3 Show in each case that there is a root of the given equation in the given interval. a x 3 = 12 4
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1 C Worksheet A Show i each case that there is a root of the equatio f() = 0 i the give iterval a f() = + 7 (, ) f() = 5 cos (05, ) c f() = e ( 6, 5) d f() = (, ) e f() = l (4 ) + (04, 05) f f() = e 9 cos 4 (0, ) Give that N 5, fid i each case the iteger N such that there is a root of the equatio f() = 0 i the iterval (N, N + ) a f() = 4 f() = l c f() = d f() = e + 4 e f() = e si f f() = ta (0) + 6 Show i each case that there is a root of the give equatio i the give iterval a = 4 [, ] e = 9 4 [, 0] c 0 l = 5 7 [047, 048] d si 4 = 7e [ 65, 6] e 4 = + 0 [ 4, ] f ta ( ) = [6, 7] 4 I each case there is a root of the equatio f() = 0 i the give iterval a a + Fid the iteger, a, such that this root lies i the iterval (, ) 0 0 a f() = (, ) f() = l (6 + ) (, ) c f() = 5 + (, ) d f() = 8 cos (, ) e f() = cosec + (5, 6) f f() = 7e + 5 (, ) 5 a O the same set of aes, sketch the graphs of y = ad y = 4 Hece, show that the equatio + 4 = 0 has eactly oe real root c Show that this root lies i the iterval (, 5) 6 f : l,, > 0 a O the same set of aes, sketch the curves y = l ad y = Hece show that the equatio f() = 0 has eactly oe real root The real root of f() = 0 is α c Fid the iteger such that < α < + 7 a O the same set of aes, sketch the curves y = e ad y = 5 Hece show that the equatio e + 5 = 0 has eactly oe egative ad oe positive real root c Show that the egative root lies i the iterval (, ) The positive root, α, is such that 0 < α < 0 +, where is a iteger d Fid the value of Solomo Press
2 C Worksheet For each equatio, show that it ca e rearraged ito the give iterative form Use this ad the give value of 0 to fid, ad Give your value of correct to 4 decimal places a = 0 + = = e = 0 + = l (8 5) 0 = c ta 5 + = 0 + = arcta (5 ) 0 = d l = 0 + = e ( + 4) 0 = 06 For each equatio, show that it ca e rearraged ito the give iterative form ad state the values of the costats a ad Use this ad the give value of 0 to fid, ad Give your value of correct to decimal places a e 6 = 0 + = a(l + ) 0 = 7 + cos = 0 + = c 6 = 0 + = a cos 0 = 08 a + 0 = d 5 l ( + ) 4 = 0 a + = e + 0 = 5 I each case, use the give iteratio formula ad value of 0 to fid a root of the equatio f() = 0 to the stated degree of accuracy Justify the accuracy of your aswers a f() = = log 0 (4 ) 0 = 044 decimal places f() = = = 05 sigificat figures c f() = si + = si 0 = 6 sigificat figures d f() = e 4 l + = 4 l (l ) 0 = 7 decimal places 4 f() = The equatio f() = 0 has a root i the iterval 4 < < a Use the iteratio formula + = ad the startig value 0 = to fid the value of this root correct to decimal places a The equatio f() = 0 ca e rearraged ito the iterative form + = Fid the values of the costats a ad i this formula The equatio f() = 0 has aother root i the iterval 0 < < c Usig the iteratio formula with your values from part ad the startig value 0 =, fid the value of this root correct to decimal places 5 f : arcsi 05 07,, 05 The equatio f() = 0 ca e rearraged ito the iterative form + = a si ( + c) a Fid the values of the costats a, ad c i this formula The equatio f() = 0 has a solutio i the iterval (0, 04) Usig the iterative formula with your values from part a ad the startig value 0 = 04, fid this solutio correct to decimal places Solomo Press
3 C Worksheet C a Show that the equatio 7 = 0 has a real root i the iterval (, 4) Usig the iterative formula + = 7 +, with 0 =, fid, ad, givig the value of correct to decimal places f() 4 cosec 5 + a Fid the values of f ad f(5) Hece show that the equatio f() = 0 has a root i the iterval (4, 5) The iterative formula + = a +, where a ad are costats, is used to fid this root si c Fid the values of a ad d Startig with 0 = 45, use the iterative formula with your values of a ad to fid further approimatios of the root, givig your fial aswer correct to decimal places y y = f() O The diagram shows the curve with equatio y = f() where Give that f(α) = 0, a show that 04 < α < 05, f : + l ( ),, > use the iterative formula + = ( + e to decimal places ), with 0 = 045, to fid the value of α correct 4 a O the same set of aes, sketch the curves y = cos ad y = for π < < π Show that the equatio cos = 0 has eactly oe positive ad oe egative real root c Show that the positive real root lies i the iterval [08, 09] d Use the iteratio formula + = root correct to decimal places 5 f() e 5 5 Show that the equatio f() = 0 a has a root i the iterval (4, 5), ca e writte as = e k, statig the value of k cos ad the startig value 0 = 08 to fid the positive k c Usig the iteratio formula + = e, with 0 = 5 ad the value of k foud i part, fid, ad Give the value of correct to decimal places Solomo Press
4 C Worksheet C cotiued 6 f : + 5, a Show that there is a solutio of the equatio f() = 0 i the iterval < < 4 Usig the iterative formula + = 5, with 0 = 4, fid,, ad 4 c Hece write dow a approimatio for this solutio of the equatio f() = 0 to a appropriate degree of accuracy Aother attempt is made to fid the solutio usig the iterative formula + = d Descrie the outcome of this attempt 7 f() = a Fid f () Hece show that the equatio f() = 0 has eactly oe real root c Show that this root lies i the iterval (, ) l (5 ) l d Use the iterative formula + = 45, with 0 =, to fid the root of f() = 0 correct to decimal places e Justify the accuracy of your aswer 8 y O A y = y = + l The diagram shows part of the curve with equatio y = + l ad the lie y = Give that the curve ad lie itersect at the poits A ad, show that a the -coordiates of A ad are the solutios of the equatio = the -coordiate of A lies i the iterval (04, 05), c the -coordiate of lies i the iterval (, 4) e, d Use the iteratio formula + = e, with 0 = 05, to fid the -coordiate of A correct to decimal places e Justify the accuracy of your aswer to part d 9 a O the same set of aes, sketch the graphs of y = 4 ad y = 5 + Show that the equatio 4 5 = 0 has eactly oe positive ad oe egative real root c Use the iteratio formula + = value of 4 correct to decimal places, with 0 = 8, to fid,, ad 4, givig the d Show that the equatio 4 a 5 = 0 ca e writte i the form = + values of a ad a, statig the e Use the iteratio formula + =, with 0 = 04 ad your values of a ad, to fid + the egative real root of the equatio correct to 4 decimal places Solomo Press
5 C Worksheet D y O y = e + 4 The diagram shows the curve y = e + 4 The curve itersects the y-ais at the poit A ad has a statioary poit at a Fid d y d () Fid a equatio for the taget to the curve at A () c Show that the -coordiate of lies i the iterval [ 04, 0] () d Usig the iteratio formula + = ( e ), with 0 = 0, fid the -coordiate of correct to decimal places The fuctio f is defied y where is measured i radias f() si ( 6) l ( + ),, The equatio f() = 0 has a root i the iterval k < < k +, where k is a positive iteger a Fid the value of k A si( 6) Use the iteratio formula + = e, with 0 = k, to fid three further approimatios for this root, givig your aswers to 4 decimal places () () A r C θ The diagram shows a sector AC of a circle, cetre C, radius r Agle AC is θ radias Give that the ratio of the area of the shaded segmet to the area of triagle AC is : 4, a show that 4θ 5 si θ = 0, use the iterative formula θ + = 5 si θ 4, with θ 0 =, to fid the value of θ correct to decimal places 4 f : e, The equatio f() = 0 ca e rearraged ito the iterative form + = l( a + ) a Fid the values of the costats a ad i this formula () The equatio f() = 0 has a solutio i the iterval (, ) Usig the iterative formula with your values from part a ad a suitale startig value, fid this solutio correct to decimal places Solomo Press
6 C Worksheet D cotiued 5 f : 9,, 0, g :, a Fid f () ad state its domai ad rage O the same set of aes, sketch the curves y = f() ad y = f () () c Show that the equatio f () + g() = 0 has a root i the iterval [, ] () d Use the iterative formula + = ( 9) 6 +, with 0 =, to fid this root correct to decimal places 6 a O the same diagram, sketch the curves y = ad y =, showig the coordiates of ay poits of itersectio with the coordiate aes () The curves itersect at the poit P Show that the -coordiate of P ca e foud y solvig the equatio + = 0 () c Use the iteratio formula + = +, with 0 = 0, to fid the -coordiate of P correct to decimal places 7 y y = 5 7 O A 5 The diagram shows the curve y = 7, 0, which crosses the -ais at the poit A, where = α, ad has a statioary poit at, where = β Show that a 4 < α < 5, < β <, c = β is a solutio to the equatio = () () d Use the iterative formula + = 4 sigificat figures , with 0 =, to fid β correct to 8 The curve with equatio y = l passes through the poit P (, ) a Fid a equatio for the ormal to the curve at P The ormal to the curve at P itersects the curve agai at the poit Q Show that the -coordiate of Q satisfies the equatio l = 0 () The -coordiate of Q is to e foud usig a iteratio of the form + = e k( ) c Fid the value of the costat k () d Usig 0 = 05, fid the -coordiate of Q correct to decimal places e Justify the accuracy of your aswer to part d () Solomo Press
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