and in each case give the range of values of x for which the expansion is valid.

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1 α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else P Kraft Further Maths A (MFPD) Assignment ζ (zeta) A Due w/b 0th October 7 DRILL Drills are the very basic techniques you need to solve maths problems. You need to practise these until you can do them quickly and accurately. Answers are not provided for drill questions. Sketch the following pairs of curves on the same axes without a calculator a) r = a( + cos θ) and r = acosθ b) r = + cos θ and r = 5cosθ c) r = 4cos θ and r = secθ d) r = sin θ and r = e) r = a( cos θ) and r = a( + cosθ) f) r = a( + cos θ) and r = a( 5 cosθ) PREPARATION Every week you will be required to do some preparation for future lessons, to be advised by your teacher.. Show that for small values of x, e e x+ x. x x. Find the series expansions, up to and including the term in x 4, of a) ln ( + x x ) b) ln ( 9+ 6x+ x ) and in each case give the range of values of x for which the expansion is valid.. a) Write down the series expansion of cos x in ascending powers of x, up to and including the term in x b) Hence, or otherwise, find the first 4 non-zero terms in the power series for sin x. 5. Find using the critical values method the complete set of values of x for which X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

2 CONSOLIDATION Show that for x >, n ( ) ( ) ln ( ) ln ln x x x+ + x + = n x x nx Find the set of values of x for which > x+ x 0. (a) By using the power series expansion for cos x and the power series expansion for ln ( + x), find the series expansion for ln ( cos x) in ascending powers of x up to and including the term in x 4. (b) Hence, or otherwise, obtain the first two non-zero terms in the series expansion for ln ( sec x ) in ascending powers of x.. (a) Find the Taylor expansion of cos x in ascending powers of 5 in π x 4. π x 4 up to and including the term (b) Use your answer to part a to obtain an estimate of cos, giving your answer to 6 decimal places. X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

3 Answers: 4 5x 7x 7x (a) x + +, < x ( ) smaller interval 4 4 x x x x (b) ln + + +, < x x 4x 8 x x 8 (a) x + + x (b) x + x (5) (6) (8) (9) - < x < or x > 8 (0a) (0b) 4 x x + + (a) x π 4 x π 4 5 x π Over the half term break do this Practice Paper completely then check the mark scheme FP PRACTICE PAPER. Using algebra, find the set of values of x for which 4 x x (b) (6 d.p.) x 5 > x. (7). (a) Find the general solution of the differential equation dy y = t + t. dt dt (b) Find the particular solution of this differential equation for which y = and (c) For this particular solution, calculate the value of y when t =. (8) = when t = 0. dt (5) () X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

4 . Figure The curve C shown in Fig. has polar equation r = a( + 5 cos θ ), π θ < π. (a) Find the polar coordinates of the points P and Q where the tangents to C are parallel to the initial line. This means, work out the values of θ for which there is a turning point, ie points at which the change of the y coordinate, as theta changes a tiny bit, is zero. ie) ie) ie) dy = 0 dθ d ( y) = 0 dθ d dθ ( r sinθ ) = 0 The curve C represents the perimeter of the surface of a swimming pool. The direct distance from P to Q is 0 m. (6) (b) Calculate the value of a. (c) Find the area of the surface of the pool. () This means work out r dθ. The limits are 0 all the way to π in order to get all the area. Or you could do 0 to π and double it, since the pool is symmetric. (6) X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

5 4. dy y + + y = 0. (a) Find an expression for. Given that y = and = at x = 0, (b) find the series solution for y, in ascending powers of x, up to an including the term in x. (c) Comment on whether it would be sensible to use your series solution to give estimates for y at x = 0. and at x = 50. () (5) (5) 5. (a) Sketch, on the same axes, the graphs with equation y = x, and the line with equation y = 5x. () (b) Solve the inequality x < 5x. () X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

6 6. (a) Use the substitution y = vx to transform the equation into the equation (4x + y)( x + y) =, x > 0 (I) x d v x = ( + v). (II) d x (b) Solve the differential equation II to find v as a function of x. (c) Hence show that y = x is a general solution of the differential equation I. ln x, where c is an arbitrary constant, x + c (5) () 7. (a) Find the value of λ for which λx cos x is a particular integral of the differential equation + 9y = sin x. (b) Hence find the general solution of this differential equation. The particular solution of the differential equation for which y = and (c) Find g(x). = at x = 0, is y = g(x). TOTAL MARKS: 75 X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

7 Answers: 4 5x 7x 7x (a) x + +, < x ( ) smaller interval 4 4 x x x x (b) ln + + +, < x x 4x 8 x x 8 (a) x + + x (b) x + x (5) (6) (8) (9) - < x < or x > 8 (0a) 4 x x (0b) 4 5 x x (a) π x π x π + + (b) (6 d.p.) FP PRACTICE PAPER Mark Schemes. (x > 0) x 5x > or x 5x = M (x + )(x ), critical values ½ and A, A x > A ft x < 0 x 5x < M Using critical value 0: ½ < x < 0 M, A ft Alt. x 5 < 0 x or (x 5)x > x M (x + )( x ) > 0 x or x(x + )(x ) > 0 M, A Critical values ½ and, x > A, A ft Using critical value 0, ½ < x < 0 M, A ft (7 marks). (a) m + 7m + = 0 (m + )(m + ) = 0 m = ½, ½t C.F. is y = Ae t + Be M, A P.I. y = at + bt + c B y = at + b, y = a X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

8 (a) + 7(at + b) + (at + bt + c) t + t M a =, a = 4 + b =, b = A c = 0, c = M, A ½t t General solution: y = Ae + Be + ( t t + ) A ft (8) ½t t (b) y = ½Ae Be + (t ) M t = 0, y = : = ½A B t = 0, y = : = + A + B one of M, A these Solve: A + B = 0, A + 6B = 4 A = 4 / 5, B = 4 / 5 M y 4 5 ½t t = ( t t + ) + (e e ) A (5) 4 ½ (c) t = : y = (e e ) + (=.445 ) B () 5 (4 marks). (a) y = r sinθ = a(sinθ + 5 sinθ cosθ ) dy = a(cosθ + dθ 5 cos θ ) M, A 5 cos θ + cosθ 5 = 0 ± cosθ =, cos θ = M, A θ = ±.07 A ft r = 4a A ft (6) (b) r sinθ = 0 M 8 a sinθ = 0, 0 a = =.795 8sinθ M, A () (c) ( + 5 cosθ ) = cosθ + 5cos θ B sin θ θ Integrate: 9θ sinθ Limits used: [...] 0 π = 8π + 5π (or upper limit: M, A 5π 9π + ) A X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

9 ½ π 0 r dθ = a (π ) 8 m M, A (6) (5 marks) 4. (a) dy dy dy y + ; + ; + = 0 marks can be awarded in(b) M A; B;B dy dy = or sensible correct alternative y B (5) (b) When x = 0 =, and = 5 MA, A ft 5 y = + x x + x... 6 M, A ft (5) (c) Could use for x = 0. but not for x = 50 as B approximation is best at values close to x = 0 B () ( marks) 5. (a) y y = 5x y = x shape points on axes B B () 5 x (b) x + = 5x - M x = 7 4 x > 7 4 A A ft () (5 marks) 6. (a) d v v + x,= (4 + v)( + v) M, M d x X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

10 (b) d v x = v + 5v + 4 v A d x d v x = (v + ) d x dv = ( v + ) x * A B, M (c) y = x + v + v = v = = ln x + c must have + c M A ln x + c ln x + c M A (5) x B () ln x + c (0 marks) X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

11 7. (a) y = λx cos x = λ cos x λx sin x M A d y = λ sin x λ sin x 9λx cos x 6λ sin x 9λx cos x + 9λx cos x = sin x λ = A A cso (b) λ 9 = 0 M λ = (±)i A y = A sin x + B cos x form M y = A sin x + B cos x + x cos x A ft on λ s (c) y =, x = 0 B = B M Aft on = A cos x B sin x + cos x 6x sin x λ s = A + A = 0 y = cos x + x cos x A ( marks) X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

12 ASSIGNMENT COVER SHEET zeta Name Maths Teacher Question Done Backpack Ready for test Notes Drill X:\Maths\TEAM - Doubles & Furthers\A doubles\assignments\mfpd\6 zeta a 7-8.doc Updated: /0/07

Further Maths A2 (M2FP2D1) Assignment ψ (psi) A Due w/b 19 th March 18

Further Maths A2 (M2FP2D1) Assignment ψ (psi) A Due w/b 19 th March 18 α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω The mathematician s patterns, like the painter s or the poet s, must be beautiful: the ideas, like the colours or the words, must fit together in a harmonious