The Liberal Party of Canada Essay Example | Topics and Well Written Essays - 1500 words

The Liberal Party of Canada - Essay Example On the one hand was the governing coalition of Liberal – Conservatives under the leadership of Sir John A. McDonald in Canada West and his French – Canadian counterpart, George – Etienne Cartier in Canada East. This disparate organization encompassed a number of distinct groups, many of which had potentially conflicting interests – Catholic and Protestants, English and French, Urban and Rural (Stephen Brooks, page 282) â€Å"The Liberal Party of Canada is always located somewhere in the middle of the political spectrum and often finds itself divided between a progressive wing and a conservative wing† (Patrick Malcolmson, Richard Myers, page 178) Base of Support The main support group for the classical liberals consisted of Industrialists, merchants and property owning- individuals while the chief supporters for the contemporary Liberal party includes several middle of the road advocacy groups within the feminist, environmental and multicultural movements, public – sector workers, middle – class intellectuals in the universities and the media, the national Liberal Party and the Bloc Quebecois; think – tanks including Canadian Policy Research Networks, the institute for Research on Public Policy and the Canada West Foundation. ( Stephen Brooks, page 40 - 41) According to the views propagated by R. B. McCullum, the Liberal Party was the â€Å"party of the middle class, with the support of the industrial workers† (Robert Kelly, page 43) Political Ideology One way of categorizing political ideas – perhaps the most popular way – is to describe them as being left wing/ right wing or centrist/ moderate.

Fixed and Floating Charge Essay Example | Topics and Well Written Essays - 1000 words

Fixed and Floating Charge - Essay Example It is imperative to observe that the stipulations fixed charge as well as equitable mortgage are every now and then utilized in replacement with each other( Rawlings, 1997). A floating charge is a charge that does not primarily fasten itself to a specific resource or property which implies the fact that the borrower is permitted to arrange the resources as he otherwise she desires devoid of having to obtain the authorization of the lender. It transmits to a varying group of present as well as prospective assets until it is which characteristically takes place when the borrower evades on the credit. The Floating charges are well thought-out as being valuable for the borrower for the reason that they permit the borrower to award the lender a safety interest, however at the equivalent time, the borrower is capable enough to utilize the asset within the standard course of trade devoid of disruption. Similar to the United Kingdom as well as other English-law provinces, the United States distinguishes the maintenance of an everlasting charge or else safety over liquid guarantee. No variations, on the other hand, are made amid the fixed as well as the floating charges within the U.S. regulation. If the groups inflowing into the safekeeping conformity used language comparable to that characteristically used in a floating charge, that affection would take place at the instance the agreement is penetrated into relatively than floating over the asset as it would underneath the English law. This appreciation of a floating security interest in surplus of the agreement pronounces the proposal that a fixed security interest is not automatically irreconcilable through the borrower having the autonomy to make use of as well as sell off the protected asset within the regular course of company. Nevertheless, beneath U.S. law, the contractual security conformity provides if the borrower is permitted to set out of the guarantee free of the safekeeping interest. If the borrower is authorized to advertise the goods, then the security interest within the goods is extinguished with the lender's interest persists during any earnings resulting from the auct ion. The Fixed charges are well thought-out particularly beneficial for the reason that they provide the creditor the precedence over privileged creditors along with holders of the floating charges when the borrower defaulting(Rawlings, 1997). Spectrum Plus: The House of Lords settled with the High Court in decision that it was an indispensable attribute of a fixed charge on book debts that the borrower's accurate to contract with mutually the debts themselves as well as their earnings be adequately constrained. In the expressions of Lord Scott of Foscote: The indispensable assessment of a book debt as a protection lies within the currency that can be attained from the debtor within the payment. This was without a doubt conflicting with the conception that a fixed charge over the debts had to confine the dealings mutually with the debts as well as their earnings. on the other hand, the House of Lords was not bound by New Bullas as well as decided

Gender Stereotypes In Trifles

Gender Stereotypes In Trifles The Nineteenth century was built around a society that contradict a strong point in the United States Constitution, that all men are created equal. Even though segregation was one of the main problems that got acknowledge, gender stereotype was often the case in most home throughout society. In the plays A Dolls House, by Henrik Ibsen, and Trifles, by Susan Glaspell, the characters show stereotype through gender. This includes the belief that women are lesser people, child like in their action, and in need of being controlled. Nora was the main female character in the play A Dolls House; she lived in a lavish home eating macaroons, drinking champagne and hosting banquets. Nora lived her life, in one sense, as a complete lie. She never thought for herself or had her own opinions. Noras father would tell her what he thought about everything, leaving her no opinion but his. If she did have an opinion of her own she kept quiet knowing he would not have agreed. She played his little doll until she moved in with Torvald, her husband. She felt as though she was passed from her father hands into Torvalds hands. Now she played the role of Torvalds little doll, pretending to take on his views of everything. This evidence of gender stereotype is shown in the way Torvald treats his wife Nora, and in the way Nora acts to please her husbands. In the first act of the play we come across the first instance of Torvald name calling his wife in a child like manner is that my little lark twittering out there?(Ibsen 796) This is the first of many names Torvald use to give his wife child like tendencies. Torvald also refer to Nora as a squirrel, a spendthrift, a songbird, and a goose. Torvald shows ownership of Nora by calling her my richest treasure, denoting his attitude toward Nora as his possession. This stereotypical oppression by Torvald serves a purpose of keeping women in their place, and keeping men on the top of the social structures. One can easily determine Nora as immature and childlike in her action, this stereotype is not only presented by Torvald, but by Nora as well. When Torvald ask Nora what did she want for Christmas she replies in a very child like manner. Nora speaking quickly you might give me money(Isben 798) when Torvald said yes Nora said Oh do Dear Torvald, please, please do! Then Ill wrap it up in beautiful gold paper and hang it on the Christmas tree. Wouldnt that be fun? (Isben 798) As one reads on Nora come home in say Oh yes, Torvald, we can squander a little now. Cant we ? Just a tiny, wee bit. Now that youve got a big salary and are going to make piles and piles of money. With this excerpt, we see a child-like attitude, not only in Noras manner of speaking with the statement Just a tiny, wee bit, but also in her attitude toward money and the unrealistic expectations of making piles and piles of money. The following example also shows Noras childish manner in her personal interactions with her husband. Her manner seems more like that of a favorite daughter, accustomed to getting her way, than that of a wife, also keeping with the stereotype concerning control by keeping oppression high. Through-out A Dolls House, men are seemingly in the dominant position, and they manipulate their power to control women in ideological sphere so that their own identity and social status may be retained and be acknowledged by the society. As exposed in A Dolls House, men are in a financially and ideologically superior position over women while women are kept in a subordinate position and are confined to their homes as they are not economically independent and have to rely on their husbands for support. What causes this situation to exist in the 19th-century Europe is the social context in which people held the notion that men were supposed to be responsible to their families and provide all the necessities that a family needed, while women were supposed to maintain their sacred duty of a good wife and mother. In the case of the male protagonist Helmer Torvald, his ideology is conditioned by social standards and his conception of manliness is based on mans social values. Gender stereotype was not only shown by the male writers, but also by the female writers as well. In the play Trifle by Susan Glaspells the stereotypes made are those of the women being concerned only with trifling things, that are loyal to the female gender, and that women must live under their spouses. Trifles is based on an actual event that took place in Iowa at the turn of the Nineteenth century. The play Trifles involves a murder case that explores gender relationships, power between the sexes, and the nature of truth. In the play Trifles, the victim in the suspect is not the main focus of gender stereotype: instead the gender stereotype is presented by the investigator in his associates, follow by their wives. In the opening act of the play the men enter the house, follow by their wives. This stereotypical action was to show the dominate status of men. In Trifles, the male characters make several assumptions concerning the female characters. These assumptions show the way in which the men view the women, and are purely gender stereotypical. The first assumption that of women being only concerned with trifling things, is seen beginning with line 120 when the sheriff say Well, can you beat the women! Held for murder and worryin about her preserves. I guess before were through she may have something more serious than her preserves to worry about. Well, women are used to worrying over trifles. These lines show the attitude toward women prevalent throughout the play. It is the mens nonchalance toward the small details that ultimately undermines the murder case. The second stereotype presented by the men concerning the women is that of being loyal to their own gender. In line 140, the County Attorney states, Ah, loyal to your sex, I see. But you and Mrs. Wright were neighbors. I suppose you were friends, too. This statement also makes an assumption concerning country women being neighborly. The County Attorney assumes that because the two women were neighbors, they were naturally friends, and spoke regularly. Because of the way the men treated the women they expresses discomfort at the mens violation of Mrs. Wright house. They didnt think about the law, instead they care more about the relationships than the rules. The action by the women shows a very stereotypical asset between the bonds they have for each other, instead of doing whats right by the law. The men try to diminish the female identity or role in the societal context by setting up in unfavorable image of female as ideologically submissive and dependent, that men are the subject or the absolute, and that women objects is existence and defined by males. The consequence of this stereotypical thinking causes the women to hide the truth from the men in Trifles. The women knew that Mr. Wright kept Mrs. Wright isolated from the world: Therefore the men remain oblivious to the action of the women hiding the truth. Their men vision and, their way of knowing narrows their focus in this play. In this case gender stereotype leads to a type of blindness on what happen in the farmhouse. If the men had discovered the evidence they would have had no doubt about on how to use it. The women discovered the evidence, but since gender stereotype was a factor it lead them to hold the truth from the men. As the women read Mrs. Wright story the understood it as if it was their story. I think the ir decision are based on gender, it springs from their maps of the territory, one fundamentally. The third stereotype the men make concerning the women can be found in line 159, when the Sheriff says I suppose anything Mrs. Peters does be all right, Do you want to see what Mrs. Peters is going to take in? Oh, I guess theyre not very dangerous things the ladies have picked out. No, Mrs. Peters doesnt need supervising. For that matter, a sheriffs wife is married to the law. It is here assumed that, because Mrs. Peters is the sheriffs wife, she will do only what she should. Mrs. Peters has been asked to bring a couple of things to Mrs. Wright in jail, and if she finds anything that may be of use to the case, she is asked to let them know. After Mrs. Peters and Mrs. Hale find the very evidence that could incarcerate Mrs. Wright, they proceed to hide it from the men, providing the greatest evidence for the loyalty among women issue, and departing from the assumptions of women being trifling and subservient. In Trifles, the women both conform to and depart from the assumptions made by the men. Mrs. Hale has found an important piece of evidence, a dead bird. The ladies decide not to allow the men to know of the true reason for the birds death. This departs from the stereotype depicting women as only being concerned with trifling things. As concerns the loyalty of one woman to another, evidence is found in line 366 that seems to allude to such a bond. Line 366 begins when the County Attorney say Is there a cat? Mrs. Hale glances in a quick covert way at Mrs. Peters. Mrs. Peters say Well, not now. Theyre superstitious, you know. They leave. These lines also show a departure from the assumption that women are subservient to their men in that they lie to the men. In line 411 one can see more evidence of the almost sisterly bond between women. Mrs. Hale says I might have known she needed help! I know how things can befor women. I tell you, its queer Mrs. Peters reply with We live close together and we live far apart. We all go through the same thingsits all just a different kind of the same thing Here one gets the idea of women depart from the stereotypical roles assigned to them by men.

Gathering at the River: Cruising on East Speedway :: Essays Papers

Gathering at the River: Cruising on East Speedway "Whither goest thou, America, in thy shiny car in the night?" -Jack Kerouac, On the Road Roll the windows down, turn the music up, and drive slowly. Now you're cruising. Cruising is the art of seeing and being seen, and in Tucson the center of this art is Speedway Boulevard. This six-lane street runs east to west through Tucson and is one of the busiest thoroughfares in the city. It hosts a mix of commercial and private buildings: small shops, offices, restaurants, grocery stores, apartment buildings and older homes, as well as the University of Arizona. Despite the apartments and occasional houses, Speedway is mostly a commercial street populated with strip malls and other businesses. Cruising is most visible along the more commercial, business-oriented East Speedway, which for the purposes of this essay is defined as the three mile stretch of road from Alvernon to Wilmot. Like most streets, Speedway was built for an entirely practical reason: to conduct automobile traffic from one place to another with a minimum of waiting. This utilitarian reason is inverted by cruising. The purpose of cruising, unlike driving, is not to arrive but to not arrive. Cruising is a social activity wherein the cars become tools for meeting other people as well as a means of getting from one place to another. The reputation of cruising, and of the nighttime Speedway, is not nearly so benign. As traffic slows and the music increases, the character of Speedway as a place - that is, a focus for human memory and experience - changes to reflect the activities and desires of the cruisers. The character of Speedway as a place is temporal and spatial; the street of Friday midnight is radically different from the street of Tuesday morning. Daytime drivers quickly change lanes in an attempt to find the one which will deliver them to their destinations the fastest. Speed and efficacy define the daytime Speedway. As the type of driver changes, so does the driver's reason for being there. The weekday commute is replaced by the weeknight cruise. The drivers also become, as a group, less diverse in terms of age and gender. The majority of cruisers appeared to be teenagers. Also, though I saw many young women, the majority of cruisers were male.

Integration

http://sahatmozac. blogspot. com ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION http://mathsmozac. blogspot. com http://sahatmozac. blogspot. com CHAPTER 3 : INTEGRATION Content Concept Map page 2 3–4 5 6 7 8–9 10 – 11 12 4. 1 Integration of Algebraic Functions Exercise A 4. 2 The Equation of a Curve from Functions of Gradients. Exercise B SPM Question Assessment Answer http://mathsmozac. blogspot. com 1 http://sahatmozac. blogspot. com Indefinite Integral a) o o a x n a dx = ax + c. xn+ 1 + c. n+ 1 b) x n dx = c ) o d x = a o x n d x = a n x + n + 1 1 + c . Integration of Algebraic Functions ) ) The [f (x)  ± g(x) ]dx = o f (x) dx  ± d o Equation of a Curve from Functions of Gradients o g(x)dx y = y = o f ‘( x ) d x c, f (x) + http://mathsmozac. blogspot. com 2 http://sahatmozac. blogspot. com INTEGRATION 1. Integration is the reverse process of differentiation. dy 2. If y is a function of x and = f ‘( x) then o f ‘( x)dx = y + c, c = c onstant. dx If dy = f ( x ), then dx o f ( x)dx = y 4. 1. Integration of Algebraic Functions Indefinite Integral a) b) o o a dx = ax + c. n a and c are constants xn+ 1 x dx = + c. n+ 1 n c is constant, n is an integer and n ? – c) o ax dx = a o ax n + 1 x dx = + c. n+ 1 n and c are constants n is an d) o [f ( x )  ± g ( x ) ]dx = o f ( x) dx  ± o g ( x)dx http://mathsmozac. blogspot. com 3 http://sahatmozac. blogspot. com Find the indefinite integral for each of the following. a ) ? 5dx b) ? x 3 dx c) ? 2 x dx 5 d) ? ( x ? 3x 2 )dx Always remember to include ‘+c’ in your answers of indefinite integrals. Solution : a) ? 5dx ? 5x ? c b) 3 ? x dx ? x3? 1 ? c 3 ? 1 x4 = ? c 4 2 c) 5 ? 2 x dx ? 2 x5? 1 ? c 5 ? 1 2 x6 = ? c 6 1 = x6 ? c 3 d) ? ( x ? 3x )dx ? ? xdx ? ? 3x 2 dx = x 2 3 x3 ? ?c 2 3 x2 = ? x3 ? c 2 Find the indefinite integral for each of the following. a) ? ? x ? 3x ? dx 2 x 4 b) ?x ? x 2 4 ? ? ? 3 ? ? dx x ? ? a) Solution : x ? 3Ãâ€"2 ? ? x 4 ?dx ? ? x 3Ãâ€"2 ? ? ? x4 ? x4 ? dx ? ? b) 2 4? ? ? 2 4? ? 3 ? 4 ? dx = ? ? 3x ? 2 ? dx x ? x ? ? ? = ? 3Ãâ€"2 ? 4 x ? 2 dx ? x ? 1 ? 3x 3 = ? 4? c 3 ? ?1 ? 4 = x3 ? ? c x ? ? x? 3 ? 3x? 2 dx ? x? 1 ? x? 2 = ? 3? c ? 2 ? ?1 ? 1 3 =? 2 ? ?c 2x x ? ? ? ? http://mathsmozac. blogspot. com 4 http://sahatmozac. blogspot. com 1. Find ? ? 3x 2 ? 4 x ? 10 dx. ? [3m] 2. Find ? ? x 2 ? 1 ? 2 x ? 3 ? dx. ? [3m] 1? ? 3. Find ? ? 2 x ? ? dx. x? ? 2 [3m] 4. Find ? ? 2x ? ? 3 ?x? 3 ? ? 2 ? dx. 4 x ? [3m] 6x ? 5 5. Integrate with respect to x. x3 [3m] 6. Find ? ?x 5 ? 4Ãâ€"2 2x 4 ? dx [3m] 3 ? ? 7. Find ? x ? 6 ? 6 ? x . x ? ? 2 [3m] 8. Integrate x 2 ? 3x ? 2 with respect to x. x ? 1 [3m] http://mathsmozac. blogspot. com 5 http://sahatmozac. blogspot. com The Equation of a Curve from Functions of Gradients dy ? f ‘( x), then the equation of the curve is dx If the gradient function of the curve is y ? ? f ‘( x ) dx c is constant. y ? f ( x) ? c, Find the equation of the curve that has the gradient function 3x ? 2 and passes through the point (2, ? 3). Solution The gradient function is 3x ? 2. dy ? 3x ? 2 dx y ? ? (3x ? 2)dx y? 3Ãâ€"2 ? 2x ? c 2 The curve passes through the point (2, ? 3). Thus, x = 2, y = ? 3. 3(2) 2 ? 3 ? ? 2x ? c 2 ? 3 ? 6 ? 4 ? c c ? 5 Hence, the equation of curve is y? 3x 2 ? 2x ? 5 2 http://mathsmozac. blogspot. com 6 http://sahatmozac. blogspot. com 1. Given that dy ? 6 x ? 2 , express y in terms of x if y = 9 when x = 2. dx 2. Given the gradient function of a curve is 4x ? 1. Find the equation of the curve if it passes through the point (? 1, 6). 3. The gradient function of a curve is given by dy 48 ? kx ? 3 , where k is a constant. dx x Given that the tangent to the curve at the point (-2, 14) is parallel to the x-axis, find the equation of the curve. http://mathsmozac. blogspot. com 7 http://sahatmozac. blogspot. com SPM 2003- Paper 2 :Question 3 (a) Given that y ? 2 x ? 2 and y = 6 when x = ? 1, find y in terms of x. dx [3 marks] SPM 2004- Paper 2 :Question 5(a) The gradient function of a curve which passes through A(1, ? 12) is 3 x 2 ? 6 x. Find the equation of the curve. [3 marks] http://mathsmozac. blogspot. com 8 http://sahatmozac. blogspot. com SPM 2005- Paper 2 :Question 2 A curve has a gradient function px 2 ? 4 x , where p is a constant. The tangent to the curve at the point (1, 3) is parallel to the straight line y + x ? 5 =0. Find (a) the value of p, [3 marks] (b) the equation of the curve. [3 marks] http://mathsmozac. blogspot. com 9 http://sahatmozac. blogspot. com 1.Find the indefinite integral for each of the following. (a) ? ? 4x 3 ? 3 x ? 2 dx ? (b) 3? x ? ? 2 2 ? 6? ? dx x3 ? 1 ? 2 ( c) (c) ? ? x 5 + 5 6x ? 3 ? ? dx ? ? x2 ? 3 (d) ? ? ? x2 ? ? ? 2 ? ? dx ? ? 2. If dy ? 4 x3 ? 4 x, and y = 0 when x = 2, find y in terms of x. dx http://mathsmozac. blogspot. com 10 http://sahatmozac. blogspot. com 3. If dp v3 ? 2v ? , and p = 0 when v = 0, find the value of p when v = 1. dv 2 4. Find the equation of th e curve with gradient 2 x 2 ? 3 x ? 1, which passes through the origin. 5. d2y dy dy Given that ? 4 x, and that ? 0, y = 2 when x = 0. Find and y in terms 2 dx dx dx of x. http://mathsmozac. blogspot. om 11 http://sahatmozac. blogspot. com EXERCISE A 1) 2) 3) 4) 5) 6) 7) 8) x ? 2 x ? 10 x ? c 3 2 SPM QUESTIONS 1) y ? x2 ? 2x ? 7 2) y ? x3 ? 3 x 2 ? 10 3) p ? 3, y ? x3 ? 2 x 2 ? 4 x4 ? x3 ? 3x ? c 2 4 3 1 x ? 4x ? ? c 3 x 4 2 x x 1 ? ? 3 ? 2x ? c 2 2 x 6 5 ? ? 2 x 2x 2 x 2 ? ?c 4 x 1 2 x3 ? 3 ? c x 2 x ? 2x ? c 2 ASSESSMENT 1) (a ) x 4 ? 3 2 x ? 2x ? c 2 2 3 (b) 3x ? ? 2 ? c x x 6 x 1 (c ) ? ?c 9 24 x 4 x3 9 (d ) ? 6x ? ? c 3 x y ? x4 ? 2 x2 ? 8 p? 7 8 2 3 3 2 x ? x ? x 3 2 2 3 x ? 2 3 EXERCISE B 1) y ? 3x 2 ? 2 x ? 1 3 x 2 24 ? 2 ? 2 2 x 2) 2) y ? 2 x 2 ? x ? 3 3) y ? 3) 4) y? 5) y? http://mathsmozac. blogspot. com 12 http://sahatmozac. logspot. com ADDITIONAL MATHEMATICS FORM 5 MODULE 5 INTEGRATION http://mathsmozac. blogspot. com 13 http://sahatmozac. blogspot. com CONTENT CONCEPT MAP INTEGRATION BY SUBSTITUTION DEFINITE INTEGRALS EXERCISE A EXERCISE B ASSESSMENT SPM QUESTIOS ANSWERS 2 3 5 6 7 8 9 10 http://mathsmozac. blogspot. com 14 http://sahatmozac. blogspot. com CONCEPT MAP INTEGRATION BY SUBSTITUTION un ? ax ? b ? dx ? ? du ? a n DEFINITE INTEGRALS If b d g(x) ? f (x) then dx b where u = ax + b, a and b are constants, n is an integer and n ? -1 OR (a) ? f (x)dx g(x)? ? g(b) ? g(a) a a (b) ? f (x)dx f (x)dx a a b b (c) ? f (x)dx f (x)dx ? ? f (x)dx a b a b c ? ax ? b ? ? ? ax ? b ? dx ? a ? n ? 1? n n ? 1 ? c, where a, b, and c are constants, n is integer and n ? -1 http://mathsmozac. blogspot. com 15 http://sahatmozac. blogspot. com INTEGRATION BY SUBSTITUTION un ? ? ax ? b ? dx ? ? a du n where u = ax + b, a and b are constants, n is an integer and n ? -1 O R ? ax ? b ? ? ? ax ? b ? dx ? a ? n ? 1? n n ? 1 ? c, where a, b, and c are constants, n is integer and n ? -1 Find the indefinite integral for each of the following. (a) ? ? 2 x ? 1? dx 3 (b) ? 4(3 x ? 5)7 dx 2 (c) ? dx (5 x ? 3)3 SOLUTION (a) ? ? 2 x ? 1? dx 3 Let u = 2x +1 du du ? 2 ? dx ? dx 2 3 3 ? du ? ? (2 x ? 1) dx ? ? u ? ? ? ? u3 = ? du 2 u 3 ? 1 = ? c 2(3 ? 1) u4 +c 8 (2 x ? 1) = +c 8 = Substitute 2x+1 and substitute dx with du dx = 2 OR (2 x ? 1) 4 ? c ? (2 x ? 1) dx ? 2(4) 3 = ? 2 x ? 1? 8 4 ?c Substitute u = 2x +1 http://mathsmozac. blogspot. com 16 http://sahatmozac. blogspot. com (b) ? 4(3 x ? 5) dx 7 (c) Let u ? 3 x ? 5 du du ? 3 ? dx ? dx 3 7 4u 7 du ? 4(3 x ? 5) dx ? ? 3 4u 8 = ? c 3(8) u8 ? c 6 (3u ? 5)8 = ? c 6 = 2 dx ? ? 2(5 x ? 3) ? 3 dx (5 x ? 3)3 Let u ? 5 x ? 3 du du ? 5 ? dx ? dx 5 ? 3 2u ? 3 du ? 2(5 x ? 3) dx ? ? 5 2u ? 3 = ? c 5(? 2) ? OR 4(3 x ? 5)8 ? c ? 4(3 x ? 5) dx ? 3(8) 7 u ? 2 ? c ? 5 1 = ? 2 5u 1 =? ?c 5(5 x ? 3)2 = = (3x ? 5)8 ? 6 DEFINITE INTEGRALS If d g ( x) ? f ( x) then dx b (a) (b) ? b a b f ( x)dx ? ? g ( x) ? ? g (b) ? g (a) a ? (c ) ? a b f ( x)dx ? ? ? f ( x)dx a b a f ( x)dx ? ? f ( x)dx ? ? f ( x)dx b a c c http://maths mozac. blogspot. com 17 http://sahatmozac. blogspot. com Evaluate each of the following ( x ? 3)( x ? 3) (a) ? 12 dx x4 1 1 (b) ? 0 dx (2 x ? 1) 2 SOLUTION (a) x2 ? 9 2 ( x ? 3)( x ? 3) ? c ? ?12 4 dx ? 1 x4 x 2 9 ? 2? x = ? 1 ? 4 ? 4 ? dx x ? ?x = ? 12 ( x ? 2 ? 9 x ? 4 )dx ? x ? 1 ? x ? 3 ? ? =? ? 9? ? ? 3 ? ?1 ? ?1 2 2 (b) ?0 1 1 1 dx ? ?0 (2 x ? 1)? 2 dx 2 (2 x ? 1) 1 = ? 0 (2 x ? 1) ? 2 dx ? (2 x ? 1) ? 1 ? =? ? ? ?1(2) ? 0 ? 1 = ? ? 2(2 x ? 1) ? 0 =? ? ? 1 1 ? 2[2(1) ? 1] ? 2[2(0) ? 1] ? 1 1 ? 1 3? = ? 3 ? ? x x ? 1 ? 1 3 ? ? 1 3? = ? 3 ? ? 3 ? ? 2 2 ? ? 1 1 ? 1 3 = ? ? ? (? 1 ? 3) 2 8 1 =? ?2 8 1 =? 2 8 1 ? 1? = ? ? 6 ? 2? 1 = 3 http://mathsmozac. blogspot. com 18 Distributed:18. 1. 09 Return:20. 1. 09 INTEGRATE THE FOLLOWING USING SUBSTITUTION METHOD. (1) ? ( x ? 1)3dx (2) ? ?4 ? 3 x ? 5 ? dx ? 5 (3) ? 1 ? 5 x ? 3? dx 4 1 ? ? (4) ? ? 5 ? x ? dx 2 ? ? ?3 1 ? ? (5) ? 5 ? 4 ? y ? dy 2 ? ? 4 3? 2 ? (6) ? ? 5 ? u ? du 2? 3 ? 5 19 http://sahatmozac. blogspot. com EXERCISE B 8 1. Evaluate ? 3 ( x3 ? 4)dx Answer : 1023. 75 2. Evaluate Answer: 3 ? ?3 1 2 x( x ? x ? 5)dx 8 83 96 ?2 ? 3. Integrate ? x ? 5 ? with respect to x ? 3 ? 4 4. Evaluate ? 1 3 1 ? ? ? 2 ? 3x ? 4 ? dx ? 1 x ? ? 1 Answer: 3 ? 2 ? ? x ? 5? ? c 10 ? 3 ? 5 Answer : 3 5. Evaluate ? 3 1 ? 2 x ? 1 2 x ? 1? dx 4 x2 6. Given that of 2 5 ? 5 2 f ( x)dx ? 10 , find the value 5 Answer: 1 6 ? ? 1 ? 2 f ( x)? dx Answer :17 http://mathsmozac. blogspot. com 20 http://sahatmozac. blogspot. com ASSESSMENT ?6 and 2. (a) ? 5(2 ? 3v) dv 4 (b) ? dx 5 3 ? 1 ? 5 x ? 1. Given that ? 2 2 1 f ( x)dx ? 3 ? 2 3 f ( x)dx ? ?7 . Find (a) the value of k if (b) ? ? kx ? f ( x)? dx ? 8 1 ? ? 5 f ( x) ? 1? dx 3 1 Answer : (a) k = (b) 48 22 3 3.Show that d ? x 2 ? 2 x 2 ? 6 x 4. . ? dx ? 3 ? 2 x ? ? 3 ? 2 x ? 2 4 Given that ? 4 0 f ( x)dx ? 3 and Hence, find the value of Answer : 1 10 ? ? 3 ? 2x ? 0 1 x ? x ? 3? ? 0 g ( x)dx ? 5 . Find 4 0 2 dx . ? f ( x)dx ? ? g ( x)dx (b) ? ?3 f ( x) ? g ( x)? dx (a) 0 4 0 4 Answer: (a) – 15 (b) 4 http://mathsmozac. blogspot. com 21 http://sahatmozac. blogspot. com SPM QUESTIONS SPM 2003 – PAPER 1, QUESTION 17 1. Given that ? SPM 2004 – PAPER 1, QUESTION 22 k n dx ? k ? 1 ? x ? ? c , 2. Given that 1 ? 2 x ? 3? dx ? 6 , where k ; -1 , find the value of k. [4 marks] ? 1 ? x ? find the value of k and n [3 marks] Answer: k = 5 5 Answer: k = ? =-3 3 5 4 SPM 2005 – PAPER 1, QUESTION 21 6 6 3. Given that ? 2 f ( x)dx ? 7 and ? 2 (2 f ( x) ? kx)dx ? 10 , find the value of k. Answer: k = 1 4 http://mathsmozac. blogspot. com 22 http://sahatmozac. blogspot. com ANSWERS EXERCISE A 1. 3 ( x + 1)4 + c 2. 60 (3 x +5) – 4 + c 3. ?20 EXERCISE B 1. 1023. 75 ? 5 x ? 3? 3 ?c 2. 3 83 96 5 4. 3? 1 ? ?5 ? x? ? c 2? 2 ? ? y? ?c ? 6 4 ?2 3 ? 2 ? 3. ? x ? 5? ? c 10 ? 3 ? 1 3 5 5. 1 6 6. 17 1 ? 5. ?10 ? 4 ? 2 ? 6. 4. 3 2 ? ? ? 5 ? 5 ? u ? ? c 3 ? ? ASSESSMENT 22 1. (a) k = 3 (b) 48 2. (a) 90(2 – 3v) +c ? 100 (b) (1 ? 5 x) ? 4 ? c 3 3. 1 10 -5 SP M QUESTIONS 1. k = ? 2. k = 5 3. = 1 4 5 3 n=-3 4. (a) – 15 (b) 4 http://mathsmozac. blogspot. com 23 http://sahatmozac. blogspot. com ADDITIONAL MATHEMATICS MODULE 6 INTEGRATION http://mathsmozac. blogspot. com 24 http://sahatmozac. blogspot. com CHAPTER 3 : INTEGRATION Content Concept Map 9. 1 Integration as Summation of Areas page 2 3 4–6 7–8 9 – 11 12 – 14 15 Exercise A 9. 2 Integration as Summation of Volumes Exercise B SPM Question Answer http://mathsmozac. blogspot. com 25 http://sahatmozac. blogspot. com a) The area under a curve which enclosed by x-axis, x = a and x = b is a) The volume generated when a curve is rotated through 360? bout the x-axis is ? ? b a y dx b) The area under a curve which enclosed by y-axis, y = a and y = b is b a Vx ? ? ? y 2 dx a b x dy b) The volume generated when a curve is rotated through 360? about the y-axis is c) The area enclosed by a curve and a straight line ? ? f ( x) ? g ( x)? dx b a Vy ? ? ? x 2 dy a b http://mathsmozac. blogspot. com 26 http://sahatmozac. blogspot. com 3. INTEGRATION 3. 1 Integration as Summation of Area y y = f(x) b a a b 0 The area under a curve which enclosed by x = a and x = b is x 0 x y = f(x) ? b a ydx The area under a curve which is enclosed by y = a and y = b isNote : The area is preceded by a negative sign if the region lies below the x – axis. ? b a xdy Note : The area is preceded by a negative sign if the region is to the left of the y – axis. The area enclosed by a curve and a straight line y y = g (x) y = f (x) a The area of the shaded region = = b b x ? ? ? f ( x) ? g ( x)? dx a b a a b f ( x)dx ? ? g ( x) http://mathsmozac. blogspot. com 27 http://sahatmozac. blogspot. com 1. Find the area of the shaded region in the diagram. y y = x2 – 2x 2. Find the area of the shaded region in the diagram. y y = -x2 + 3x+ 4 x -1 0 4 0 x http://mathsmozac. blogspot. com 28 http://sahatmozac. logspot. com 3. Find the area of the shaded region y y=2 4. Find the area of the shaded region in the diagram. y y = x2 + 4x + 4 0 x = y2 x -2 -1 0 2 x http://mathsmozac. blogspot. com 29 http://sahatmozac. blogspot. com 5. Find the area of the shaded region in the diagram y 1 x = y3 – y x 6. y y = ( x – 1)2 0 0 x x=k -1 Given that the area of the shaded region in 28 the diagram above is units2. Find the 3 value of k. http://mathsmozac. blogspot. com 30 http://sahatmozac. blogspot. com 3. 2 Integration as Summation of Volumes y y=f(x) The volume generated when a curve is rotated through 360? about the x-axis is 0 a b xVx ? ? ? y 2 dx a b y y=f(x) The volume generated when a curve is rotated through 360? about the y-axis is b a 0 x Vy ? ? ? x 2 dy a b http://mathsmozac. blogspot. com 31 http://sahatmozac. blogspot. com y y=x(x+1) Find the volume generated when the shaded region is rotated through 360? about the x-axis. x 0 Answer : x=2 ? ? ? y 2 dx 0 2 Volume generated ? ? ? x 2 ? x ? 1? dx 2 2 0 ? ? ? ( x 4 ? 2 x3 ? x 2 )dx 0 2 ? x 5 2 x 4 x3 ? ? ? ? ? 4 3 ? 0 ? 5 2 25 2(2)4 23 ? ? ? ? ? ? ? ? 0? 5 4 3? ? 256 1 ? ? @ 17 ? units 3 . 15 15 y y ? 6 ? x2 The figure shows the shaded region that is enclosed by the curve y ? ? x 2 , the x-axis and the y-axis. Calculate the volume generated when the shaded region is revolved through 360? about y-axis. 0 Answer : Given y ? 6 ? x 2 substitute x ? 0 into y ? 6 ? x Then, y ? 6? 0 y? 6 2 x Volume generated ? ? ? x 2 dy 0 6 ? ? ? ? 6 ? y ? dx 6 0 ? y2 ? ? ? ?6 y ? ? 2 ? 0 ? 62 ? ? 6(6) ? 2 ? 18? units 3 . ? ? ? ? 0? ? ? 6 http://mathsmozac. blogspot. com 32 http://sahatmozac. blogspot. com 1. y y = x (2 – x) 0 x The above figure shows the shaded region that is enclosed by the curve y = x (2 – x) and x-axis. Calculate the volume generated when the shaded region is revolved through 360? bout the y-axis. [4 marks] http://mathsmozac. blogspot. com 33 http://sahatmozac. blogspot. com 2. y R (0, 4) Q (3, 4) P (0, 2) y? = 4 (x + 1) 0 x=3 x The f igure shows the curve y ? ( x ? 2) 2 . Calculate the volume generated when the shaded region is revolved through 360? about the x-axis. http://mathsmozac. blogspot. com 34 http://sahatmozac. blogspot. com 3. y R (0, 4) x y ? ? 3? x 0 x=k The above figure shows part of the curve y ? ? 3 ? x and the straight line x = k. If the volume generated when the shaded region is revolved through 1 360? about the x-axis is 12 ? units3 , find the value of k. 2 http://mathsmozac. logspot. com 35 http://sahatmozac. blogspot. com SPM 2003- Paper 2 :Question 9 (b) Diagram 3 shows a curve x ? y 2 ? 1 which intersects the straight line 3 y ? 2 x at point A. y 3 y ? 2x 3y ? 2x x ? y2 ? 1 ?1 0 x Diagram 3 Calculate the volume generated when the shaded region is involved 360? about the y-axis. [6 marks] http://mathsmozac. blogspot. com 36 http://sahatmozac. blogspot. com SPM 2004- Paper 2 :Question 10 Diagram 5 shows part of the curve y ? y 3 ? 2 x ? 1? 2 which passes through A(1, 3). A(1,3) y? 0 a) b) Di agram 5 3 ? 2 x ? 1? 2 x Find the equation of the tangent to the curve at the point A. [4 marks] A egion is bounded by the curve, the x-axis and the straight lines x=2 and x= 3. i) Find the area of the region. ii) The region is revolved through 360? about the x-axis. Find the volume generated, in terms of ? . [6 marks] http://mathsmozac. blogspot. com 37 http://sahatmozac. blogspot. com SPM 2005- Paper 2 :Question 10 In Diagram 4, the straight line PQ is normal to the curve y ? straight line AR is parallel to the y-axis. y x2 ? 1 at A(2, 3). The 2 y? x2 ? 1 2 A(2, 3) 0 R Diagram 4 Find (a) (b) (c) Q(k, 0) x the value of k, [3 marks] the area of the shaded region, [4 marks] the volume generated, in terms of ? when the region bounded by the curve, the y-axis and the straight line y = 3 is revolved through 360? about y-axis. [3 marks] http://mathsmozac. blogspot. com 38 http://sahatmozac. blogspot. com EXERCISE A EXERCISE B 1. 1 1 ? unit 2 15 1. 1 1 units 2 3 5 units 2 6 2. 2. 20 3 6 ? unit 3 5 k ? ?2 3. 3. 2 2 units 2 3 2 units 2 3 SPM QUESTIONS SPM 2003 Volume Generated ? 52 ? units3 15 4. 24 SPM 2004 i) Area ? 1 units 2 5 49 ? units3 1125 5. 1 units 2 2 k? 4 ii) Volume Generated ? 6. SPM 2005 a) k ? 8 1 b) Area ? 12 units2 3 c) Volume Generated ? 4? units? http://mathsmozac. blogspot. com 39

Equality in Mare’s War Essay

Any normal grandmother would visit you with open arms, chocolate chip cookies, and comment on how amazing you look. However, in Mare’s War, by Tanita Davis, Mare is not your average grandma (unless you consider normal grandmas to wear auburn wigs, stiletto heels, and padded push up bras). Therefore, Mares granddaughters, Octavia and Tali, are dreading to accompany their 80 year old grandmother on a road trip across the country. As the three travel further into their journey, they begin to build a closer relationship and understand Mare’s past. Mare’s War was written in a particular fashion to express two different point of views. One is in the past representing Mare’s perspective, while the other is given from her 15 year-old granddaughter Octavia, during the present. Ultimately, both sides represent the struggle of equality: Mare’s battle of racism in the past, and Tali’s judgemental thoughts of her sister, today. In Mare’s War the 6888th battalion wasn’t the only battle Mare was fighting in, but also the battle of discrimination. Throughout the novel, prejudice between blacks and whites were clearly expressed during the 40’s. However, the most expressed struggle regarding equality is when Mare’s Army Corporation is stationed to another location. This new area yells out discrimination by having separate tables to eat at between blacks and whites and having separate water fountains. After vigorous training, all the black girls are lined up to get their drinks at the water fountain. Mare happens to notice the white fountain is empty. Although she knows it is wrong, Mare bravely stepped out of her line and goes straight to the white water fountain. After doing so, other black women decide, they too, would drink out of the white water fountain. This daring move made by Mare shows other black women that it is possible to express subtle ways of yearning for equality. Steps similar to this have shaped our generation today to have an equal foundation for any ethnicity or race. Fast forwarding to the future, 17 year-old Tali always finds a way to make Octavia feel worthless. When they were younger they had a close relationship with one another, but once Tali grew older she made her friends a priority and treated her sister like a stranger. This unfair rank between friends and family makes Octavia wistful for her sister to make her higher, or at least equal, to her friends. As the road trip continues to proceed, Octavia tries different ways for her sister to look highly of her. One night when Mare falls asleep in another room, Octavia agrees for Tali to put on makeup and dress her up for dinner. Although Octavia is a goody good, she plays it cool when Tali orders an alcoholic beverage. Although Tali is very judgemental, that night she looked at her sister highly and treated her equivalent to how she would treat her friends. From there on out Octavia and Tali began creating a connection as sisters. Whether it being a widespread problem like racism, or an individual conflict, Mare’s War presented how to deal with overcoming and expressing equality. Overall, this novel shares many life lessons and has a unique structure for portraying the past and the present.