Chapter 4

Use the Monotonicity Theorem to prove each statement if \(0\frac{1}{y}\)

Show that \(f(x)=\sin 2 x\) satisfies a Lipschitz condition with constant 2 on the interval \((-\infty, \infty)\). See Problem \(38 .\)

Consider a general quadratic curve \(y=a x^{2}+b x+c\). Show that such a curve has no inflection points.

$$ \int \frac{\sinh x}{1+\cosh x} d x $$

Show that the relative rate of change of any polynomial approaches zero as the independent variable approaches infinity.

A function \(f\) is said to be nondecreasing on an interval \(I\) if \(x_{1}

Show that the curve \(y=a x^{3}+b x^{2}+c x+d\) where \(a \neq 0\), has exactly one inflection point.

Prove that if the relative rate of change is a positive constant then the function must represent exponential growth.

Prove that, if \(f\) is continuous on \(I\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x) \geq 0\) on the interior of \(I\), then \(f\) is nondecreasing on \(I\). Similarly, if \(f^{\prime}(x) \leq 0\), then \(f\) is nonincreasing on \(I\).

Consider a general quartic curve \(y=a x^{4}+b x^{3}+\) \(c x^{2}+d x+e\), where \(a \neq 0\). What is the maximum number of inflection points that such a curve can have?

Prove that if the relative rate of change is a negative constant then the function must represent exponential decay.

Suppose that the cubic function \(f(x)\) has three real zeros, \(r_{1}, r_{2}\), and \(r_{3}\). Show that its inflection point has \(x\)-coordinate \(\left(r_{1}+r_{2}+r_{3}\right) / 3\). Hint: \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right)\left(x-r_{3}\right)\).

Prove that if \(f(x) \geq 0\) and \(f^{\prime}(x) \geq 0\) on \(I\), then \(f^{2}\) is nondecreasing on \(I\).

Assume that (1) world population continues to grow exponentially with growth constant \(k=0.0132,(2)\) it takes \(\frac{1}{2}\) acre of land to supply food for one person, and (3) there are \(13,500,000\) square miles of arable land in the world. How long will it be before the world reaches the maximum population? Note: There were \(6.4\) billion people in 2004 and 1 square mile is 640 acres.

In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), investigate how the extremum and inflection points depend on the value of \(c\). Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=\frac{c x}{4+(c x)^{2}} $$

The Census Bureau estimates that the growth rate \(k\) of the world population will decrease by roughly \(0.0002\) per year for the next few decades. In \(2004, k\) was \(0.0132\). (a) Express \(k\) as a function of time \(t\), where \(t\) is measured in years since \(2004 .\) (b) Find a differential equation that models the population \(y\) for this problem. (c) Solve the differential equation with the additional condition that the population in \(2004(t=0)\) was \(6.4\) billion. (d) Graph the population \(y\) for the next 300 years.

I have enough pure silver to coat 1 square meter of surface area. I plan to coat a sphere and a cube. What dimensions should they be if the total volume of the silvered solids is to be a maximum? A minimum? (Allow the possibility of all the silver going onto one solid.)

Use the Mean Value Theorem to prove that $$ \lim _{x \rightarrow \infty}(\sqrt{x+2}-\sqrt{x})=0 $$

In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), investigate how the extremum and inflection points depend on the value of \(c\). Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=\frac{1}{\left(c x^{2}-4\right)^{2}+c x^{2}} $$

Consider \(f(x)=A x^{2}+B x+C\) with \(A>0\). Show that \(f(x) \geq 0\) for all \(x\) if and only if \(B^{2}-4 A C \leq 0\).

Let \(f(x)=\sin x+\cos (x / 2)\) on the interval \(I=(-2,7)\). (a) Draw the graph of \(f\) on \(I\). (b) Use this graph to estimate where \(f^{\prime}(x)<0\) on \(I\). (c) Use this graph to estimate where \(f^{\prime \prime}(x)<0\) on \(I\). (d) Plot the graph of \(f^{\prime}\) to confirm your answer to part (b). (e) Plot the graph of \(f^{\prime \prime}\) to confirm your answer to part (c).

Use the Mean Value Theorem to show that $$ |\sin x-\sin y| \leq|x-y| $$

In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), investigate how the extremum and inflection points depend on the value of \(c\). Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=\frac{1}{x^{2}+4 x+c} $$

Let \(E\) be a differentiable function satisfying \(E(u+v)=E(u) E(v)\) for all \(u\) and \(v\). Find a formula for \(E(x)\).

Consider \(f(x)=A x^{3}+B x^{2}+C x+D\) with \(A>0\). Show that \(f\) has one local maximum and one local minimum if and only if \(B^{2}-3 A C>0\).

Suppose that in a race, horse \(A\) and horse \(B\) begin at the same point and finish in a dead heat. Prove that their speeds were identical at some instant of the race.

In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), investigate how the extremum and inflection points depend on the value of \(c\). Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=c+\sin c x $$

Prove the formula $$ \int\left[f(x) g^{\prime}(x)+g(x) f^{\prime}(x)\right] d x=f(x) g(x)+C $$

Using the same axes, draw the graphs for \(0 \leq t \leq 100\) of the following two models for the growth of world population (both described in this section). (a) Exponential growth: \(y=6.4 e^{0.0132 t}\) (b) Logistic growth: \(y=102.4 /\left(6+10 e^{-0.030 t}\right)\) Compare what the two models predict for world population in 2010,2040 , and 2090 . Note: Both models assume that world population was \(6.4\) billion in \(2004(t=0)\).

Let \(f^{\prime}(x)=x^{3}-5 x^{2}+2\) on \(I=[-2,4]\). Where on \(I\) is \(f\) increasing?

What conclusions can you draw about \(f\) from the information that \(f^{\prime}(c)=f^{\prime \prime}(c)=0\) and \(f^{\prime \prime \prime}(c)>0\) ?

Prove the formula $$ \int \frac{g(x) f^{\prime}(x)-f(x) g^{\prime}(x)}{g^{2}(x)} d x=\frac{f(x)}{g(x)}+C $$

An object thrown from the edge of a 100 -foot cliff follows the path given by \(y=-\frac{x^{2}}{10}+x+100\). An observer stands 2 feet from the bottom of the cliff. (a) Find the position of the object when it is closest to the observer. (b) Find the position of the object when it is farthest from the observer.

Let \(f^{\prime \prime}(x)=x^{4}-5 x^{3}+4 x^{2}+4\) on \(I=[-2,3]\). Where on \(I\) is \(f\) concave down?

Use the Mean Value Theorem to show that the graph of a concave up function \(f\) is always above its tangent line; that is, show that $$ f(x)>f(c)+f^{\prime}(c)(x-c), \quad x \neq c $$

Let \(g(x)\) be a function that has two derivatives and satisfies the following properties: (a) \(g(1)=1\); (b) \(g^{\prime}(x)>0\) for all \(x \neq 1\); (c) \(g\) is concave down for all \(x<1\) and concave up for all \(x>1\) (d) \(f(x)=g\left(x^{4}\right)\); Sketch a possible graph of \(f(x)\) and justify your answer.

The earth's position in the solar system at time \(t\) can be described approximately by \(P(93 \cos (2 \pi t), 93 \sin (2 \pi t))\), where the sun is at the origin and distances are measured in millions of miles. Suppose that an asteroid has position \(Q(60 \cos [2 \pi(1.51 t-1)], 120 \sin [2 \pi(1.51 t-1)])\). When, over the time period \([0,20]\) (i.e., over the next 20 years), does the asteroid come closest to the earth? How close does it come?

Prove that if \(|f(y)-f(x)| \leq M(y-x)^{2}\) for all \(x\) and \(y\) then \(f\) is a constant function.

Let \(H(x)\) have three continuous derivatives, and be such that \(H(1)=H^{\prime}(1)=H^{\prime \prime}(1)=0\), but \(H^{\prime \prime \prime}(1) \neq 0\). Does \(H(x)\) have a local maximum, local minimum, or a point of inflection at \(x=1\) ? Justify your answer.

An advertising flyer is to contain 50 square inches of printed matter, with 2-inch margins at the top and bottom and 1-inch margins on each side. What dimensions for the flyer would use the least paper?

Give an example of a function \(f\) that is continuous on \([0,1]\), differentiable on \((0,1)\), and not differentiable on \([0,1]\), and has a tangent line at every point of \([0,1]\).

In each case, is it possible for a function \(F\) with two continuous derivatives to satisfy the following properties? If so sketch such a function. If not, justify your answer. (a) \(F^{\prime}(x)>0, F^{\prime \prime}(x)>0\), while \(F(x)<0\) for all \(x\). (b) \(F^{\prime \prime}(x)<0\), while \(F(x)>0\). (c) \(F^{\prime \prime}(x)<0\), while \(F^{\prime}(x)>0\).

\text { Find } \int f^{\prime \prime}(x) d x \text { if } f(x)=x \sqrt{x^{3}+1}

John traveled 112 miles in 2 hours and claimed that he never exceeded 55 miles per hour. Use the Mean Value Theorem to disprove John's claim. Hint: Let \(f(t)\) be the distance traveled in time \(t\).

Use a graphing calculator or a CAS to plot the graphs of each of the following functions on the indicated interval. Determine the coordinates of any of the global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. Restrict the \(y\)-axis window to \(-5 \leq y \leq 5\). (a) \(f(x)=x^{2} \tan x ;\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (b) \(f(x)=x^{3} \tan x ;\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (c) \(f(x)=2 x+\sin x ;[-\pi, \pi]\) (d) \(f(x)=x-\frac{\sin x}{2} ;[-\pi, \pi]\)

Brass is produced in long rolls of a thin sheet. To monitor the quality, inspectors select at random a piece of the sheet, measure its area, and count the number of surface imperfections on that piece. The area varies from piece to piece. The following table gives data on the area (in square feet) of the selected piece and the number of surface imperfections found on that piece. $$ \begin{array}{ccc} \hline \text { Piece } & \begin{array}{c} \text { Area in } \\ \text { Square Feet } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Surface Imperfections } \end{array} \\ \hline 1 & 1.0 & 3 \\ 2 & 4.0 & 12 \\ 3 & 3.6 & 9 \\ 4 & 1.5 & 5 \\ 5 & 3.0 & 8 \\ \hline \end{array} $$ (a) Make a scatter plot with area on the horizontal axis and number of surface imperfections on the vertical axis. (b) Does it look like a line through the origin would be a good model for these data? Explain. (c) Find the equation of the least-squares line through the origin. (d) Use the result of part (c) to predict how many surface imperfections there would be on a sheet with area \(2.0\) square feet.

A car is stationary at a toll booth. Eighteen minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Sketch a possible graph of \(v\) versus \(t\). Sketch a possible graph of the distance traveled \(s\) against \(t\). Use the Mean Value Theorem to show that the car must have exceeded the 60 mile per hour speed limit at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

Translate each statement from the following newspaper column into a statement about derivatives. (a) In the United States, the ratio \(R\) of government debt to national income remained unchanged at around \(28 \%\) up to 1981 , but (b) then it began to increase more and more sharply, reaching \(36 \%\) during 1983 .

Each of the following functions is periodic. Use a graphing calculator or a CAS to plot the graph of each of the following functions over one full period with the center of the interval located at the origin. Determine the coordinates of any of the global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. (a) \(f(x)=2 \sin x+\cos ^{2} x\) (b) \(f(x)=2 \sin x+\sin ^{2} x\) (c) \(f(x)=\cos 2 x-2 \cos x\) (d) \(f(x)=\sin 3 x-\sin x\) (e) \(f(x)=\sin 2 x-\cos 3 x\)

Prove the formula $$ \begin{array}{r} \int f^{m-1}(x) g^{n-1}(x)\left[n f(x) g^{\prime}(x)+m g(x) f^{\prime}(x)\right] d x \\ =f^{m}(x) g^{n}(x)+C \end{array} $$