Chapter 1

In order to solve an inequality on a graphics calculator, we can graph a corresponding function and determine where it is positive and where it is negative. In Exercises \(51-52\) use the zoom feature of a graphics calculator to find an approximate solution of the inequality. Zoom until successive values of the \(x\) coordinate have identical first three digits. $$ \begin{aligned} &x^{3}+1 \geq-x-2 \text { (Hint: Let } f(x)=x^{3}+x+3 \text { , and determine } \\ &\text { where } f(x) \geq 0 . \text { ) } \end{aligned} $$

Solve the equation. $$ |6 x+5|=0 $$

Find a two-point equation of the given line. The line containing \((-2,4)\) and \((-1,3)\)

Find a formula for the function \(A\) that expresses the area of an equilateral triangle in terms of the length of one of its sides.

Let \(a\) be a real number and \(f(x)=a-x\). Show that \(f(f(x))=x\) for all \(x\).

During the 1950 's, scientists devised an experimental formula relating the energy \(E\) (in ergs) of an earthquake or explosion to the Richter scale magnitude \(M\) of the occurrence. The formula that arose is $$ \log E=11.4+1.5 M $$ During the Gulf War of 1991 , the United Nations forces used explosives amounting to 90 kilotons. Using the fact that a kiloton of explosives releases approximately \(10^{20}\) ergs of energy, determine the magnitude \(M\) of an earthquake that would release the same amount of encrgy.

In order to solve an inequality on a graphics calculator, we can graph a corresponding function and determine where it is positive and where it is negative. In Exercises \(51-52\) use the zoom feature of a graphics calculator to find an approximate solution of the inequality. Zoom until successive values of the \(x\) coordinate have identical first three digits. $$ 4 x^{3}+4 x<3 $$

Solve the equation. $$ |3-4 x|=2 $$

Find a two-point equation of the given line. The line containing \(\left(-\frac{3}{2},-\frac{1}{2}\right)\) and \(\left(\frac{1}{2}, 2\right)\)

Let \(f(x)=1 /(1-x)\). Show that \(f(f(f(x)))=x\) for all \(x\) different from 0 and 1 .

The noise level of a whisper is about 30 decibels, and that of ordinary conversation is around 50 decibels. Determine the ratio of the intensity of a whisper to that of conversation.

a. Let \(f(x)=x^{2}\) and \(g(x)=f(x+3)\). Using a suitable translation, sketch the graph of \(g\). b. Let \(f(x)=|x|\) and \(g(x)=f(x-2)\). Sketch the graph of \(g\)

Solve the equation. $$ |x|=|x|^{2} $$

Sketch the region in the plane satisfying the given conditions. \(x>0\)

Let \(f(x)=a x+b\), where \(a\) and \(b\) are constants, and let \(p\) be any real number. Show that if $$ g(x)=f(x+p)-f(x) $$ then \(g\) is a constant function.

Determine the number of decibels that corresponds to cach of the following intensities. a. \(10^{-12}\) (threshold of hearing) b. \(10^{-11}\) (rustling leaves) c. \(10^{-2}\) (power mower) d. 10 (jackhammer)

Let \(f\) be a function, and let \(g(x)=f(x)-3 .\) What is the relationship between the graphs of \(f\) and \(g\) ?

Solve the equation. $$ |x|=|1-x| $$

Sketch the region in the plane satisfying the given conditions. \(y \leq 0\)

In order to discover the height above ground of a ten-story apartment, Pat drops a ball from its balcony. If the ball hits the grass below after \(2.5\) seconds, determine the height (in meters) from which the ball was dropped.

Let \(c\) be any number, and define a function \(f\) by \(f(x)=c x\). Show that \(f(x+y)=f(x)+f(y)\) for all numbers \(x\) and \(y\).

Suppose that \(f\) is a function and that \(g(x)=f(x)+d\) for all \(x\) in the domain of \(f\), where \(d\) is a constant. What is the relationship between the graphs of \(f\) and \(g\) ?

Solve the equation. $$ |x+1|^{2}+3|x+1|-4=0 $$

Sketch the region in the plane satisfying the given conditions. \(y>x\) (Hint: Consider first the line \(y=x .\) )

Suppose you walk along a straight path from your dormitory to the mathematics building, which is 300 meters away. After 100 meters you pass the Student Union. Assuming that you walk at the rate of 1 meter per second, sketch the graph of your distance from the Student Union as a function of time \(t\), with \(t=0\) corresponding to the instant you leave your dormitory.

Let \(f\) be a function, and let \(g(x)=\frac{1}{2}[f(x)+f(-x)]\) and \(h(x)=\frac{1}{2}[f(x)-f(-x)]\) a. Show that \(g\) is an even function. b. Show that \(h\) is an odd function. c. Show that \(f=g+h .\) (Thus every function can be written as the sum of an even and an odd function.)

What is the ratio of the intensity of a given sound to that of one that is 100 decibels higher?

Let \(f(x)=|x-1|+|x+1|\). a. Determine whether the graph of \(f\) is symmetric with respect to either axis or the origin. b. Find alternative expressions for \(f(x)\) in the three cases \(x<-1,-1 \leq x \leq 1\), and \(x>1\), and use this information to sketch the graph of \(f\).

Solve the equation. $$ |x-2|^{2}-|x-2|=6 $$

Sketch the region in the plane satisfying the given conditions. \(x<-y\)

Let $$ T(x)=\left\\{\begin{array}{ll} 2 x & \text { for } 0 \leq x \leq \frac{1}{2} \\ 2(1-x) & \text { for } \frac{1}{2}

The human ear can just barely distinguish between two sounds if one is \(0.6\) decibels higher than the other. What is the ratio of the intensity of one sound to that of another sound that is lower than the first and is just barely distinguishable from the first sound?

Suppose the graph of an equation is symmetric with respect to both axes. Prove that it is symmetric with respect to the origin. Is the converse true?

Solve the equation. $$ |x+4|=|x-4| $$

Sketch the region in the plane satisfying the given conditions. \(x<0\) and \(y<0\)

A ball is thrown downward from the roof of a building 30 meters high, with an initial speed of 5 meters per second. a. Find the approximate height of the ball after \(1 / 2\) second and after 1 second. b. Approximately how long does it take for the ball to reach a window 10 meters above the ground?

Halley's Law states that the barometric pressure \(p(t)\) in inches of mercury at \(t\) miles above sea level is given by $$ p(t) \approx 29.92 e^{-0.2 t} \quad \text { for } t \geq 0 $$ Find the barometric pressure a. at sea level b. 5 miles above sea level c. 10 miles above sea level

Suppose the graph of an equation is symmetric with respect to the \(y\) axis and the origin. Is it necessarily symmetric with respect to the \(x\) axis? Explain.

Solve the equation. $$ |x-1|=|2 x+1| $$

Sketch the region in the plane satisfying the given conditions. \(x>0\) and \(y<0\)

A tank has the form of a right circular cylinder with hemispherical ends. Its volume \(V\) is 100 cubic meters. a. Find the length \(L\) of the cylinder in terms of the radius of the hemispheres. b. Find the length \(L\) (to the nearest centimeter) if the radius of the hemispheres is 2 meters. c. How much longer (to the nearest centimeter) would the cylindrical portion of the tank need to be if the radius of the hemispheres were 1 meter instead of 2 meters?

Let \(f(x)=c x-c x^{2}\), where \(c\) is a constant. By calculating a large number (up to 100 , if necessary) of iterates, make a reasonable guess for the behavior of the \(n\)th iterate of \(f\) at \(0.3\) for large values of \(n\). $$ 2.5 $$

Suppose that a living organism died \(t\) years ago. The number \(t\) frequently can be assessed by carbon-14 dating. If \(p\) percent of the original \(\mathrm{C}^{14}\) in the organism is now present, then \(t\) is given approximately by $$ t=-\frac{\ln (p / 100)}{0.000124} $$ Find the approximate age of each of the following objects with the given value of \(p\). a. mammal tusk, where \(p=1\) b. wooden post, where \(p=60\)

Suppose \(c>0 .\) If \(f(c-x)=f(c+x)\) for all \(x\), what property of symmetry does the graph of \(f\) have?

Solve the inequality. $$ |x-2|<1 $$

Sketch the region in the plane satisfying the given conditions. \(x<2\) and \(y>4\)

The period \(T\) of a pendulum of length \(L\) that swings under the influence only of gravity is given approximately by \(T=2 \pi \sqrt{L / g}\), where \(g=9.8\) meters per second squared. a. Write \(L\) as a function of \(T\). b. The length of the Foucault pendulum at the Smithsonian Institution is approximately \(21.8\) meters. Determine its period.

The gravitational force \(F\) that the earth exerts on a unit mass depends on whether the mass is inside or outside the earth. Let \(R\) denote the radius of the earth and \(r\) the distance of the unit mass from the center of the earth. Then \(F\) is given by $$ F=\left\\{\begin{array}{ll} \frac{G M r}{R^{3}} & \text { for } 0

Show that the points twice as far from the point \((2,-3)\) as from the point \((-1,0)\) form a circle, and find the center and radius of that circle.

Solve the inequality. $$ |x-4|<0.1 $$