6 Fascinating Facts About Tent Caterpillars

Portia in Shakespeares The Merchant of Venice and...

Portia in Shakespeares The Merchant of Venice and Abigail of Marlowes the Jew of Malta Portia and Abigail are two characters with very different values. Portia in Shakespeare’s The Merchant of Venice remained true to her religion, and her father’s wishes throughout the play. Abigail, on the other hand, changed religions and disobeyed her father. However, the writers used these two women to make similar statements about religion. Portia represented the quintessential Christian. Abigail of Marlowe’s The Jew of Malta, was more of an ethically ambiguous character, but it can still be argued that she was the most principled character in the play. Both Shakespeare and Marlowe used the daughter character to represent the ideal human†¦show more content†¦Already Portia clearly puts her sense of duty ahead of her desires, as a good Christian should. Later, in the trial scene, Portia again showed her love of the law, but attempted to use the law with mercy. She gave Shylock a choice as she judged the trial. She said he may choose either to be merciful to Antonio or the court would abide by the law- word for word. Shylock did not accept her offer because he felt that he was entitled to the justice of the law. Portia would never have broken the law, but she was able to find a way to use it to her advantage. â€Å"This bond doth give thee here no jot of blood; The words expressly are ‘a pound of flesh,’† (Merchant of Venice, IV.i.306-307). Portia found the loophole in the bond that none of the other characters could see. It was a triumphant moment for her because she was able to release Antonio from his debt while still being utterly just and fair. â€Å"He shall have justice and his bond,† she explained to the court (Merchant of Venice, IV.i.339). She is was the same time more lenient and s tricter to the letter of the law than Shylock and defeated him on his own terms as well as by Christian terms. When she offered Shylock a chance to be merciful before she used the full extent of the law against him, Shakespeare’s audience would have seen that as evidence ofShow MoreRelated Father-Daughter Relationships in Sidney’s The Countess of Pembroke’s Arcadia, Marlowe’s The Jew of Malta, and Shakespeare’s The Merchant of Venice3187 Words   |  13 PagesFather-Daughter Relationships in Sidney’s The Countess of Pembroke’s Arcadia, Marlowe’s The Jew of Malta, and Shakespeare’s The Merchant of Venice Justification for the subjugation of females to males during the sixteenth century came from a variety of sources. Ranging from the view that God gave Adam authority over Eve as penalty for the fall, to a belief in the superiority of a husbands’ physical strength over that of his wife, attempts at rationalization of the restricted freedom of women

Integration Free Essays

string(665) " 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\." http://sahatmozac. blogspot. com ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION http://mathsmozac. We will write a custom essay sample on Integration or any similar topic only for you Order Now 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 = constant. dx If dy = f ( x ), then dx o f ( x)dx = y 4. 1. Integration of Algebraic Functions Indefinit e 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 ? ? ? = ? 3à â€"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 the 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 ? 2 x ? 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. You read "Integration" in category "Essay examples" 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 ? a x ? 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://mathsmozac. 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 SPM 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 is Note : 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 x Vx ? ? ? 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 figure 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 wh ich 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) Diagram 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 Fin d (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 How to cite Integration, Essay examples

SLR digital camera Essay Example For Students

SLR digital camera Essay On a SLR digital camera it should have a dial that turns the camera on manual, the M setting is when the photographer have complete control unless the camera has preset limitations that doesnt allow it to do certain things. In the M setting it should allow the user to change the shutter speed and the aperture, when taking a picture that one needs to stop all action or time in the picture, it should be shot at a speed of 125 and greater, depending on your subject, to give water its moving effect you would need to shoot the photo of 60 and less to create the effect, while changing these setting you will have to compensate the other settings such as aperture, aperture is the opening of the camera, which controls the amount of light the passes the shutter, by increasing the size more light will come into the camera, having the film more exposed. The aperture is measured by f-stops the higher the f-stops the smaller the opening in which the light goes in, for example f-stop 2. 8 is bigger than 3. 2, the f-stop not only adjust the light coming through, but also the focal point of the image, and helps create a sense of depth in the photo. When taking a picture with depth one wants to see how far you are from the subject the closer the person the wider the aperture, hence a lower f-stop, when you take a picture of a subject far away one you should have a higher f-stop, a smaller aperture/opening. By doing this it will help focus the picture to a certain point, or to focus as much as possible in the scene, to make a sharper image, so that the focusing in the lens can produce a photo with more depth, because it shortens/lengthen the focal area. In most cameras there is a presetting that is called depth of view mode, what that does is that it will chose the best setting to take a picture where everything is in focus. Also in the mode dial, there should be a setting called Av, which is aperture priority, the camera will then adjust its setting around what the user set for the aperture, and how exposed the picture will be, this setting allows the photographer take a picture with the desired f-stop, without having the photographer adjust the shutter for a proper exposure. There is also another setting called Tv, which is shutter priority, where the camera sets its own setting according to the shutter speed that the, user had set, this allows the user to set the desired shutter speed for different effects, while lightening the load of information the user has to take in. In many cameras the setting may have different names because this essay is based on a Canon camera. For most SLR cameras the lenses normal cost way more than the lens, because normal the camera breaks down faster than a normal lenses. So if one decides to replace your old SLR oneyou shwouldd like to find a camera that is compatible with the lenses one already owns. When walking in to a camera shop it is better to walk in blind, ask a lot of questions, and let the salesman show you around, this will see if they know their stuff, did I mention ask a lot of questions? So how does one pick a good lens? In a normal 35mm film format, a 50mm lens would be the equivalent of what your eyes sees and is a 1:1 conversion, but on most digital SLR because of prices, and quality most of the digital SLR have a film conversion of 1. 3x/1. 6x and making a lower mm lens to be more telephoto than it would be than in a normal camera. When looking for a good camera lens one should first look at what kind of connection the camera uses to attach on to theiryour camera, hopefully they are compatible first, and then look for what effects or shots that theyyou want to take. There is no point in doing a portrait shot with a macro or wide angle lens, there are many different kinds of lenses they are ultra wide-angle zoom, standard zoom, telephoto, telephoto zoom, macro, tilt-angle lens, super telephoto etc. Next see if the optics of the lens match the cost and also what you want to be doing with the lens, a 28-70mm lens could have more desirable things than another lens but one might lack. A couple features, such as a special lens coating, or floating aspherical lenses,that one might so you want to see if the lens meets what your expectation, there is nothing worse than getting ripped off of the price. Then of course what should also be a concern is the maximum f-stop of the lens that oneyou would like to get. Reading an Advert EssayIn the golden mean is similar to the rule of third but instead of splitting the photo in to nine equal sections it divides it into it first splits it into nine unequal sections then splitting it with four lines, to use the golden mean is like to use the rule of thirds placing objects of importance in the intersecting areas the 4 secondary lines are used to align other lines to it, such as the line of a street. When taking a photo it is also useful to observe the textures in the photo to create an interesting photo, textures can greatly create depth of a photo. A good example of a good texture is a split piece wood, if shot in at an angle can give good results. The best way of utilizing textures is in black and white photography, because of instead of having colour that emphasizes the saturation or that hues, which can be distracting, black and white emphasizes the textures. In most pictures complexity can be overwhelming, if used in a wrong way, because there might be things in the photo, which could distracts the viewer from the main point of the picture, especially if the picture has more than one subject of interest. Complexity can work both ways, it allows the viewer to be mesmerized instead of just letting a viewer skim past the photo, and a good example of this is machinery where there are a lot of gears. The thing that occurs in photo, which are complex, and not too distracting, all have the trait that the picture has components, which are mutually the same. Simplicity is very important in a picture because it allows the picture to convey the message much more effectively because of the lack in distractions. In simplicity it is best to leave things by them selves instead of bring the whole scene in your picture, say if there is a chapel, and there is also a parking lot, with a telephone booth, it is better to just take a picture of the chapel, and leave the other component for other pictures. The depth of a picture is important to a picture because it adds dimension to a photo instead of having the photo look 2D. You can create depth in many ways having a large aperture that would narrow the amount of the picture that will be focused, by manipulating the things in focus, it will make a more 3D look in the picture. By having a foreground, a midground, and a background, it will make a picture with more depth than just having one or leaving out one of these categories. Lines are just lines until you actually bend them and move them a straight and vertical line is too plain and too normal, but if you bend them, then the sky is the limit. A S curve is pleasant to the eye and combining a S curve with it starting from one side and leading away from the picture, is quite a combination. Usually curved lines are much preferred than straight lines, and having lines leading off to the side is also a good thing to have. Shapes are just lines which are closed, to form a shape. The rule of the lines also applies to shapes, lines that have curves, is better than straight lines. Shapes can be more constructive than lines because of the shapes have more than one way of looking at it. In the modern age of digital cameras, the skill involved in taking a good picture has decreased. But I hope that this essay might have given you an insight to the expansive world of photography. There is a growing number of people that is starting to use a S.L. R. camera because the age of digital camera, letting people take photo without having consequences which was involved in film such as developing and the cost of buying film and developing. This essay was meant to teach you about a camera and how to become a better photographer, but not to go out and buy a new camera, the camera you have now should beis sufficient, its much better to spend more time, and money with your family, a camera is only a camera, its who that takes the picture that counts.

Review of Anatomy of Grey Play Essay Example

Review of Anatomy of Grey Play Essay Anatomy of Grey Play Review I went and saw Anatomy of Gray at Orange Community Play House in Orange, Texas. I went on a a Thursday night and it was performed by the Orange Community Players. The play was ultimately about a town who does not have a doctor. A young woman prays and wishes for a doctor to come to the town because the community was starting to get sick. When the doctor arrives and begins to help everyone but falls in love with the young ladys mother. Everyone in the community starts to get sick and is begining to die. The mother makes the daughter and the doctor leave the town which is the only way to stay alive. The actors in the play were extremely believable. They accomplished this by nailing all of their lines and moevmentsin the performance. The most successful actress was the young women who fell in love with the doctor and eventually moved from the small town to stay alive. The old lady sang really good, I really enjoyed her part in the performance. The actors were listening to each other and were in perfect sync. The director did a great job with the entire show. We will write a custom essay sample on Review of Anatomy of Grey Play specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Review of Anatomy of Grey Play specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Review of Anatomy of Grey Play specifically for you FOR ONLY $16.38 $13.9/page Hire Writer All the elements in the show went perfectly with the actors, the sets, and the lighting. The whole stage was used and the performers moved around throughout the entire stage. The pace of the production seemed to be right. I thought it might have dragged out a little but picked up in the secound act. The type of theatre that was used was a picture fame stage. The stage was fairly small but perfect for the production that was being put on. It was a simple theatre with a older look to it. The stage sat directly in front of the audience seating. The play space was also a little small but worked out great for the performance. The production was perfect the way it was setup. The scenery was pretty clever throughout the performance. There was a stack of hay which made me think that it was back in the olden days when they rode horses and medicine did not really exist. It looked real like i was in this old town. It effected the production by allowing the perforers to easily move around and provided enough information for the viewer to understand what was going on. he scenery made me visualise me being with the performers on the stage. The costumes were a little dull but i guess thats how people dressed back in those days. the information that the costumes portrayed helped me understand the performance to the fullest. The costumes really brought out the characters personality and social status. My favorite costume was when the man was in that diaper or the under ware it was quite funny. I thought the lighting was constructed alrig ht and changed on time and really influenced the scene it was trying to portray. Although, I was not very impressed because it seemed like typical lighting you would find in any play. Nothing really stood out or grabbed my attention as much as I feel it should have. I did although enjoy the thunder and lightning and when the man came out with the under ware on. I don’t feel as if it influenced the play as much as it could have. But I still think it was a good job. The goals portrayed by the characters was affectionate, hatred, and love. I felt they all did a good job at what they were suppose to do. Even though i thought that some of the performance was pretty funny. The best character was the older lady she had a dry sense of humor which i absolutely loved. I just felt because the play itself was rather boring, I found it hard to follow and to figure out what was actually going on. I thought the play was good but not great. I found it boring and hard to follow. The characters were good at their job and so was the director. I just thought the plot was slow and hard to follow. It was entertaining in some aspects, it just wasn’t for me. I didn’t have a favorite scene or part of the performance. But as for everyone involved i thought they did a really good job.

Basic English Curriculum for Teaching ESL

Basic English Curriculum for Teaching ESL The following grammar points will provide students with a solid base to build their English speaking and comprehension skills. Specific points are included in notes for the various grammar points. Grammar These are important grammar objectives for basic English courses. Present simple/present continuous (present progressive): Contrast between habits and temporary actions.Past simplePast continuous: Focus on use with the past simple to describe interrupted actions in the past.Present perfect: Focus on the use of present perfect for unfinished time- i.e. the duration form. Focus should also include adverbs commonly used with the present perfect, such as since, for, just, already, and yet.Future with will: Contrast this form with future intentions form- i.e. future with going to.Future with going to: Contrast this form with future predictions form- i.e. future with will.Present continuous (present progressive): Use for future intentions and plans, discuss similarities to future with going to.First conditional (real conditional): Used for probable or realistic situations.Modal verbs of deduction: Use of must be, might be, and cant be in the present.Some or any: Call to attention the irregular use of some in requests and offers.Quantifiers: too, enough, a lot of, a few, much, many (in question and negative forms), and others. Prepositions of place: in front of, opposite, behind, between, across, and other terms.Prepositions of movement: straight on, on your right, past the house, into, out of, and other prepositionsCommon phrasal verbs: get on with, look after, fed up with, put off, make up, and other verbs.Verb and gerund: like doing, enjoy doing, go swimming, etc.Verb and infinitive: hope to do, want to do, manage to do, and other examples.Basic verb and preposition combinations: listen to, arrive at, go through, and other combinations.Comparatives and superlatives: taller than, more beautiful than, as tall as, happier than, the tallest, the most difficult, etc. Listening Skills Listening skills should include the ability to understand and act on basic information in the following situations: Personal information: name, address, telephone number, nationality, etc.Telling timeNumbers: cardinal and ordinalSimple directions and prepositions of placeSpellingSimple descriptions of people and places Vocabulary These are some topics and categories of vocabulary that are important to learn at the beginner stages: Descriptions of people, such as looks, character, and familyFood, drink, and restaurantsLikes and dislikesHome, rooms, furnitureTown and countryShops and shoppingWeatherTime, the seasons, months, weeks, days, and related termsFilms and televisionLeisure and interestsHolidays, travel, and hotels Language Functions Language functions concern chunks of language which provide essential phrases for everyday use. Introductions and Greetings: How do you do? Pleased to meet you. How are you? Asking for Information: How do you spell ____? How do you pronounce? Where is the nearest bank? What does X mean? Offering: Can I help you? Would you like some ____? Requesting: May I have a coffee? Could you help me? Inviting: Would you like to come with me? Suggesting: Shall we go out this evening?Lets have some lunch.Why dont we play some tennis? Asking for Descriptions: What is he like?What does it look like? Buying and Selling: Which size are you?How much does it cost? Asking for Directions: Excuse me, where is the train station?Where is the nearest bank? Giving Advice: You should see a doctor.I think he should work harder.

Cue vs. Queue

Cue vs. Queue Cue vs. Queue Cue vs. Queue By Maeve Maddox The first time I read the following passage, I thought the use of the word queue must be arcane computer science jargon, but when I read it a second time, I realized the spelling queue in this context was just a mistake: This happens to be one of our common gripes: that once the battery is end-of-life, we are pretty much forced to get a replacement unit since the battery takes a  queue  from Apple and is sealed inside the casing. The passage is from a product review. The writer is criticizing the fact that the consumer cannot replace the tool’s battery because, like the batteries in some Apple devices, it’s sealed. The tool’s manufacturers have â€Å"taken their cue† from Apple. Take one’s cue: use someone else’s behavior as a model for one’s own. The mistaken use of queue for cue was a new one for me, so I did a web search to see if this reviewer had company. He has plenty of company. And don’t pounce on the Americans; the misuse is global. I found examples on sites in the U.K., Australia, and Jamaica, as well as in the U.S. and in at least one printed book. Deron is on half speed one game, full speed another game. Everyone else takes their queue from that. Telstra are dead in the water. Hopefully the .au government takes their queue and busts it in half. The populace takes their queue from people of influence who are either silent on the matter†¦[or] are outrightly vocalising their contempt†¦ If the bride and groom are uptight and nervous the whole wedding takes their queue from them.   Cue and queue are both pronounced like the name of the letter Q. Queue, a French borrowing with the literal meaning of â€Å"tail,† is sometimes spelled cue in the context of hair: cue/queue: A long roll or plait of hair worn hanging down behind like a tail, from the head or from a wig; a pigtail. The expression â€Å"to take one’s cue† is from the figurative use of a theatrical term: cue: The concluding word or words of a speech in a play, serving as a signal or direction to another actor to enter, or begin his speech. The OED lists several historical spellings of cue in the sense of an actor’s signal, but the list does not include the spelling queue: kew, ku, quew, q, quue, que, Q, qu, kue. The word is spelled cue in A Midsummer’s Dream (1600); Bottom the Weaver is bossing his fellow actors: Curst be thy stones, for thus deceiuing mee. [deceiving me]. Deceiuing mee is Thisbyes cue: she is to enter now, and I am to spy Her through the wall. In British usage, a queue is a line of people or vehicles waiting to proceed or be served. In computer use, a queue is a list of data items or commands that are attended to in a certain order, such as a queue of documents waiting to be printed. Next time you have occasion to write the expression â€Å"take one’s cue,† take your cue from this article and remember to spell it cue. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Misused Words category, check our popular posts, or choose a related post below:How to Structure A Story: The Eight-Point ArcIs There a Reason â€Å"the Reason Why† Is Considered Wrong?1,462 Basic Plot Types