Notes

We are given the data set (0,0),(1,2),(2,16),(3,52),(4,118),(5,223) and we are asked to fit these in a power model `y=ax^b ` : (1) The easiest way is to input the data in Excel or a graphing utility and perform a power regression yielding `y=2.04x^2.93 ` with a,b rounded to two decimal places. (2) To determine if the data set will fit a power model well we can plot (lnx,lny) and determine if the plot is approximately a straight line. Taking the natural logarithm of the coordinates yields the points (0,.6931),(.6931,2.773),(1.0986,3.9512),(1.3862,4.7707),(1.6094,5.4072). These points appear to lie on a line. Performing linear regression on these points yields the line y=.7146+2.929x So we have `lny=lna+blnx ` Exponentiating both sides with base e gives us: `y=e^(lna)*e^(blnx)=a*(e^(lnx))^b=ax^b ` where `a=e^(.7146) ` and b=2.929 So the power model is `y~~2.04x^(2.93) `

The parasympathetic nervous system is part of the autonomic nervous system whose activities in the body maintain both homeostasis and balance. It is active during the “rest and digest” physiological state. It normally serves to increase the glandular and intestinal activity, slow down the heart rate, decrease the blood pressure, as well as relax the sphincter muscles. It enhances food processing and the elimination of wastes under normal circumstances by stimulating and regulating digestion control. The neurons in this system make use of a neurotransmitter known as acetylcholine that allows for communication to take place from one cell to the next. The cholinergic fibers secrete acetylcholine. The parasympathetic nervous system, therefore, brings the body to a state of low activity where food digestion, as well as resting, occurs.

The decision by the Soviet Union to pull out of World War I was a decision that concerned the Allies. The Soviet Union signed the Treaty of Brest-Litovsk in 1918. The fighting against Germany in the Soviet Union would stop. The Soviet Union gave up a significant amount of land. Finland, Ukraine, and Georgia became independent. The Soviet Union gave Germany and Austria-Hungary land in Poland, Latvia, Lithuania, and Estonia. This treaty also impacted the Allies. Since Russia was one of the Allied Powers, the departure of Russia from fighting in World War I allowed Germany to concentrate its fighting on the Western Front. Germany was able to move troops from the Eastern Front to the Western Front. This gave Germany a better chance of winning the war. Fortunately, the United States had joined the war and helped the Allies. The Central Powers were eventually defeated, and the Treaty of Brest-Litovsk was nullified by the terms of the Versailles Treaty.

At first glance, it may seem as though Sandel, the fighter who opposes Tom King, is the antagonist of the story. Roughly two-thirds into the story, it even states  The third round began as usual, with Sandel doing all the leading, and delivering all the punishment.  However, even though Tom King does not fight back, it turns out Sandel is not the true antagonist of "A Piece of Steak," as evidenced by this quote a little bit further on: His gloves had already been removed, and Sandel, bending over him, was shaking his hand. He bore no ill-will toward the man who had put him out and he returned the grip with a heartiness that made his battered knuckles protest. "A Piece of Steak" is atypical of many stories because it does not have a person who is the antagonist; instead, the antagonist is simply old age. In the beginning of the story, Tom King sees Stowsher Bill crying in the dressing room and cannot fathom why such a prized fighter would be bent over in tears.  As t...

Both photosynthesis and rusting (formation of rust) are chemical reactions and represent chemical changes.  A chemical change involves a change in chemical composition of the reactants. That is, the products are different from the reacting species.  During the chemical reaction of photosynthesis, plants convert carbon dioxide and water into glucose and oxygen. The following chemical reaction takes place during photosynthesis: `6CO_2 + 6H_2O + sunlight -> C_6H_12O_6 + 6O_2` One can see that, in this process, the reactants are different from the products. Similarly, during the process of rusting, solid iron is converted to Fe(III) oxide, Fe2O3. This is the process which causes the loss of iron and the formation of the red powdery material known as rust.  In comparison to a chemical change, a physical change does not involve any change in the chemical composition of the reacting species.  Hope this helps. 

The part of the plot that you are asking about is in part one of Jean Craighead George's story.  At this point in the book, Julie is referred to by her Eskimo name, Miyax.  She is completely lost, which is bad in normal situations, but Miyax's situation is not normal.  She is on the North Slope of Alaska, which is about as barren of a wasteland as you can get.   The barren slope stretches for three hundred miles from the Brooks Range to the Arctic Ocean, and for more than eight hundred miles from the Chukchi to the Beaufort Sea. No roads cross it; ponds and laces freckle its immensity. In addition to being a barren wasteland, it's full of wolves.  That's scary, but Miyax is not scarred of the wolves . . . at all.  In fact, she believes that the wolves can help her.  She knows that the wolves are better hunters and killers than her, so Miyax is hoping to figure out a way to communicate with the wolves.  She wants to tell the wolves that she is hungry and needs some of th...

For the given integral problem: `int_(-pi/2)^(pi/2) cos(x)/(1+sin^2(x)) dx` , we can evaluate this applying indefinite integral formula: `int f(x) dx = F(x) +C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x)` `C` as the constant of integration. From the basic indefinite integration table, the  problem resembles one of the formula for integral of rational function:  `int (du)/(1+u^2)= arctan (u) +C` . For easier comparison, we may apply u-substitution by letting: `u = sin(x)` then `du =cos(x) dx` . Since `x=+-pi/2` then `u=+-1` `int_(-pi/2)^(pi/2) cos(x)/(1+sin^2(x)) dx                            `=int_-1^1 (du)/(1+u^2)`                           `= arctan(u) |_-1^1 `                              `=arctan(1)-arctan(-1)`                           `=pi/4- (-pi/4)`                                         `=pi/4+pi/4`                                         ` =(2pi)/4`                                         `= pi/2`