(ii) Use Simpson s rule with two strips to find an approximation to Use your answers to parts (i) and (ii) to show that ln 2.

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1 C umerical Methods. June 00 qu. 6 (i) Show by calculation that the equation tan = 0, where is measured in radians, has a root between.0 and.. [] Use the iteration formula n+ = tan + n with a suitable starting value to find this root correct to decimal places. You should show the outcome of each step of the process. [4] (iii) Deduce a root of the equation sec = 0. []. Jan 00 qu. (i) Find, in simplified form, the eact value of 0 60 d. 0 [] Use Simpson s rule with two strips to find an approimation to 0 60 d. 0 [] (iii) Use your answers to parts (i) and to show that ln. 6 []. Jan 00 qu. 8 (i) The curve y = can be transformed to the curve y = + by means of a stretch parallel to the y-ais followed by a translation. State the scale factor of the stretch and give details of the translation. [] It is given that is a positive integer. By sketching on a single diagram the graphs of y = + and y =, show that the equation + = has eactly one real root. [] (iii) A sequence,,,... has the property that n+ = ( 6 n + ). For certain values of and, it is given that the sequence converges to the root of the equation + =. (a) Find the value of the integer for which the sequence converges to the value.907 (correct to 4 decimal places). [] (b) Find the value of the integer for which, correct to 4 decimal places, =.60 and 4 =.68. [] 4. FP Jan 00 qu part i) It is given that f() = sin. (i) The iteration n+ = sin n, with = 0.87, is to be used to find a real root, α, of the equation f() = 0. Find, and 4, giving the answers correct to 6 decimal places. []. June 009 qu. 4 It is given that a a (e + e )d = 00, where a is a positive constant. (i) Show that a = ln(00 + e a e a ). 9 [] Use an iterative process, based on the equation in part (i), to find the value of a correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process. 6. June 009 qu. 8

2 The diagram shows the curves y = ln and y = ln( 6). The curves meet at the point P which has -coordinate a. The shaded region is bounded by the curve y = ln( 6) and the lines = a and y = 0. (i) Give details of the pair of transformations which transforms the curve y = ln to the curve y = ln( 6). [] Solve an equation to find the value of a. [4] (iii) Use Simpson s rule with two strips to find an approimation to the area of the shaded region. [] 7. Jan 009 qu. (i) Use Simpson s rule with four strips to find an approimation to ln d, 4 giving your answer correct to decimal places. [4] Deduce an approimation to ln( 0 ) d. [] 8. Jan 009 qu. 6 4 The function f is defined for all real values of by f() = +. The graphs of y = f() and y = f () meet at the point P, and the graph of y = f () meets the -ais at Q (see diagram). (i) Find an epression for f () and determine the -coordinate of the point Q. []

3 State how the graphs of y = f() and y = f () are related geometrically, and hence show that the -coordinate of the point P is the root of the equation = +. [] (iii) Use an iterative process, based on the equation = +, to find the -coordinate of P, giving your answer correct to decimal places. [4] 9. FP Jan 009 qu. part i) It is given that α is the only real root of the equation + 8 = 0 and that.8 < α <. (i) The iteration n+ = 8 n, with =.9, is to be used to find α. Find the values of, and 4, giving the answers correct to 7 decimal places. [] 0. June 008 qu. 4 The gradient of the curve y = + 9 at the point P is 00. (i) Show that the -coordinate of P satisfies the equation = [] Show by calculation that the -coordinate of P lies between 0. and 0.4. [] (iii) Use an iterative formula, based on the equation in part (i), to find the -coordinate of P correct to 4 decimal places. You should show the result of each iteration. []. Jan 008 qu. The sequence defined by =, n+ = n converges to the number α. (i) Find the value of α correct to decimal places, showing the result of each iteration. [] Find an equation of the form a + b + c = 0, where a, b and c are integers, which has α as a root. []. June 007 qu. 6 a (i) Given that ( 6e + )d = 4, show that a = ln( a ). [] 0 6 Use an iterative formula, based on the equation in part (i), to find the value of a correct to decimal places. Use a starting value of and show the result of each iteration. [4]. Jan 007 qu. (a) It is given that a and b are positive constants. By sketching graphs of y = and y = a b on the same diagram, show that the equation + b a = 0 has eactly one real root. [] (b) Use the iterative formula n+ = n, with a suitable starting value, to find the real root of the equation + = 0. Show the result of each iteration, and give the root correct to decimal places. [4] 4. Jan 007 qu. 8 ( ) ( ).

4 8 The diagram shows the curve with equation y = e. The curve has maimum points at P and Q. The shaded region A is bounded by the curve, the line y = 0 and the line through Q parallel to the y-ais. The shaded region B is bounded by the curve and the line PQ. (i) Show by differentiation that the -coordinate of Q is. [] Use Simpson s rule with 4 strips to find an approimation to the area of region A. Give your answer correct to decimal places. [4] (iii) Deduce an approimation to the area of region B. []. June 006 qu. The equation + 4 = 0 has one real root. (i) Show by calculation that this real root lies between and. [] Use the iterative formula = n+. 7 n, 6. Jan 006 qu. 7 with a suitable starting value, to find the real root of the equation + 4 = 0 correct to decimal places. You should show the result of each iteration. [] The diagram shows the curve with equation y = cos. (i) Sketch the curve with equation y = cos ( ), showing the coordinates of the points where the curve meets the aes. [] By drawing an appropriate straight line on your sketch in part (i), show that the equation cos ( ) = has eactly one root. [] (iii) Show by calculation that the root of the equation cos ( ) = lies between.8 and.9. [] (iv) The sequence defined by =, n+ = + cos converges to a number α. Find the value of α correct to decimal places and eplain why α is the root of the equation cos ( ) =. [] n

5 7. Jan 006 qu. 8 The diagram shows part of the curve y = ln( ) which meets the -ais at the point P with coordinates (, 0). The tangent to the curve at P meets the y-ais at the point Q. The region A is bounded by the curve and the lines = 0 and y = 0. The region B is bounded by the curve and the lines PQ and = 0. (i) Find the equation of the tangent to the curve at P. [] Use Simpson s Rule with four strips to find an approimation to the area of the region A, giving your answer correct to significant figures. [4] (iii) Deduce an approimation to the area of the region B. [] 8. June 00 qu. 8 y y =e P y = ( + 8) O The diagram shows part of each of the curves y = e and y = ( + 8). The curves meet, as shown in the diagram, at the point P. The region R, shaded in the diagram, is bounded by the two curves and by the y-ais. (i) Show by calculation that the -coordinate of P lies between. and.. [] Show that the -coordinate of P satisfies the equation = ln( + 8). [] (iii) Use an iterative formula, based on the equation in part, to find the -coordinate of P correct to decimal places. [] (iv) Use integration, and your answer to part (iii), to find an approimate value of the area of the region R. [] 9. June 00 qu. 4 (a) The diagram shows the curve y =. The region R, shaded in the diagram, is bounded by the curve and by the lines =, = and y = 0. The region R is rotated completely about the -ais. Find the eact volume of the solid formed. [4] (b) Use Simpson s rule, with 4 strips, to find an approimate value for ( + ) d, giving your answer correct to decimal places. [4]