The Pros and Cons of Abortion Essay -- Abortion Essays

The topic of abortion is one of the most controversial issues in today’s society. Thousands of abortions take place every single day, and yet public opinion remains at a standstill as to whether or not abortion is ethical or not. According to a poll in 2013, fifty-four percent of the American public believes that the practice of abortion should be legal in all or most cases (â€Å"Public Opinion on Abortion†) Abortion has been defined as â€Å"the act of removing a human embryo or fetus from the uterus of a pregnant woman prior to the completion of the full term of pregnancy†(Rich, Wagner, and Geraldine). There are very strong opinions for and against this issue, but no one can deny the vast gray area of abortion. A person’s stance on the situation is often determined by how he views the fetus: a part of the mother’s body or as a human being. Abortion continues to be a moral issue because people have various views on the rights of the fetus and mo ther, the circumstances of the pregnancy, and their own religious convictions concerning the issue. In the most recent study, 1.21 million abortions took place in the United States in the year 2008 (â€Å"About Abortion†). According to a study performed by the Guttmacher Institue, twenty-one percent of all pregnancies in the United States end in abortions (â€Å"Induced Abortion†). Fifty percent of pregnancies in the United States are unplanned (â€Å"Induced Abortion†). Of those unplanned pregnancies, four out of ten result in abortions (â€Å"Induced Abortion†). When analyzing the statistics of women who receive abortions, one must become aware that outside circumstances might contribute to a woman’s decision to have an abortion. For instance, women who are one hundred percent below the federal poverty level ac... ...rtion Federation: History of Abortion. National Abortion Federation, n.d. Web. 17 Mar. 2014. "Public Opinion on Abortion." Public Opinion on Abortion. Pew Research Center, July 2013. Web. 17 Mar. 2014. Rich, Alex K. Wagner, Geraldine. "Abortion: An Overview." Points Of View: Abortion (2013): 1. Points of View Reference Center. Web. 16 Mar. 2014. Rubio, Marco. "Why Abortion is Bad for America." Human Life Review Winter 2012 2012: 19-26. ProQuest Health Management. Web. Voegeli Jr., William J. â€Å"A Critique of the Pro-Choice Argument† Review of Politics Vol. 43, no. 4 (Oct., 1981) , Pp. 560-571 Published by: Cambridge University Press for the University of Notre Dame Du Lac on Behalf of Review of Politics Stable Print. "World Abortion Laws 2009 Fact Sheet." Center for Reproductive Rights. Center for Reproductive Rights, 2013. Web. 18 Mar. 2014.

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

Now That This Paper Has Evaluated Aquinas’S Summa Contra

Now that this paper has evaluated Aquinas’s Summa Contra Gentiles, it will move on to evaluate his next important work. In the years 1265–1274 Aquinas wrote what is considered one of his most prominent works, The Summa Theologiae. In Summa Theologiae (also known as Suma Theologica or simply Summa), Aquinas gave five proofs for the existence of God. This paper will first tell why these proofs are necessary then describe the proofs in themselves. These proofs are necessary because Aquinas believed that the existence of God is not self-evident. A self-evident proposition is one in which the predicate forms part of what is meant by the subject (PUT, 103). Meaning that â€Å"God exists† is not self-evident because we cannot grasp divine essence†¦show more content†¦Therefore anything that is in the process of changing cannot change itself so one thing is changed by another which in turn is changed by yet another (Clark, 122). Eventually, this stream of chang e has to stop somewhere or else there would be no first cause of change and consequently no subsequent causes. So when we come to the first cause that is not changed by anything else, Aquinas believed it is what we understand to be God (Clark, 122-123). The second proof is derived from the nature of causation. Aquinas thought that in the natural world we find causes in a natural order of succession. We never see something causing itself because if we did then it would be pre-existing and this would be impossible (Clark, 123). Every first cause impacts an intermediate (there can be many intermediates) which then impacts a last. You cannot take out any one cause without getting rid of its effects (Clark, 123). So you cannot take out the first cause without losing the intermediates and last causes that follow. Thus Aquinas thought that we must suppose a first cause, which is God (Clark, 123). The third proof addresses the issue of what is unnecessary and what is unnecessary. Our experience has shown us that in life there are things that are necessary and things that are unnecessary. Things that are

How Neil Armstrong Became the First Man on the Moon

For thousands of years, man had looked to the heavens and dreamed of walking on the moon. On July 20,  1969, as part of the Apollo 11 mission, Neil Armstrong became the very first to accomplish that dream, followed only minutes later by Buzz Aldrin. Their accomplishment placed the United States ahead of the Soviets in the Space Race and gave people around the world the hope of future space exploration. Fast Facts: First Moon Landing Date: July 20, 1969Mission: Apollo 11Crew: Neil Armstrong, Edwin Buzz Aldrin, Michael Collins Becoming the First Person on the Moon When the Soviet Union launched Sputnik 1 on October 4, 1957, the United States was surprised to find themselves behind in the race to space. Still behind the Soviets four years later, President John F. Kennedy gave inspiration and hope  to the American people in his speech to Congress on May 25, 1961 in which he stated, I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the Earth. Just eight years later, the United States accomplished this goal by placing Neil Armstrong and Buzz Aldrin on the moon. Portrait of American astronauts, from left, Buzz Aldrin, Michael Collins, and Neil Armstrong, the crew of NASAs Apollo 11 mission to the moon, as they pose on a model of the moon, 1969. Ralph Morse / Getty Images Take Off At 9:32 a.m. on July 16, 1969, the Saturn V rocket launched Apollo 11 into the sky from Launch Complex 39A at the Kennedy Space Center in Florida. On the ground, there were over 3,000 journalists, 7,000 dignitaries, and approximately a half million tourists watching this momentous occasion. The event went smoothly and as scheduled. CAPE KENNEDY, UNITED STATES - JULY 16, 1969: Composite 5 frame shot of the gantry retracting while the Saturn V boosters lift off to carry the Apollo 11 astronauts to the Moon.   Ralph Morse / Getty Images After one-and-a-half orbits around Earth, the Saturn V thrusters flared once again and the crew had to manage the delicate process of attaching the lunar module (nicknamed Eagle) onto the nose of the joined command and service module (nicknamed Columbia). Once attached, Apollo 11 left the Saturn V rockets behind as they began their three-day journey to the moon, called the translunar coast. A Difficult Landing On July 19, at 1:28 p.m. EDT, Apollo 11 entered the moons orbit. After spending a full day in lunar orbit, Neil Armstrong and Buzz Aldrin boarded the lunar module and detached it from the command module for their descent to the moons surface. As the Eagle departed, Michael Collins, who remained in the Columbia while Armstrong and Aldrin were on the moon, checked for any visual problems with the lunar module. He saw none and told the Eagle crew, You cats take it easy on the lunar surface. Members of the Kennedy Space Center control room team rise from their consoles to see the liftoff of the Apollo 11 mission 16 July 1969.   NASA / Getty Images As the Eagle headed toward the moons surface, several different warning alarms were activated. Armstrong and Aldrin realized that the computer system was guiding them to a landing area that was strewn with boulders the size of small cars. With some last-minute maneuvers, Armstrong guided the lunar module to a safe landing area. At 4:17 p.m. EDT on July 20, 1969, the landing module landed on the moons surface in the Sea of Tranquility with only seconds of fuel left. Armstrong reported to the command center in Houston, Houston, Tranquility Base here. The Eagle has landed. Houston responded, Roger, Tranquility. We copy you on the ground. You got a bunch of guys about to turn blue. Were breathing again. Walking on the Moon After the excitement, exertion, and drama of the lunar landing, Armstrong and Aldrin spent the next six-and-a-half hours resting and then preparing themselves for their moon walk. At 10:28 p.m. EDT, Armstrong turned on the video cameras. These cameras transmitted images from the moon to over half a billion people on Earth who sat watching their televisions. It was phenomenal that these people were able to witness the amazing events that were unfolding hundreds of thousands of miles above them. This grainy, black-and-white image taken on the Moon shows Neil Armstrong about to step off the Eagle lander and onto the surface of the Moon for the first time. NASA   Neil Armstrong was the first person out of the lunar module. He climbed down a ladder and then became the first person to set foot on the moon at 10:56 p.m. EDT. Armstrong then stated, Thats one small step for man, one giant leap for mankind. A few minutes later, Aldrin exited the lunar module and stepped foot on the moons surface. Working on the Surface Although Armstrong and Aldrin got a chance to admire the tranquil, desolate beauty of the moons surface, they also had a lot of work to do. NASA had sent the astronauts with a number of scientific experiments to set up and the men were to collect samples from the area around their landing site. They returned with 46 pounds of moon rocks. Armstrong and Aldrin also set up a flag of the United States. Armstrong and Aldrin unfurl the US flag on the moon, 1969. Apollo 11, the first manned lunar landing mission, was launched on 16 July 1969 and Neil Armstrong and Edwin Aldrin became the first and second men to walk on the moon on 20 July 1969. The third member of the crew, Michael Collins, remained in lunar orbit. Oxford Science Archive / Getty Images While on the moon, the astronauts received a call from President Richard Nixon. Nixon began by saying, Hello, Neil and Buzz. I am talking to you by telephone from the Oval Office of the White House. And this certainly has to be the most historic telephone call ever made. I just cant tell you how proud we are of what you have done. Time to Leave After spending 21 hours and 36 minutes upon the moon (including 2 hours and 31 minutes of outside exploration), it was time for Armstrong and Aldrin to leave. To lighten their load, the two men threw out some excess materials like backpacks, moon boots, urine bags, and a camera. These fell to the moons surface and were to remain there. Also left behind was a plaque which read, Here men from the planet Earth first set foot upon the moon. July 1969, A.D. We came in peace for all mankind. Apollo 11 lunar module rising above the moon to rendezvous with command module before heading home, with half Earth visible over horizon in background. Time Life Pictures / NASA / Getty Images   The lunar module blasted off from the moons surface at 1:54 p.m. EDT on July 21, 1969. Everything went well and the Eagle re-docked with the Columbia. After transferring all of their samples onto the Columbia, the Eagle was set adrift in the moons orbit. The Columbia, with all three astronauts back on board, then began their three-day journey back to Earth. Splash Down Before the Columbia command module entered the Earths atmosphere, it separated itself from the service module.  When the capsule reached 24,000 feet, three parachutes deployed to slow down the Columbias descent. At 12:50 p.m. EDT on July 24, the Columbia safely landed in the Pacific Ocean, southwest of Hawaii. They landed just 13 nautical miles from the U.S.S. Hornet that was scheduled to pick them up. astronauts wait in life raft for a helicopter to lift them to the U.S.S. Hornet after successful splashdown July 24th. Astronauts Neil Armstrong, Michael Collins, and Buzz Aldrin successfully completed moon mission. Theyre wearing isolation garments.   Bettmann / Getty Images Once picked up, the three astronauts were immediately placed into quarantine for fears of possible moon germs. Three days after being retrieved, Armstrong, Aldrin, and Collins were transferred to a quarantine facility in Houston for further observation. On August 10, 1969, 17 days after splashdown, the three astronauts were released from quarantine and able to return to their families. The astronauts were treated like heroes on their return. They were met by President Nixon and given ticker-tape parades. These men had accomplished what men had only dared to dream for thousands of years—to walk on the moon.

How Does Conrad Link His Physical Exploration to a...

The â€Å"Heart of Darkness† is a tale of passage and discovery, not only into the heart of Africa, but into the heart of our human mind. Written by Joseph Conrad, this novel follows Marlow’s expedition into the unknown depths of the Congo in search of Kurtz and his adored wisdom. Conrad links Marlow’s physical journey to a psychological quest of discovery into evil and darkness inside each one of us. Through the impassable landscape, the language barrier between the colonists and the natives, and embodiment of Kurtz this idea is portrayed. We are given a glimpse of what mankind is capable of, how destructive and hostile we can be. But the question this novel probes at is to what length can we restrain ourselves from revealing our inner†¦show more content†¦The psychological discovery which this journey of Marlow represents; is man’s burning desire to overpower and conquer all that it sees. But here in the depths of the unknown, the trees are the kings. They outnumber the humans significantly, â€Å"trees, trees millions of trees†¦Ã¢â‚¬  and demonstrate that we cannot have control over all things, that there is a higher power which governs even us. Marlow’s journey into the unknown is predominantly in search of the idealism of meeting Kurtz. Before we finally encounter Kurtz in the story, an admirable reputation of him and all he represents is formed. Marlow is so fixated on the wisdom and greatness of the man that he becomes more of a God than a person. He is a representation of darkness and is the focal point of the play to some degree. He is a personification of his surroundings, part of the African earth, an ominous shape in the foliage of trees. Kurtz and Marlow are the only two characters named in the entire story, which places emphasis on them and distinctly dehumanises the other characters in the story. Conrad amplifies Kurtz’s significance through waiting to the third chapter to finally reveal him to Marlow and the audience. We feel as we have earnestly waited for an eternity to finally meet him. His entrance on a stretcher carried by the African natives is an image of significant connotation. He is desc ribed initially by Marlow as â€Å"an insoluble probability†. HeShow MoreRelatedLibrary Management204752 Words   |  820 Pages. . . . . . . . . . 240 Performance Appraisals. . . . . . . . . . . . . . . . . . . . . . . . 241 Why Appraisals Are Done . . . . . . . . . . . . . . . . . . . . . . 242 When to Do Appraisals . . . . . . . . . . . . . . . . . . . . . . . . 243 Who Does the Appraisals?. . . . . . . . . . . . . . . . . . . . . . 244 Problems in Rating. . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Methods of Performance Appraisal . . . . . . . . . . . . . . . 246 The Performance Appraisal Review ProcessRead MoreProject Mgmt296381 Words   |  1186 PagesAuthors Erik W. Larson ERIK W. LARSON is professor of project management at the College of Business, Oregon State University. He teaches executive, graduate, and undergraduate courses on project management, organizational behavior, and leadership. His research and consulting activities focus on project management. He has published numerous articles on matrix management, product development, and project partnering. He has been honored with teaching awards from both the Oregon State University MBA

Early Civilizations Matrix Essay Free Essays

Using your readings and outside beginnings. finish the undermentioned matrix. Be certain to turn to the undermentioned in your matrix: Provide names. We will write a custom essay sample on Early Civilizations Matrix Essay or any similar topic only for you Order Now rubrics. day of the months. brief descriptions of of import events. and other inside informations. as necessary. Note the inside informations of cardinal political. socioeconomic. technological. artistic. musical. architectural. philosophical. and literary developments for each civilisation listed in the tabular array. which were evidenced in the humanistic disciplines. Properly mention the beginnings you use in finishing this matrix. CivilizationPoliticssSociety and EconomicssTechnologyArtMusicArchitectureDoctrineLiteratureBuddhismBuddism is the 5th largest faith in the universe.Prioritizing goods was of import to their economic system.Opportunities for the spread of the Dharma.Art media was created.Buddism music was inspired by buddism.Churchs were made for the spread of Buddism.Buddism doctrine was learning Budda.Bibles of Buddism and literary texts in Buddism.Early on Middle AgesA powerful cardinal authorities.Life centered around subsistence and security.Mechanical redstem storksbills were invented.Gothic art from the dark ages.Goliards originated the in-between ages.Roman arch system enabled contructors to back up heavier rocks. History of Christian doctrine.Theological plants were dominant signifier of literature.High Middle AgesThe first European enlargements out of Europe.Alps began to settle new lands cal great clearences.The hourglass was created.Romanesque. the first knowing manner since the Roman Empi re. Western music was popular.Churchs were built to distribute Christianity.Christian doctrine was popular.Robert Henryson is a modern-day English poet.Late Middle AgesTreaty of Caltabelotta ends the war of the Sicilian Vespers. System of utilizing unfastened Fieldss helped the economic system.The water wheel and the Cathedral were created.Renaissance Human was portion of the art universe so.Western music was common.Romanesque manner was besides used in this epoch.Albertus Magnus’s Dominican colleague of philosopy and divinity. Didactic literature prose renditions of authoritative plants.Ancient GreecePrime curates of Greece is the caput of authorities.The importance of importing goods.Rotary Millss were created.Scultures and Vases.Folk music was popular music.Urban development and life infinites.Socrates. Plato. and Aristole were philosophers.Epic verse forms of Homer were popular.Ancient RomanAncient Rome was a Italic civilisation.Focus was on agriculture and trade.Civil ap plied scientists and constructions like the Pantheon.Ocular humanistic disciplines were created.The Tibia. a woodwind instrument.Pantheon was created.Political doctrine was invented with Plato.Horace was popular in the literary universe.ChinaHan dynasty came to power.Horses advanced growing with trade.Horses pulled supplies and goods.Spouted Ritual Wine Vessel thirteenth centuryMusic Bureau was created 120 BCE.Broad eaves for the roof.The Book of Changes is the usher to construing the workings of the Universe. IndiaModel of the fundamental law is political relations in India.The economic system of India is the 10th largest in the universe.Science was admired in India.Indian art was popular.Indian music was listened to excessively.Buildings and schools were built.Indian doctrine was popular.Literature produced on the Indian suncontinent.HebraismThere was a batch of disappreement among the Jews politically. . Judaic economic theory that we posit is the ineffiency of authorities and the dangers of concentrated power. Papermaking was brought to the Middle East. . Ocular humanistic disciplines. the king of beastss on Torah drapes.Tunes of the Judaic peopleMany theaters were built.Teaching relation of Juddism.Judaic literature contributed to the national linguistic communication of many states. Early ChristianChristianity is matked as moral power.Christian societies were communal.Radio was created.Paleochristian art produced by Christians.Christian music was popular.Churchs were built for Christianity.Christian divinity and mediaeval philosopgy. MuslimismThe laminitis Mohammad his political philosophy.There were self-identified Islamic groups have varied throughout history. Digital engineering was created.Abstract Mosaic Art was popular.Religious music was popular.Secular and spiritual manners.Christian doctrine was in Islam.Muslim literature the topographic point of Muslim power. Use a list format of complete sentences instead than paragraphs. Do non copy and paste from outside resources. The following are two illustrations of the degree of item and certification expected for this assignment: Ancient Egyptian political relations: The brotherhood of Upper and Lower Egypt by Narmer in 3150 B. C. E. is commemorated in a 2-foot high slate known as the pallet of Narmer ( Sayre. 2013. pp. 32–33 ) . Ancient Grecian architecture: Minoan society: The three-story castle at Knossos was a labyrinthine masonry construction with tonss of suites and corridors built around a cardinal courtyard ( Sayre. 2013. pp. 43–44 ) . How to cite Early Civilizations Matrix Essay, Essay examples

Junk Food free essay sample

Hello, my name is Daniela and I am here today to share my opinion about banning junk food, and hopefully I change your opinion, if you disagree with me. Well, junk food is food that’s very high in fat, sugar, and calories. It hardly has protein, vitamins, or minerals. I know it may be very tasty and good, but it’s not so good for your body. Eating junk food can cause your brain to get addicted to it in a way like drugs do. You’ll constantly think you need it, when you don’t. If you go a couple weeks without junk food and you’re use to eating it, you’ll want it even more, which is bad. Poor eating habits as a kid, become worse when you’re an adult. Eating it for years can cause obesity. Over 31. 8% of children in America are obese or overweight. We will write a custom essay sample on Junk Food or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page This all leads to heart problems, diabetes, and other health problems. It can also cause cavities and other dental problems. Junk food also has a lack of oxygen, which is what feeds the brain and the rest of the body. Lack of oxygen causes fatigue and lack of concentration. A lot of studies also show that kid’s grades are sliding down very fast, due to being tired and lack of concentration. They can’t keep all the information they’re learning in their head. Parents should monitor their kid’s diet, but at school they can’t. Therefore the school should help the students in deciding what’s better for their nutrition. Kids do what they see other people do no matter if it’s bad or good. They see us eating a lot of junk food; they’ll eat a lot of junk food. They see us being healthy; they’ll want to become healthy. So why teach them that eating a lot of junk food is healthy? At home, maybe some parents may not care what their kid eats; therefore they should at least have one healthy meal provided by the school. Many students like to go and socialize with their friends during lunch since they can’t really talk or see each other any other times during school. Therefore, they’ll want to just grab a quick snack, like a type of junk food, and just leave to go talk with their friends. They aren’t eating a good meal and are just putting junk in their body. It’s much better if the school bans it and replaces the junk food with a healthy snack so at least the person can be eating something good for them. What if the school makes you take P. E. more than you’re required to since they decided to keep junk food? I know many of you hate it. So I don’t know about you, but I rather be eating healthy, than be trying to lose those calories I got from the junk food. I rather not even be taking P. E. another year just because they decided to keep that junk. Also, students are taught in health classes about healthy eating, and if the schools promotes and sells junk food, then the school contradicts its purpose of teaching this. The money you waste on junk food funds most of the cool and new stuff we get for our school. The school board should be paying for this stuff. They’re basically making money off your poor health choices. I hope you see things my way now and think about supporting the idea of banning junk food in school. Thank you for your listening!