RVHS H2 Math Differentiation Maclaurin series Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Maclaurin Series & Small angle approximation 1 Differentiation: Maclaurin Series & Small angle approximation 1. AJC/II/1 A curve with equation ( )fyx= passes through the point (0,1) and satisfies the relation 22 d(1 ) 1 d yx xy xx+ + = + . By further differentiation of this result, find the Maclaurin’s series for y in ascending powers of x, up to and including the term in 2x . [3] Deduce the approximate value of 0.01 0 d dd y xx , giving your answer correct to 3 decimal places. [2] 2. ACJC/I/11 (i) Given that cot 2 4yx =+  , find an equation relating d d y x and y . [2] Hence show that 232 32 d d d dd y y ykyx x dx  =+   , where k is a constant to be found. [2] (ii) Find the Maclaurin’s series for y up to and including the term in 3x if x is sufficiently small for powers of x higher than 3x to be neglected. [2] (iii) Using the Maclaurin’s expansion for y, estimate the value of 2 13cos ec 50   , giving your answer in the form 2a b c++ , where ,ab and c are constants to be determined. [4] 3. IJC/I/9 Given that ( )sin ln 1 3yx=−  , prove that ( ) ( ) 22 2 dd1 3 3 1 3 9 0 dd yyx x y xx − − − + = . Hence find Maclaurin’s series for y, up to and including the term in 3x . [6] Deduce an expansion for ( )cos ln 1 3 x− up to and including the term in 2x . [2]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Maclaurin Series & Small angle approximation 2 4. PJC/II/1 The diagram shows triangle ABC. It is given that the height AD is h units, 6ABD = and  ACD =  4 + x . Show that if x is sufficiently small for x3 and higher powers of x to be neglected, then 2 (1 3 2 + 2 )BC h x x + − . [5] 5. TPJC/2010/II/5 Given that ( )ln 1 2yx=+ , show that ( ) 2 2 dd1 2 2 0 dd yyx xx + + = . [1] (i) By repeated differentiation of this result, find the Maclaurin’s expansion for y up to and including the term in 5x . [5] (ii) By writing down the Maclaurin’s expansion of ( )ln 1 2 x− , show that 351 2 8 32ln 2 21 2 3 5 x x x xx +    = + + + −     . [2] (iii) Use the series in part (ii) for 1 4x= , find the exact value of ( ) 210 1 2 1 2 rr r  +=  + . [3] 6. AJC/2010/I/3 Given that xy 2sinln 1−= , show that 2d 1 4 2d y xyx −= . By repeated differentiation of this result, (i) Find the series expansion of y in ascending powers of x, up to and including the term in 3x . [4] (ii) Hence deduce the approximate value of 3e  , giving your answer in the form of ( ) 1 3 , ,8 a b a b+  where a and b are to be determined. [2] A B D C h

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Maclaurin Series & Small angle approximation 3 7. ACJC/2011/I/5 Given that f( )yx= where 11tan 2 tan 4yx −− =+ for 0.4 0.4x−   , show that ( ) ( ) 22 d1 2 1 d yxy x+ = + . By further differentiation of this result, find the Maclaurin’s series for y up to and including the term in 3.x Denote the above Maclaurin’s series for y by g( )x . Find the range of values of x for which the value of g( )x differs from f( )x by less than 0.5. 8. NYJC/2012/I/6 (a) In a triangle with vertices A, B and C, angle BAC is a right-angle and angle ABC = 3 x − . (i) Show that 1 3 tan 3 tan AB x AC x += − . [1] (ii) Hence, show that when x is small enough for x2 and higher powers of x to be neglected, then AB a bxAC + , where a and b are exact constants to be determined. [3] (b) A curve is defined by the equation ( ) 22 d11 d yx xy xx+ + = + and (0, 1) is a point on the curve. (i) Find the Maclaurin’s expansion of y up to and including the term in x2. [3] (ii) Hence, find the series expansion of e y , up to and including the term in x2. [3] 9. ACJC/2014/1/3 The curve ( )fyx= passes through the point ( )0,1 and satisfies the equation d 6 2 d cos 2 yy xx −= . Find the Maclaurin’s series of ( )f x , up to and including the term in 3x . [4] Using standard results given in the List of Formulae (MF 27), express 1 sin cos x x − as a power series of x , up to and including the term in 3x . [3] Using the two power series you have found, show to this degree of approximation, that ( )f x can be expressed as ( )tan2 sec2a x x b −+ where a and b are constants to be determined. [2]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Maclaurin Series & Small angle approximation 4 10. JJC/2014/II/1 (a) Given that e cosyx x+ = , show that d tan 1 0d y xx+ + = . (i) By further differentiation of this result obtain the Maclaurin’s series for y in terms of x, up to and including the term in x2. [3] (ii) Let the result in (i) be h( )x . Find the set of values of x for which h( )x is within 0.2 of the value of y. [2] (b) Expand, in ascending powers of x, 3 n xa − where a and n are integers and 0n , up to and including the term in x2. [2] It is given that the coefficient of x is four times the coefficient of x2 and the constant in the expansion is 1 4 . Find a and n. [3] 11. AJC/2010/II/1 Find the values of a and b if the expansion of 2 9 1 ax bx + + in ascending powers of x up to and including the term in 2x is 2353 6xx++ . With these values of a and b, state the range of values of x for which expansion is valid. [6] 12. MJC/2010/I/4 (i) Express ( ) ( )( ) 4f 1 3 2 xx xx −= ++ in partial fractions. Hence, expand ( )f x in ascending powers of x, up to and including the term in 3x . [4] (ii) State the range of values of x for which this expansion is valid. [1] (iii) Find the coefficient of nx in this expansion. [2] 13. IJC/2013/I/8 Find the expansion of 1 2 212 4 x x  +  − in ascending powers of x, up to and including the term in 2x . [4] (i) Find the set of values of x for which the expansion is valid. [2] (ii) By putting 1 4x= , show that 30 a b , where a and b are integers to be determined. [2]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Maclaurin Series & Small angle approximation 5 14. PJC/2011/I/2 (i) Find the first three terms in the expansion of 2 1 4 x+ in ascending powers of x . [3] (ii) Hence find the first four terms in the expansion of 2 1 4 x x + + . [2] (iii) State the set of values of x for which this expansion is valid. [1] 15. ACJC/2013/I/4 In the triangle PQR, PQ = 3, QR= 2 and angle PQR= 4  + radians. Given that  is a sufficiently small angle, show that ( ) 1 22 25 6 3PR a b c     + +  + + , for constants a, b and c to be determined. [5] 16. AJC/2015/I/5 In the triangle ABC, angle radians,BAC = angle radians6ACB = and 3.AC= Given that θ is sufficiently small, show that 2 2 23 , 2 2 3 AB a b c     + + +− where a, b and c are constants to be determined in exact form. [7] 17. JJC/2018/2/1 Given that sinf ( ) e xx = , use the standard series to find the series expansion for f ( )x in