Notes

Helen Keller faced many challenges in her life. She was deaf and blind, and for several years she could not communicate effectively. After she learned to communicate using the manual alphabet, Helen was still not satisfied. She wanted to learn how to speak. She attended a special program for this purpose. Helen wished to broaden her horizons because she thought, "one who is entirely dependent on the manual alphabet has always a sense of restraint, of narrowness" ( The Story of My Life , Chapter XIII). Shortly after Helen learned to speak, she was accused of plagiarism because of a short story she wrote. This accusation left a deep impact on her. She lost her confidence as a writer for a while. Helen also faced other challenges. Her father passed away, and she lost dear friends. Despite these challenges, Helen did not give up. She worked hard, even graduating from college with honors. There are many lessons to be learned from Helen Keller. One lesson is to never give up. Other...

To evaluate the given integral problem:` int 2/(7e^x+4)dx` , we may apply u-substitution using:  `u= e^x ` then `du = e^x dx` .  Plug-in `u = e^x` on `du= e^x dx` , we get: `du = u dx` or `(du)/u =dx` The integral becomes: `int 2/(7e^x+4)dx =int 2/(7u+4)* (du)/u`                          `=int 2/(7u^2+4u)du`  Apply the basic properties of integration:` int c*f(x) dx= c int f(x) dx` . `int 2/(7u^2+4u)du =2int 1/(7u^2+4u)du` Apply completing the square: `7u^2+4u =(sqrt(7)u+2/sqrt(7))^2 -4/7` `2int 1/(7u^2+4u)du =2int 1/((sqrt(7)u+2/sqrt(7))^2 -4/7)du` Let `v =sqrt(7)u+2/sqrt(7)` then `dv = sqrt(7) du`  or `(dv)/sqrt(7) = du` . The integral becomes:  `2int 1/(7u^2+4u)du =2 int 1/(v^2 -4/7) *(dv)/sqrt(7)` Rationalize the denominator: `2 int 1/(v^2 -4/7) *(dv)/sqrt(7) *sqrt(7)/sqrt(7)` `= 2 int (sqrt(7)dv)/ ( 7*(v^2 -4/7))` `=2 int (sqrt(7)dv)/ ( 7v^2 -4)` From the table of integrals, we may apply `int dx/(x^2-a^2) = 1/(2a)ln[(u-a)/(u+a)]+C`  Let:` w = sqrt(7)v` then `dw = sqrt(7) dv` `2int...

Despite living in the early 15th century, Joan of Arc's story remains incredibly relevant to young people today. Like most young people today, Joan was born into ordinary circumstances. She was a peasant, so it was highly unlikely that her life would have any noticeable effect on world history. However, Joan believed that she could help rescue France from its English invaders. Although many historians believe she was insane (she reportedly heard voices which told her God was calling her to save France), modern young people can emulate the courage and valor it took for her, a teenage peasant girl, to lead an army of men against the imposing English army. Unfortunately, Joan was eventually captured and executed, but her fame continues to endure. Modern young people may never lead their nation into battle, but, like Joan of Arc, they might have the opportunity to effect change in their communities. A modern young person could, for example, start a canned food drive to help stock a loc...

Before he even meets Gollum, Bilbo has already shown himself to be brave, resilient and capable of taking initiative.  When Chapter 5 begins, Bilbo wakes up in the dark, disoriented and alone. He has been dropped and left behind in the goblin tunnels. Though scared, he manages not to panic.  He draws his little sword, finds that it glows in the dark, and decides that to "go forward" is the "only thing to do." It is under these circumstances that he steps into the edge of Gollum's lake, which, as far as Bilbo is concerned, is the end of his road. Now he really has no idea what to do. It is then that Gollum approaches him and starts speaking to him. Bilbo draws his sword, which shows that he is  prepared (if not very well-equipped) to defend himself.  Gollum, nervous of the sword and curious about Bilbo, suggests they sit and tell riddles. Bilbo, the author tells us, is anxious to agree, until he found out more about the creature, whether he was quite alone, wheth...

What an interesting question! Bard's immediate reward for killing Smaug, the dragon, is the freedom of Dale, his city. However, forthcoming events transpire which allow Bard the ability to restore both Dale and Laketown to their former glories. To recap, immediately after Smaug's death, the men of Laketown clamor to make Bard king of Esgaroth (or Laketown). However, the Master of Laketown isn't too keen on this development. He argues that Laketown has always "elected masters from among the old and wise, and have not endured the rule of mere fighting men." Bard moves on, but he eventually meets with Thorin to bargain for part of the treasure Smaug had hoarded when he was alive. Bard reasons that since Smaug had stolen much of the wealth from Dale, the people of Dale deserve some part of the treasure that the dwarves now claim for themselves. At this point, Bard also speaks up for the people of Laketown, whose city was destroyed by the dragon. Bard's position is...

To solve the equation `log_3(x)=log_9(6x)`, we may apply logarithm properties. Apply the logarithm property: `log_a(b)= (log_c(b))/log_c(a)` on `log_3(x)` , we get: `(log_9(x))/(log_9(3))=log_9(6x)` Let `3 =sqrt(9) = 9^(1/2)` `(log_9(x))/(log_9(9^(1/2)))=log_9(6x)` Apply the logarithm property: `log(x^n)= n*log(x) ` and `log_a(a)=1 ` on `log_9(9^(1/2))` . `(log_9(x))/(1/2log_9(9))=log_9(6x)` `(log_9(x))/(1/2*1)=log_9(6x)` `(log_9(x))/(1/2)=log_9(6x)` `log_9(x)*(2/1)=log_9(6x)` `2log_9(x)=log_9(6x)` Apply the logarithm property: `log(x*y)=log(x)+log(y)` on `log_9(6x)` . `2log_9(x)=log_9(6)+log_9(x)` `2log_9(x)-log_9(x)=log_9(6)` `(2-1)log_9(x)=log_9(6)` `log_9(x)=log_9(6)` Apply the logarithm property:`a^(log_a(x))=x` on both sides. `9^(log_9(x))=9^(log_9(6))` `x=6` Check: Plug-in `x=6` on `log_3(x)=log_9(6x).` `log_3(6)=?log_9(6*6)` `log_3(6)=?log_9(36)` `1.631~~1.631` Final Answer: `x=6` is a real solution for the equation `log_3(x)=log_9(6x)` .

Grandma Dowdel doesn't like the idea of feeding Mary Alice's cat table scraps because she believes that cats should be able to hunt for their own food. Also, since Grandma Dowdel has very little food to spare, table scraps are hard to come by. Additionally, the idea of cat food in a can is strange to Grandma Dowdel. It seems a little unnatural to her, considering that she's always believed all cats have natural hunter's instincts. Her personal philosophy about cats is that "they'd eat anything they could bite." In the story, Mary Alice's cat, Bootsie, eventually learns how to live like a country cat. She camps out in the cob-house and lies in wait for birds and field mice. After a time, Bootsie becomes "sleek and round," due to her country diet.