IGCSE DIFFERENTIATION

These are questions on differentiation

Transcript

1. GCSE DIFFERENTIATION [ESTIMATED TIME: 75 minutes] (+ IGCSE) EXAM QUESTION PRACTICE 1. [2 marks] y = 2x 3 + 3x 2 − 5 1 Find dy dx ........................................................ (2) (Total for Question 4 is 8 marks) 2. Leave blank [8 marks] 2 x 217. Differentiate with respect A curve has equation y = xto – 4x + 1. 5x 2 this curve find (a) (a) For ........................................................ dy (i) , d x 3 (b) x .......................... (ii) the coordinates of the turning point. (c) 3 ........................................................ x ........................................................ ..........................(3) (4) (Total for Question 4 is 8 marks) (b) State, with a reason, whether the turning point is a maximum or a minimum. Differentiate with respect to x ....................................................................................................................................... 5 (a) ....................................................................................................................................... x (2) ........................................................ (c) Find the equation of the line of symmetry of the curve y = x2 – 4x + 1 (b) 3 x ........................................................ .......................... (2) Q17 (c) x(x + 4) (Total 8 marks) Contains questions which have been reproduced with the kind permission of Pearson Education Limited UK -1- ........................................................ (4) Questions compiled by: @Maths4Everyone
2. 3. [6 marks] 15 For the curve with equation y = 4x3 – 2x + 5 (i) find dy dx 71+O1! ()+.V OXE ) * F J I .............................................................. (ii) find the coordinates of the two points on the curve where the gradient of the curve is 1 = ! "#1!.'/(12%!+21!,#1!'>-V7:!-K!1)1/1.,%!$.!1+*#!&+2,!-K!,#1!i1..!`$+82+/9 ! L$M! [$.3!.L)M ! !99999999999999999999999999 ! L$$M! [$.3!!.L*3 M ! !99999999999999999999999999 L$$$M![$.3!.L) % * % *3 M ! ! !99999999999999999999999999 L$OM![$.3!.L)3 & *3 M ! ) and (............... QOX , ............... ) (............... , !9............... 9999999999999999999999999 ! (Total for Question 15 is 6 marks) H;&/,<)!)-,:N0J 4. I"E! C!*'2O1!#+%!1W'+,$-.!" ^ !J P F!; \ E! P = ! L+M! [$.3!,#1!82+3$1.,!-K!,#1!*'2O1!+,!L;R!PJM9 Do NOT write in this space. ! !99999999999999999999999999 )H!J ! L(M! D#+,!3-1%!0-'2!+.%412!,-!&+2,!L+M!,1))!0-'!+(-',!,#1!&-$.,!L;R!!JMj ! ! 16 !999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 HOJ *P43131A01624*H;&/,<)S)-,:N0J QI" [5 marks]
3. 5. [4 marks] 15 A particle moves along a straight line. The fixed point O lies on this line. The displacement of the particle from O at time t seconds is s metres, where s = t 3 – 6t + 3 (a) Find an expression for the velocity, v m/s, of the particle at time t seconds. v = ............................................... (2) (b) Find the acceleration of the particle at time 5 seconds. y = 2x 3 + 3x 2 − 5 1 Find dy dx 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s ........................................................ (2) (2) (Total is is 8 marks) (Totalfor forQuestion Question415 4 marks) 6. [3 marks] 216 Differentiate with respect x Make r the subject of the to formula A = 4r 2 – ʌU 2 where r is positive. (a) 5x 2 ........................................................ (b) 3 x ........................................................ (c) x r = ........................... (Total for Question 16 is 3 marks) ........................................................ (3) (Total for Question 4 is 8 marks) Do NOT write in this space. *P43130A01520* 15 Turn over
4. 7. Leave blank 15. The diagram shows the graph of y = x3 – 12x + 17 A is the maximum point on the curve. C is the minimum point on the curve. The curve crosses the y axis at B. A y B C O x For the equation y = x3 – 12x + 17 (a) find dy , dx .......................... (2) (b) find the gradient of the curve at B, .......................... (2) (c) find the coordinates of A and C. A (.......... , ..........) C (.......... , ..........) (4) (Total 8 marks) 16 *N23069A01620* Q15 [8 marks]
5. 8. [4 marks] 19 A particle is moving in a straight line which passes through a fixed point O. The displacement, s metres, of the particle from O at time t seconds is given by s = 10 + 9t 2 – t 3 (a) Find an expression for the velocity, v m/s, of the particle at time t seconds. v = ...................................................... (2) (b) Find the time at which the acceleration of the particle is zero. ...................................................... (2) seconds (Total for Question 19 is 4 marks) 9. [4 marks] 320 Differentiate PTR and QTSwith are respect chords to of xa circle. Q 5 (a) PT =x3 cm. P ST = 10 cm. 3 cm RT = 15 cm. (b) QT =3 x xcm. Diagram NOT accurately drawn x cm ........................................................ T 10 cm 15 cm S R ........................................................ (c) x(x + 4) Calculate the value of x. ........................................................ (4) (Total for Question 4 is 8 marks) x = ...................................................... 4 Differentiate with respect to x (Total for Question 20 is 2 marks) (a) x(x 2 − x) 16 *P42933A01620* ........................................................
6. 10. 1'#S'+ B.#&Y OTF* K+@,(S'+A#*+'Z,#5%-&+++*+_+#I+]+I#E+T+EQ#+ + 6#7+ ^%&2+ 2* 2# ( * + ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; I#K 6B7+ ^%&2+5A'+@--(2%&#5'*+-?+5A'+5,(&%&8+=-%&5*+-?+5A'+@,(S'; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ISK ROT I<'0-=*V*.-;N1K !"#$$$%&%-$'(! GJ <?;(*'L8; [8 marks]
7. 11. Leave blank 19. A particle moves in a straight line through a fixed point O. The displacement, s metres, of the particle from O at time t seconds is given by [4 marks] s = t3 í 5t2 + 8 (a) Find an expression for the velocity, v m/s, of the particle after t seconds. 3 Differentiate with respect to x v = ....................................... (2) 5 (a) x (b) Find the time at which the acceleration of the particle is 20 m/s2. ........................................................ (b) 3 x ........................................................ (c) x(x + 4) ................................ seconds ........................................................ (2) (4) Q19 (Total for Question 4 is4 8marks) marks) (Total 12. 4 [3 marks] Differentiate with respect to x (a) x(x 2 − x) ........................................................ (b) 1 x ........................................................ (3) (Total for Question 4 is 8 marks) *H34885A01924* 19 Turn over
8. 13. [7 marks] 21 y = x 2 – 16 x (a) Find dy dx dy = ....................................................... dx (3) y P x O The graph shows part of the curve with equation y = x 2 – 16 x The point P is the turning point of the curve. (b) Work out the coordinates of P. ( ................ , ................ ) (4) (Total for Question 21 is 7 marks) TOTAL FOR PAPER IS 100 MARKS *P44393A02324* 23
9. " !+^++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; H;&/,<)I)-,:N0J 14. [6 marks] QZE) H+=#(5%@.'+$-N'*+%&+#+*5(#%8J5+.%&'+5J(-,8J+#+?%"'2+=-%&5+); + 0J'+2%*=.#@'$'&5+-?+5J'+=#(5%@.'+?(-$+)+#5+5%$'+0+*'@-&2*+%*+1+$'5('*K+OJ'(' 1+^+0+C+G+\0+]+TD + 6#7+ `%&2+ 21 20 + + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; HIJ 6M7+ `%&2+5J'+N'.-@%5/+-?+5J'+=#(5%@.'+OJ'&+0+^+X + + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+$Y* HIJ 6@7+ `%&2+5J'+#@@'.'(#5%-&+-?+5J'+=#(5%@.'; +;;;;;;;;;;;;;;;;;;;;;;;;;;;;+$Y*C HIJ + H;&/,<)W)-,:N0J 15. 5 SQZ [3 marks] Differentiate with respect to x ⎛ 5⎞ (a) x 2 ⎜ 2x + ⎟ x⎠ ⎝ !"#$%#&'(-$#%! TW ;>:')&K7: ........................................................ (c) 5 x ........................................................ (3) (Total for Question 4 is 8 marks) 1 ThisIsATestQuestion ........................................................ (2)
10. 16. [8 marks] 14 A cm2 y cm x cm The diagram shows a rectangular photo frame of area A cm2. The width of the photo frame is x cm. The height of the photo frame is y cm. The perimeter of the photo frame is 72 cm. (a) Show that A = 36x – x2 (3) dA (b) Find dx .............................................................. (2) (c) Find the maximum value of A. A = .............................................................. (3) (Total for Question 14 is 8 marks) 14 *P40612A01420*
11. 17. [8 marks] 14 A farmer has 180 metres of fencing. With the 180 metres of fencing, he makes an enclosure divided into eight equal, rectangular pens. The fencing is used for the perimeter of each pen. y x The length of each pen is x metres and the width of each pen is y metres. (a) (i) Show that y = 18 – 1.2x The total area of the enclosure is A m2. (ii) Show that A = 144x – 9.6x2 (3) (b) Find dA dx ............................................................... (2) (c) Find the maximum value of A. A = ...................................... (3) (Total for Question 14 is 8 marks) 14 *P43028A01424*