leq x \leq 1 .\) The resulting plot will illustrate the gain in smoothness that results from averaging. $$ f(x)=\left\\{\begin{array}{lll} 0 & \text { if } & x<0 \\ 1 & \text { if } & 0 \leq x \end{array}\right. $$

A flashlight reflector is made of an aluminum alloy whose mass density is \(3.74 \mathrm{~g} / \mathrm{cm}^{3} .\) The reflector occupies the solid region that is obtained when the region bounded by \(y=\) \(2.05 \sqrt{x}+0.496, y=2.05 \sqrt{x}+0.546, x=0,\) and \(x=2.80 \mathrm{~cm}\) is rotated about the \(x\) -axis. What is the mass of the reflector?

In each of Exercises \(85-88\), a function \(f\) is given. Let \(A_{f}(c)\) denote the average value of \(f\) over the interval \([c-1 / 4, c+1 / 4]\) Plot \(y=f(x)\) and \(y=A_{f}(x)\) for \(-1 \leq x \leq 1 .\) The resulting plot will illustrate the gain in smoothness that results from averaging. $$ f(x)=\left\\{\begin{array}{ccc} x^{2} & \text { if } & x<0 \\ x^{2}+x & \text { if } & 0 \leq x \end{array}\right. $$

The equation of the St. Louis Gateway Arch is $$ y=693.8597-34.38365\left(e^{k x}+e^{-k x}\right) $$ for \(k=0.0100333\) and \(-299.2239

The rate of elimination of alcohol from the bloodstream is proportional to the amount \(A\) that is present. That is, $$ \frac{d A}{d t}=-\frac{1}{k} A $$ where \(k\) is a time constant that depends on the drug and the individual. If \(k\) is \(1 / 2\) hour for a certain person, how long will it take for his blood alcohol content to reduce from \(0.12 \%\) to \(0.06 \% ?\)

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=\sqrt{x} \exp (-x) \quad I=[0,4] $$

The rate of elimination of alcohol from the bloodstream is proportional to the amount \(A\) that is present. That is, $$ \frac{d A}{d t}=-\frac{1}{k} A $$ where \(k\) is a time constant that depends on the drug and the individual. If \(k\) is \(1 / 2\) hour for a certain person, how long will it take for his blood alcohol content to reduce from \(0.12 \%\) to \(0.06 \% ?\)

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=\sin \left(\pi\left(x^{2}-x^{3}\right)\right) \quad I=[0,1] $$

Suppose that \(\alpha\) and \(\beta\) are positive constants. The differential equation $$ P^{\prime}(t)=\alpha \cdot e^{-\beta t} \cdot P(t) $$ for a positive function \(P\) is known as the Gompertz growth equation. (It is named for Benjamin Gompertz \((1779-1865)\), a self-educated scholar of wide- ranging interests.) Find an explicit formula for \(P(t) .\) Use your explicit solution to show that there is a number \(P_{\infty}\) (known as the carrying capacity) such that $$ \lim _{t \rightarrow \infty} P(t)=P_{\infty} $$ Show that the Gompertz growth equation may be written in the form $$ P^{\prime}(t)=k \cdot P(t) \cdot \ln \left(\frac{P_{\infty}}{P(t)}\right) $$

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=x /\left(16+x^{3}\right) \quad I=[2,4] $$

In forestry, the function \(E(T)=C \cdot e^{\alpha T /(T+\beta)}\) has been used to model water evaporation as a function of temperature \(T .\) Here \(\alpha\) and \(\beta\) are constants with \(\alpha>2\) and \(\beta>0 .\) Show that $$ E^{\prime}(T)=\frac{\alpha \beta}{(T+\beta)^{2}} E(T) $$ and $$ E^{\prime \prime}(T)=\frac{2((\alpha / 2-1) \beta-T)}{(T+\beta)^{4}} E(T) $$ Deduce that \(E\) is an increasing function of \(T\) with a sigmoidal (or \(S\) -shaped) graph. What is the point of inflection of the graph of \(E ?\)

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=\frac{\ln (x)}{1+x} \quad I=[1, e] $$

Suppose that \(h_{\infty}, k,\) and \(q\) are positive constants with \(q<1\) Show that the Chapman-Richards function, defined by $$ h(t)=h_{\infty} \cdot(1-\exp (-q k t))^{1 / q} $$ is the solution of the initial value problem $$ h^{\prime}(t)=k \cdot h(t) \cdot\left(\left(\frac{h_{\infty}}{h(t)}\right)^{q}-1\right), h(0)=0 $$ which is used in forest management to model tree growth. Here \(t\) represents time, and \(h\) measures tree height (or some other growth indicator).

Solve the given initial value problem for \(y(x)\). Determine the value of \(y(2)\). $$ d y / d x=\left(1+x^{2}\right) /\left(1+y^{4}\right) \quad y(0)=0 $$

Solve the given initial value problem for \(y(x)\). Determine the value of \(y(2)\). $$ d y / d x=\left(x^{2}\right) /(1+\sin (y)) \quad y(0)=0 $$

Solve the given initial value problem for \(y(x)\). Determine the value of \(y(2)\). $$ y^{2} \cdot d y / d x=(1+y) /(1+2 x) \quad y(0)=0 $$

Solve the given initial value problem for \(y(x)\). Determine the value of \(y(2)\). $$ d y / d x=x /\left(1+e^{y}\right) \quad y(0)=0 $$

Suppose that in a certain ecosystem, \(x \cdot 100\) denotes the number of predators, and \(y \cdot 1000\) denotes the number of prey. Suppose further that the relationship between the two population sizes satisfies the following LotkaVolterra equation: $$ \frac{d y}{d x}=\frac{y \cdot(6-2 x)}{x \cdot(4 y-5)} $$ Separate variables, and solve this equation. If the initial prey population is \(1500,\) and the initial predator population is \(200,\) what are the possible sizes of the population of prey when the predator size is \(300 ?\)