Chapter 4

\(23-36=\) Find the critical numbers of the function. $$h(p)=\frac{p-1}{p^{2}+4}$$

A cone-shaped paper drinking cup is to be made to hold 27 \(\mathrm{cm}^{3}\) of water. Find the height and radius of the cup that will use the smallest amount of paper.

Use the guidelines of this section to sketch the curve. $$y=2 x-\tan x, \quad-\pi / 2< x <\pi / 2$$

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(h(x)=5 x^{3}-3 x^{5}\)

Find \(f\) $$f^{\prime \prime}(\theta)=\sin \theta+\cos \theta, \quad f(0)=3, \quad f^{\prime}(0)=4$$

\(23-36=\) Find the critical numbers of the function. $$F(x)=x^{4 / 5}(x-4)^{2}$$

A cone with height \(h\) is inscribed in a larger cone with height \(H\) so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when \(h=\frac{1}{3} H .\)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(F(x)=x \sqrt{6-x}\)

Find \(f\) $$f^{\prime \prime}(t)=3 / \sqrt{t}, \quad f(4)=20, \quad f^{\prime}(4)=7$$

\(23-36=\) Find the critical numbers of the function. $$g(x)=x^{1 / 3}-x^{-2 / 3}$$

Use the guidelines of this section to sketch the curve. $$y=\sec x+\tan x, \quad 0< x <\pi / 2$$

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(G(x)=5 x^{2 / 3}-2 x^{5 / 3}\)

The figure shows the sun located at the origin and the earth at the point \((1,0) .\) (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU \(\approx 1.496 \times 10^{8} \mathrm{km} .\) . There are five locations \(L_{1}, L_{2}, L_{3}, L_{4},\) and \(L_{5}\) in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravitational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed at one of these libration points.) If \(m_{1}\) is the mass of the sun, \(m_{2}\) is the mass of the earth, and \(r=m_{2} /\left(m_{1}+m_{2}\right),\) it turns out that the \(x\) -coordinate of \(L_{1}\) is the unique root of the fifth-degree equation $$p(x)=x^{5}-(2+r) x^{4}+(1+2 r) x^{3}-(1-r) x^{2}$$ $$+2(1-r) x+r-1=0$$ and the \(x\) -coordinate of \(L_{2}\) is the root of the equation $$p(x)-2 r x^{2}=0$$ Using the value \(r \approx 3.04042 \times 10^{-6},\) find the locations of the libration points (a) \(L_{1}\) and \((\mathrm{b}) L_{2}\) .

Find \(f\) $$f^{\prime \prime}(x)=x^{-2}, \quad x>0, \quad f(1)=0, \quad f(2)=0$$

If a resistor of \(R\) ohms is connected across a battery of \(E\) volts with internal resistance \(r\) ohms, then the power (in watts) in the external resistor is $$P=\frac{E^{2} R}{(R+r)^{2}}$$ If \(E\) and \(r\) are fixed but \(R\) varies, what is the maximum value of the power?

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(C(x)=x^{1 / 3}(x+4)\)

\(23-36=\) Find the critical numbers of the function. $$f(\theta)=2 \cos \theta+\sin ^{2} \theta$$

Find \(f\) $$f^{\prime \prime}(t)=2 e^{t}+3 \sin t, \quad f(0)=0, \quad f(\pi)=0$$

For a fish swimming at a speed \(v\) relative to the water, the energy expenditure per unit time is proportional to \(v^{3} .\) It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current \(u(u

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(f(x)=\ln \left(x^{4}+27\right)\)

At \(2 : 00\) PM a car's speedometer reads 30 \(\mathrm{mi} / \mathrm{h}\) . At \(2 : 10 \mathrm{PM}\) it reads 50 \(\mathrm{mi} / \mathrm{h}\) . Show that at some time between \(2 : 00\) and \(2 : 10\) the acceleration is exactly 120 \(\mathrm{mi} / \mathrm{h}^{2}\)

Given that the graph of \(f\) passes through the point \((1,6)\) and that the slope of its tangent line at \((x, f(x))\) is \(2 x+1\) find \(f(2)\) .

\(23-36=\) Find the critical numbers of the function. $$f(x)=x^{2} e^{-3 x}$$

In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle \(\theta\) is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area \(S\) is given by $$S=6 s h-\frac{3}{2} s^{2} \cot \theta+\left(3 s^{2} \sqrt{3} / 2\right) \csc \theta$$ where \(s,\) the length of the sides of the hexagon, and \(h,\) the height, are constants. (a) Calculate \(d S / d \theta\) (b) What angle should the bces prefer? (c) Determine the minimum surface area of the cell (in terms of \(s\) and \(h )\) . Note: Actual measurements of the angle \(\theta\) in bechives have been made, and the measures of these angles seldom differ from the calculated value by more than \(2^{\circ}\) .

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(f(\theta)=2 \cos \theta+\cos ^{2} \theta, \quad 0 \leqslant \theta \leqslant 2 \pi\)

Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider \(f(t)=g(t)-h(t),\) where \(g\) and \(h\) are the position functions of the two runners.

Find a function \(f\) such that \(f^{\prime}(x)=x^{3}\) and the line \(x+y=0\) is tangent to the graph of \(f .\)

A boat leaves a dock at \(2 : 00 \mathrm{PM}\) and travels due south at a speed of 20 \(\mathrm{km} / \mathrm{h}\) . Another boat has been heading due east at 15 \(\mathrm{km} / \mathrm{h}\) and reaches the same dock at \(3 : 00 \mathrm{PM}\) . At what time were the two boats closest together?

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(S(x)=x-\sin x, \quad 0 \leqslant x \leqslant 4 \pi\)

\(23-36=\) Find the critical numbers of the function. $$f(x)=x^{-2} \ln x$$

A number \(a\) is called a fixed point of a function \(f\) if \(f(a)=a .\) Prove that if \(f^{\prime}(x) \neq 1\) for all real numbers \(x\) then \(f\) has at most one fixed point.

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=12+4 x-x^{2}, \quad[0,5]$$

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=1+\frac{1}{x}-\frac{1}{x^{2}}\)

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=5+54 x-2 x^{3}, \quad[0,4]$$

A woman at a point \(A\) on the shore of a circular lake with radius 2 mi wants to arrive at the point \(C\) diametrically opposite \(A\) on the other side of the lake in the shortest possible time. She can walk at the rate of 4 \(\mathrm{mi} / \mathrm{h}\) and row a boat at 2 \(\mathrm{mi} / \mathrm{h} .\) How should she proceed?

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=\frac{x^{2}-4}{x^{2}+4}\)

A particle is moving with the given data. Find the position of the particle. $$v(t)=\sin t-\cos t, \quad s(0)=0$$

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=2 x^{3}-3 x^{2}-12 x+1, \quad[-2,3]$$

Find an equation of the line through the point \((3,5)\) that cuts off the least area from the first quadrant.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=\sqrt{x^{2}+1}-x\)

A particle is moving with the given data. Find the position of the particle. $$v(t)=1.5 \sqrt{t}, \quad s(4)=10$$

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=x^{3}-6 x^{2}+5, \quad[-3,5]$$

At which points on the curve \(y=1+40 x^{3}-3 x^{5}\) does the tangent line have the largest slope?

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=\frac{e^{x}}{1-e^{x}}\)

A particle is moving with the given data. Find the position of the particle. $$a(t)=10 \sin t+3 \cos t, \quad s(0)=0, \quad s(2 \pi)=12$$

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=3 x^{4}-4 x^{3}-12 x^{2}+1, \quad[-2,3]$$

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve \(y=3 / x\) at some point?

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=e^{-x^{2}}\)

A particle is moving with the given data. Find the position of the particle. $$a(t)=t^{2}-4 t+6, \quad s(0)=0, \quad s(1)=20$$