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