Chapter 4

Show that the inflection points of the curve \(y=x \sin x\) lie on the curve \(y^{2}\left(x^{2}+4\right)=4 x^{2}.\)

A high-speed bullet train accelerates and decelerates at the rate of 4 \(\mathrm{ft} / \mathrm{s}^{2}\) . Its maximum cruising speed is 90 \(\mathrm{mi} / \mathrm{h}\) . (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?

\(53-56\) (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. $$f(x)=x \sqrt{x-x^{2}}$$

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. $$f(x)=4 x^{4}-32 x^{3}+89 x^{2}-95 x+29$$

Show that the curve \(y=(1+x) /\left(1+x^{2}\right)\) has three points of inflection and they all lie on one straight line.

\(53-56\) (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. $$f(x)=x-2 \cos x, \quad-2 \leqslant x \leqslant 0$$

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. $$f(x)=x^{6}-15 x^{5}+75 x^{4}-125 x^{3}-x$$

A rain gutter is to be constructed from a metal sheet of width 30 \(\mathrm{cm}\) by bending up one-third of the sheet on each side through an angle \(\theta .\) How should \(\theta\) be chosen so that the gutter will carry the maximum amount of water?

Show that the curves \(y=e^{-x}\) and \(y=-e^{-x}\) touch the curve \(y=e^{-x} \sin x\) at inflection points.

Between \(0^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\) , the volume \(V\) (in cubic centimeters of 1 \(\mathrm{kg}\) of water at a temperature \(T\) is given approximately by the formula $$V=999.87-0.06426 T+0.0085043 T^{2}-0.0000679 T^{3}$$ Find the temperature at which water has its maximum density.

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. $$f(x)=6 \sin x+\cot x, \quad-\pi \leqslant x \leqslant \pi$$

Show that \(\tan x>x\) for \(0< x<\pi / 2 .[\)Hint : Show that \(f(x)=\tan x-x\) is increasing on \((0, \pi / 2) . ]\)

An object with weight \(W\) is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle \(\theta\) with the plane, then the magnitude of the force is $$F=\frac{\mu W}{\mu \sin \theta+\cos \theta}$$ where \(\mu\) is a positive constant called the coefficient of friction and where 0\(\leqslant \theta \leqslant \pi / 2 .\) Show that \(F\) is minimized when \(\tan \theta=\mu\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. $$f(x)=e^{x}-0.186 x^{4}$$

(a) Show that \(e^{x} \geqslant 1+x\) for \(x \geqslant 0\) . (b) Deduce that \(e^{x} \geqslant 1+x+\frac{1}{2} x^{2}\) for \(x \geqslant 0\) . (c) Use mathematical induction to prove that for \(x \geqslant 0\) and any positive integer \(n\) $$e^{x} \geqslant 1+x+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}$$

A painting in an art gallery has height \(h\) and is hung so that its lower edge is a distance \(d\) above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best vicw? (In other words, where should the observer stand so as to maximize the angle \(\theta\) subtended at his eye by the painting?

A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function $$S(t)=-0.00003237 t^{5}+0.0009037 t^{4}-0.008956 t^{3} +0.03629 t^{2}-0.04458 t+0.4074$$ where \(t\) is measured in years since August of \(1993 .\) Estimate the times when sugar was cheapest and most expensive during the period \(1993-2003 .\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. $$f(x)=1+\frac{1}{x}+\frac{8}{x^{2}}+\frac{1}{x^{3}}$$

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three \(x\) -intercepts \(x_{1}, x_{2},\) and \(x_{3},\) show that the \(x\) -coordinate of the inflection point is \(\left(x_{1}+x_{2}+x_{3}\right) / 3.\)

The Hubble Space Telescope was deployed April \(24,1990,\) by the space shuttle Discovery. A model for the velocity of the shuttle during this mission, from liftoff at \(t=0\) until the solid rocket boosters were jettisoned at \(t=126 \mathrm{s},\) is given by $$v(t)=0.001302 t^{3}-0.09029 t^{2}+23.61 t-3.083$$ (in feet per second). Using this model, estimate the absolute maximum and minimum values of the acceleration of the shuttle between liftoff and the jettisoning of the boosters.

Produce graphs of \(f\) that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. $$f(x)=\frac{1}{x^{8}}-\frac{2 \times 10^{8}}{x^{4}}$$

For what values of \(c\) does the polynomial \(P(x)=x^{4}+c x^{3}+x^{2}\) have two inflection points? One inflection point? None? Illustrate by graphing \(P\) for several values of \(c .\) How does the graph change as \(c\) decreases?

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. $$f(x)=\sqrt{x^{4}+c x^{2}}$$

Prove that if \((c, f(c))\) is a point of inflection of the graph of \(f\) and \(f^{\prime \prime}\) exists in an open interval that contains \(c\) , then \(f^{\prime \prime}(c)=0 .\) [Hint. Apply the First Derivative Test and Fermat's Theorem to the function \(g=f^{\prime} . ]\)

When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. \(X\) -rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity \(v\) of the airstream is related to the radius \(r\) of the trachea by the equation $$v(r)=k\left(r_{0}-r\right) r^{2} \quad \frac{1}{2} r_{0} \leqslant r \leqslant r_{0}$$ where \(k\) is a constant and \(r_{0}\) is the normal radius of the trachea. The restriction on \(r\) is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than \(\frac{1}{2} r_{0}\) is prevented (otherwise the person would suffocate). (a) Determine the value of \(r\) in the interval \(\left[\frac{1}{2} r_{0}, r_{0}\right]\) at which \(v\) has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of \(v\) on the interval? (c) Sketch the graph of \(v\) on the interval \(\left[0, r_{0}\right] .\)

Show that 5 is a critical number of the function \(g(x)=2+(x-5)^{3}\) but \(g\) does not have a local extreme value at \(5 .\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. $$f(x)=x^{3}+c x$$

Show that if \(f(x)=x^{4},\) then \(f^{\prime \prime}(0)=0,\) but \((0,0)\) is not an inflection point of the graph of \(f .\)

Prove that the function \(f(x)=x^{101}+x^{51}+x+1\) has neither a local maximum nor a local minimum.

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. $$f(x)=e^{-c / x^{2}}$$

Show that the function \(g(x)=x|x|\) has an inflection point at \((0,0)\) but \(g^{\prime \prime}(0)\) does not exist.

If \(f\) has a local minimum value at \(c,\) show that the function \(g(x)=-f(x)\) has a local maximum value at \(c .\)

Suppose that \(f^{\prime \prime \prime}\) is continuous and \(f^{\prime}(c)=f^{\prime \prime}(c)=0,\) but \(f^{\prime \prime \prime}(c)>0 .\) Does \(f\) have a local maximum or minimum at \(c ?\) Does \(f\) have a point of inflection at \(c ?\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. $$f(x)=\ln \left(x^{2}+c\right)$$

Prove Fermat's Theorem for the case in which \(f\) has a local minimum at \(c .\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. $$f(x)=c x+\sin x$$

Suppose \(f\) is differentiable on an interval \(I\) and \(f^{\prime}(x)>0\) for all numbers \(x\) in \(I\) except for a single number \(c .\) Prove that \(f\) is increasing on the entire interval \(I.\)

Investigate the family of curves given by the equation \(f(x)=x^{4}+c x^{2}+x .\) Start by determining the transitional value of \(c\) at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of \(c\) at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.

For what values of \(c\) is the function $$f(x)=c x+\frac{1}{x^{2}+3}$$ increasing on \((-\infty, \infty) ?\)