Differentiation. Each correct answer in this section is worth two marks. 1. Differentiate 2 3 x with respect to x. A. 6 x

Transcription

1 Differentiation Paper 1 Section A Each correct answer in this section is worth two marks. 1. Differentiate 2 3 with respect to. A. 6 B C D Ke utcome Grade Facilit Disc. Calculator Content Source D 1.3 C NC C2, C3 HSN 091 hsn.uk.net Page 1

2 2. A function f is defined b f () = 3 + k Given that f (2) = 26, what is the value of k? A. 3 B. 7 2 C. 5 D. 10 Ke utcome Grade Facilit Disc. Calculator Content Source A 1.3 C NC C1 HSN A function f is defined b f () = k + 9. Given that f ( 1) = 13, what is the value of k? A. 7 2 B. 2 C. 5 D. 11 Ke utcome Grade Facilit Disc. Calculator Content Source B 1.3 C NC C1 HSN 020 hsn.uk.net Page 2

3 4. Given f () = , find the rate of change of f when = 2. A. 28 B. 31 C. 39 D. 43 Ke utcome Grade Facilit Disc. Calculator Content Source D 1.3 C NC C1, C1 HSN Given that f () = , what is the rate of change of f when = 8? A B C D. 130 Ke utcome Grade Facilit Disc. Calculator Content Source A 1.3 C NC C2, C6 HSN 115 hsn.uk.net Page 3

4 6. Differentiate 3 2 with respect to. A. 3 2 B C D. 2 3 Ke utcome Grade Facilit Disc. Calculator Content Source B 1.3 C CN C3 HSN What is the gradient of the tangent to the curve = at = 2? A B. 39 C. 52 D. 55 Ke utcome Grade Facilit Disc. Calculator Content Source C 1.3 C NC C4 HSN 02 hsn.uk.net Page 4

5 8. A curve has d d = Find the -values of the points on the curve where the tangent has a gradient of 4. A. 4 and 1 B. 1 and 4 C. 5 and 0 D. 0 and 5 Ke utcome Grade Facilit Disc. Calculator Content Source C 1.3 C CN C4 HSN A curve satisfing d = 4 has a tangent at = 3. d What is the gradient of an line perpendicular to this tangent? A. 12 B C D. 12 Ke utcome Grade Facilit Disc. Calculator Content Source B 1.3 C CN C4, G5 HSN 07 hsn.uk.net Page 5

6 10. A function is defined b f () = What is the largest range of -values for which f () is strictl increasing? A. < 9 4 B. > 9 4 C. 1 2 < < 4 D. < 1 2, > 4 Ke utcome Grade Facilit Disc. Calculator Content Source B 1.3 C CN C7 HSN A function is defined b f () = What is the largest range of -values for which f () is strictl decreasing? A. < 0 B. > 0 C. < 6 1 D. > 1 6 Ke utcome Grade Facilit Disc. Calculator Content Source C 1.3 C CN C7 HSN 142 hsn.uk.net Page 6

7 12. A curve has d d = What are the -values of the curve s stationar points? A. 3 and 2 B. 3 and 2 C. 3 and 2 D. 3 and 2 Ke utcome Grade Facilit Disc. Calculator Content Source C 1.3 C NC C8 HSN 067 hsn.uk.net Page 7

8 13. The curve with d d = , has a stationar point at = 2. What is the nature of this stationar point? A. maimum turning point B. minimum turning point C. rising point of infleion D. falling point of infleion Ke utcome Grade Facilit Disc. Calculator Content Source C 1.3 C NC C9 HSN 08 hsn.uk.net Page 8

9 14. The curve with d d = 2 6 9, has a stationar point at = 3. What is the nature of this stationar point? A. maimum turning point B. minimum turning point C. rising point of infleion D. falling point of infleion Ke utcome Grade Facilit Disc. Calculator Content Source D 1.3 C NC C9 HSN 051 [END F PAPER 1 SECTIN A] hsn.uk.net Page 9

10 Paper 1 Section B 15. A function f is defined b the formula f () = ( 1) 2 ( + 2) where R. (a) Find the coordinates of the points where the curve with equation = f () crosses the - and -aes. 3 (b) Find the stationar points of this curve = f () and determine their nature. 7 (c) Sketch the curve = f () Given f () = 3 2 (2 1), find f ( 1). 3 hsn.uk.net Page 10

11 17. The point P( 1, 7) lies on the curve with equation = Find the equation of the tangent to the curve at P Find d d where = Differentiate 2 ( + 2) with respect to. 4 hsn.uk.net Page 11

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13 22. Find the coordinates of the point on the curve = where the tangent to the curve makes an angle of 45 with the positive direction of the -ais. 4 Part Marks Level Calc. Content Answer U1 C3 4 C NC G2, C4 (2, 4) 2002 P1 Q4 1 sp: know to diff., and differentiate 2 pd: process gradient from angle 3 ss: equate equivalent epressions 4 pd: solve and complete 1 d d = m tang = tan 45 = = 1 4 (2, 4) 23. hsn.uk.net Page 13

14 24. The graph of a function f intersects the -ais at ( a, 0) and (e, 0) as shown. There is a point of infleion at (0, b) and a maimum turning point at (c, d). Sketch the graph of the derived function f. ( a, 0) 3 (e, 0) = f () (0, b) (c, d) Part Marks Level Calc. Content Answer U1 C3 3 C CN A3, C11 sketch 2002 P1 Q6 1 ic: interpret stationar points 2 ic: interpret main bod of f 3 ic: interpret tails of f roots at 0 and c (accept a statement to this effect) min. at LH root, ma. between roots both tails correct 25. If = 2, show that d d = Find f (4) where f () = 1. 5 hsn.uk.net Page 14

15 27. Given that = 2 2 +, find d d ( and hence show that 1 + d ) = 2. 3 d 28. If f () = k and f (1) = 14, find the value of k. 3 hsn.uk.net Page 15

16 A compan spends thousand P pounds a ear on advertising and this results in a profit of P thousand pounds. A mathematical model, illustrated in the diagram, suggests that P and are related b P = for Find the value of which gives the (12, 0) maimum profit. 5 Part Marks Level Calc. Content Answer U1 C3 5 C NC C11 = P1 Q6 1 ss: start diff. process 2 pd: process 3 ss: set derivative to zero 4 pd: process 5 ic: interpret solutions 1 dp d = or dp d = dp d = dp d = 0 4 = 0 and = 9 5 nature table about = 0 and = 9 hsn.uk.net Page 16

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19 33. Find the -coordinate of each of the points on the curve = at which the tangent is parallel to the -ais Calculate, to the nearest degree, the angle between the -ais and the tangent to the curve with equation = at the point where = 2. 4 hsn.uk.net Page 19

20 Find the equation of the tangent to the curve = at the point where = Find the equation of the tangent to the curve = at the point where = 1. 4 hsn.uk.net Page 20

21 39. Find the equation of the tangent to the curve with equation = at the point where = A ball is thrown verticall upwards. The height h metres of the ball t seconds after it is thrown, is given b the formula h = 20t 5t 2. (a) Find the speed of the ball when it is thrown (i.e. the rate of change of height with respect to time of the ball when it is thrown). 3 (b) Find the speed of the ball after 2 seconds. Eplain our answer in terms of the movement of the ball For what values of is the function f () = increasing? 5 hsn.uk.net Page 21

22 The point P( 2, b) lies on the graph of the function f () = (a) Find the value of b. 1 (b) Prove that this function is increasing at P. 3 hsn.uk.net Page 22

23 44. A sketch of the graph of = f () where f () = is shown below. The graph has a maimum at A and a minimum at B(3, 0). A = f () B(3, 0) (a) Find the coordinates of the turning point at A. 4 (b) Hence sketch the graph of = g() where g() = f ( + 2) + 4. Indicate the coordinates of the turning points. There is no need to calculate the coordinates of the points of intersection with the aes. 2 (c) Write down the range of values of k for which g() = k has 3 real roots. 1 Part Marks Level Calc. Content Answer U1 C3 (a) 4 C NC C8 A(1, 4) 2000 P1 Q2 (b) 2 C NC A3 sketch (translate 4 up, 2 left) (c) 1 A/B NC A2 4 < k < 8 1 ss: know to differentiate 2 pd: differentiate correctl 3 ss: know gradient = 0 4 pd: process 5 ic: interpret transformation 6 ic: interpret transformation 7 ic: interpret sketch 1 d d =... 2 d d = = 0 4 A = (1, 4) translate f () 4 units up, 2 units left 5 sketch with coord. of A ( 1, 8) 6 sketch with coord. of B (1, 4) 7 4 < k < 8 (accept 4 k 8) hsn.uk.net Page 23

24 45. A curve has equation = (a) Find algebraicall the coordinates of the stationar points. 6 (b) Determine the nature of the stationar points A curve has equation = Prove that this curve has no stationar points. 5 hsn.uk.net Page 24

25 47. Find the points on the curve with equation = at which the tangent to the curve makes an angle of 45 with the positive direction of the -ais. 6 Part Marks Level Calc. Content Answer U1 C3 6 C NC G2, C4 = 2, 1 B sp: know to differentiate 2 pd: differentiate 3 pd: process gradient from angle 4 ss: equate equivalent epressions 5 pd: process quadratic equation pd: complete 6 1 d d = 2 d d = m tang. = tan 45 = = 1 5 ( + 2)( 1) = 0 6 = 2, The graph of = 2 f ( + 1) is shown below. = 2 f ( + 1) The points where = 1 and = 3 are stationar points. Sketch the graph of = f (). 3 Part Marks Level Calc. Content Answer U1 C3 3 B CN A3, C11 sketch B ic: interpret transformation 2 ic: interpret stationar points 3 ic: completed sketch 1 graph in the question shows f translated 1 place to the left 2 stationar values of 2 and 4 give roots at 2 and 4 3 derived graph is a concave up parabola hsn.uk.net Page 25

26 49. A curve has equation = 3 + a 2 + b 2. When = 3, the tangent to the curve has equation 6 = 20. When = 2, the tangent to the curve makes an angle of 135 with the positive direction of the -ais. (a) Determine the values of a and b. 9 (b) Find the -coordinates of the curve s stationar points, in the form = 4 ± c. 3 3 Part Marks Level Calc. Content Answer U1 C3 (a1) 4 B CN C1, G4, C4 AT063 (a2) 5 B CN G2, C4, G8 a = 4, b = 3 (b) 3 C CN C8 (4 ± 7)/3 1 ss: know to differentiate 2 pd: differentiate 3 ic: interpret equation of line 4 pd: process 5 ic: interpret angle 6 pd: process 7 ss: start to solve simultaneous equations 8 pd: find one variable 9 pd: find other variable 10 ss: know to solve derivative = 0 11 ss: strateg (e.g. quadratic formula) 12 pd: complete 1 d/d = 2 = a + b 3 m tgt = 6 4 6a + b = 21 5 m tgt = tan 135 = 1 6 4a + b = a + b = 21, 4a + b = 13 8 a = 4 9 b = = 0 11 = (8 ± )/6 12 = (4 ± 7)/3 50. Given = 3 + 1, show that d d 3( 1) =. 3 Part Marks Level Calc. Content Answer U1 C3 3 C CN C1, A6 proof E pd: differentiate 2 ss: start to show result 3 ic: complete result 1 1 d d = ( 1) = 3( ) 3 complete hsn.uk.net Page 26

27 51. The point A( 1, p) lies on the curve with equation = (a) Find the value of p. 1 (b) Is the curve increasing or decreasing at A? Justif our answer. 3 Part Marks Level Calc. Content Answer U1 C3 (a) 1 C CN A6 p = 12 B (b) 3 C CN C1, C7 increasing 1 ic: evaluate p 2 sp: know to differentiate, and diff. 3 pd: evaluate derivative at A 4 ic: conclusion p = 12 d d = at A, d d = 9 > 0 so increasing d d hsn.uk.net Page 27

28 52. A rectangular garden has length l metres and breadth b metres. r b l Four semi-circular flower beds, of radius r metres, are to be made on one side of the garden. The remainder of the garden will be covered in gravel. The total perimeter of the garden is 30 metres. (a) Write down, in terms of l, epressions for b and r. 2 (b) Hence show that the area of garden available for gravel is given b 15l ( 1 + π 32) l 2. 2 (c) Find the value of l for which the area available for gravel is a maimum and 30 show that the corresponding value of r is π Part Marks Level Calc. Content Answer U1 C3 (a) 2 C CN CGD b = 15 l, r = l/8 AT090 (b) 2 C CN CGD proof (c) 4 C CN C11 l = π, proof 1 ic: interpret relationships 2 ic: interpret relationships 3 ic: start to find area 4 pd: complete 5 ss: start optimisation strateg 6 pd: complete method 7 ic: justif nature 8 ic: evaluate r b = 15 l 2 r = l/8 1 3 Area beds = πl 2 /32 or Area garden = (15 l)l 4 Area gravel = 15l (1 + π 32 )l2 5 A(l) = 32+π ( 32 l 2 ( π l) (l 6 = 32+π ) ) 32+π (32+π) 2 7 ma. when l = π 8 r = 30/(32 + π) or hsn.uk.net Page 28 5 stat. pts where A (l) = 0 6 l = π 7 l 240/(32 + π) A (l) Questions + marked 0 c SQA 8 r = 30/(32 All+ others π) c Higher Still Notes

29 53. A rectangle is formed under the graph of = 6 2, as shown in the diagram. = (a) Show that the area A of the rectangle is given b A() = for 0 < < 6. 2 (b) Hence find the value of which maimises the area of the rectangle, and the corresponding area. 5 Part Marks Level Calc. Content Answer U1 C3 (a) 2 B NC C11 Proof B (b) 5 C NC C11 A( 2) = 8 2 is ma. 1 ic: identif base and height 2 pd: form stated equation 3 ss: know to solve A () = 0 4 pd: find A () 5 pd: complete 6 ic: justif nature 7 ic: find and state area 1 base: 2, height: A() = 2(6 2 ) = at stat. pts. A () = 0 4 A () = = A () + 0 so = 2 gives ma. 7 A( 2) = 8 2 hsn.uk.net Page 29

30 54. A clindrical water tank, with solid top and bottom, has radius r metres and height h metres. The surface area of the tank is 4 square metres. h (a) (i) Find an epression for h in terms of r. (ii) Hence show that the volume, V cubic metres, of the tank is given b V = r(2 πr 2 ). 4 (b) Find the eact value of r for which the volume V is a maimum. 5 Part Marks Level Calc. Content Answer U1 C3 (a) 4 C CN CGD h = 2 πr r, proof AT093 (b) 5 C CN C11, C3, C8, C9 r = 2 3π r 1 ss: use area facts 2 pd: process 3 ss: use volume facts 4 ic: complete proof 5 6 pd: arrange in standard form pd: differentiate 7 ss: set derivative to zero 8 pd: process 9 ic: justification of nature 1 2πr 2 + 2πrh = 4 2 h = 2 πr r V = πr 2 h 4 V = r(2 πr 2 ) 3 5 V = 2r πr 3 6 dv dr = 2 3πr2 7 dv dr = 0 8 r = 2 3π 9 r 2/(3π) dv/dr Differentiate with respect to, where = 0. 3 Part Marks Level Calc. Content Answer U1 C3 3 C CN C3, C1, C E pd: epress in standard form 2 pd: differentiate non-negative powers 3 pd: differentiate negative power hsn.uk.net Page 30

31 56. Find the coordinates of the point on the curve = at which the tangent has a gradient of Part Marks Level Calc. Content Answer U1 C3 3 C CN C4, C1 (2, 15) E pd: differentiate 2 ss: know to equate derivative with 12 3 ic: complete for coordinates 1 1 d d = = 12 3 = 2, = A curve has equation = 4 3, where 0. Find the equation of the tangent to the curve at the point where = 4. 4 Part Marks Level Calc. Content Answer U1 C3 5 C CN C5, C3, C = 0 E ic: find corresponding -coord. 2 ss: epress in standard form 3 pd: differentiate fractional power 4 ss: find gradient of tangent 5 ic: state equation of tangent 1 = 9 2 = d d = at = 9, d d = = 2 9 ( 9) hsn.uk.net Page 31

32 58. The straight line = l() passes through the points (p, 0) and (p 2, 2), where p is a constant. (a) Determine d in terms of p. 2 d (b) Hence, or otherwise, find the values of p for which the gradient of the line is undefined. 2 Part Marks Level Calc. Content Answer U1 C3 (a) 2 B CN C6, G2 d d = 2 p 2 p (b) 2 B CN A6 p = 0, 1 B ss: know that d d = m l 2 pd: complete 3 ic: know to check for denom. = 0 4 pd: complete d d = m l p 2 p 3 undefined when p 2 p = 0 4 p = 0, 1 or 3 undefined if p = p 2 (i.e. points have same -coord.) 59. A function f is defined for = 0 b f () = Find the rate of change of f () when = 2. 3 Part Marks Level Calc. Content Answer U1 C3 3 C CN C6, C E pd: epress in differentiable form and differentiate 2 ic: know to evaluate derivative 3 pd: complete evaluation 1 f () = so f () = = rate of change is f (2) hsn.uk.net Page 32

33 60. Find the values of for which the curve with equation = is decreasing. 4 Part Marks Level Calc. Content Answer U1 C3 4 C CN C7 1 < < 1 B ss: know to consider d d < 0 2 pd: find d d 3 pd: process 4 ic: complete 1 solve d d < 0 2 d d = ( 1)( + 1) < < < 1 or 3 stat. values are 1, 1 4 nature table, leading to 1 < < Show that the curve with equation = is never increasing. 4 Part Marks Level Calc. Content Answer U1 C3 4 C CN C7 Proof B sd: know to use derivative, and differentiate 2 pd: process 3 ss: show that d d is never positive 4 ic: conclusion or d d = d d = 3( 3)2 ( 3) 2 0, so d d 0 so curve is never increasing 3 onl stat. value is 3 4 nature table, showing curve is never increasing hsn.uk.net Page 33

34 62. A function f is such that f () = a + a. (a) Given that = 2 is a stationar value of f, find the value of a. 3 (b) Hence determine whether f is increasing, decreasing or stationar when = 1. 2 Part Marks Level Calc. Content Answer U1 C3 (a) 3 C CN C8 a = 4 B (b) 2 C CN C7 decreasing 1 ss: use f ( 2) = 0 2 pd: process f ( 2) 3 pd: solve for a 1 f ( 2) = a 3 a = 4 4 ss: find f ( 1) 5 ic: conclusion 4 5 f ( 1) = 1 f ( 1) < 0 so f is decreasing 63. A function f is defined for b f () = Find the range of values of for which f () is strictl increasing. 4 Part Marks Level Calc. Content Answer U1 C3 4 C CN C7 < 2, > pd: find f () 2 ss: know to consider f () < 0 3 pd: process 4 ic: complete 1 1 f () = < 0 3 ( + 2)( 2) < 0 4 < 2, > 2 or 2 stat. pts. = ± f () hsn.uk.net Page 34

35 64. A curve has equation = 3 a , where a R is a constant. (a) State the restriction on the value of a. 1 (b) Hence show that the curve alwas has two distinct stationar points. 5 Part Marks Level Calc. Content Answer U1 C3 (a) 1 B CN A6 a = 0 B (b) 1 B CN A6 (b) 4 C CN C8 Proof 1 ic: interpret equation 2 ss: know to differentiate 3 pd: differentiate 4 pd: start to solve d d = 0 5 pd: obtain solutions of d d = 0 6 ic: justif distinctness 1 a = d d = d d = 3 a ( 3 a + 8) = 0 5 = 0 or = 8 3 a 6 these are distinct since a = Find the stationar points of the curve with equation = and justif their nature. 7 Part Marks Level Calc. Content Answer U1 C3 7 C NC C8, C9 ma. at (0, 19), min. at B (1, 20) 1 ss: know to differentiate 2 pd: differentiate 3 ss: set derivative to zero 4 pd: solve for 5 pd: evaluate the corresponding -values 6 ss: know to justif (e.g. nature table) 7 ic: interpret the stat. pts 1 2 d d = d d = d d = 0 4 = 0 or = 1 5 = 19 or = d d ma. at (0, 19), min. at (1, 20) hsn.uk.net Page 35

36 66. (a) Find the stationar points of the curve with equation = and justif their nature. 7 (b) (i) Show that ( 2) 2 ( + 1) = (ii) Hence sketch the graph of = Part Marks Level Calc. Content Answer U1 C3 (a) 7 C NC C8, C9 ma. at (0, 4), min. at (2, 0) B (bi) 1 C NC A6 Proof (bii) 3 B NC C10 Sketch 1 ss: know to differentiate 2 pd: differentiate 3 ss: set derivative to zero 4 pd: solve for 5 pd: evaluate the corresponding -values 6 ss: know to justif (e.g. nature table) 7 ic: interpret the stat. pts pd: epand and complete 9 ic: state -ais intersections 10 ic: state -ais intersections 11 ic: sketch d d = d d = d d = 0 4 = 0 or = 2 5 = 4 or = d d ma. at (0, 4), min. at (2, 0) ( )( + 1) and complete 9 -ais: ( 1, 0), (2, 0) 10 -ais: (0, 4) 11 sketch 8 [END F PAPER 1 SECTIN B] hsn.uk.net Page 36

37 Paper 2 1. A curve has equation = 16, > 0. Find the equation of the tangent at the point where = 4. 6 Part Marks Level Calc. Content Answer U1 C3 6 C CN C4, C5 = P2 Q2 1 ic: find corresponding -coord. 2 ss: epress in standard form 3 ss: start to differentiate 4 pd: diff. fractional negative power 5 ss: find gradient of tangent 6 ic: write down equ. of tangent 1 (4, 4) stated or implied b d d = m =4 = 2 6 ( 4) = 2( 4) hsn.uk.net Page 37

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39 3. A goldsmith has built up a solid which consists of a triangular prism of fied volume with a regular tetrahedron at each end. The surface area, A, of the solid is given b A() = ( ) where is the length of each edge of the tetrahedron. Find the value of which the goldsmith should use to minimise the amount of gold plating required to cover the solid. 6 Part Marks Level Calc. Content Answer U1 C3 6 A/B CN C11 = P2 Q6 1 ss: know to differentiate 2 pd: process 3 ss: know to set f () = 0 4 pd: deal with 2 5 pd: process 6 ic: check for minimum 1 A () = ( ) or A () = or = A () ve 0 +ve so = 2 is min. hsn.uk.net Page 39

40 4. The shaded rectangle on this map represents the planned etension to the village hall. It is hoped to provide the largest possible area for the etension. The Vennel 6 m Village hall 8 m Manse Lane The coordinate diagram represents the right angled triangle of ground behind the hall. The etension has length l metres and breadth b metres, as shown. ne corner of the etension is at the point (a, 0). (a) (i) Show that l = 5 4 a. (0, 6) l b (a, 0) (8, 0) (ii) Epress b in terms of a and hence deduce that the area, A m 2, of the etension is given b A = 3 4a(8 a). 3 (b) Find the value of a which produces the largest area of the etension. 4 Part Marks Level Calc. Content Answer U1 C3 (a) 3 A/B CN CGD proof 2002 P2 Q10 (b) 4 A/B CN C11 a = 4 1 ss: select strateg and carr through 2 ss: select strateg and carr through 3 ic: complete proof 4 ss: know to set derivative to zero 5 pd: differentiate 6 pd: solve equation 7 ic: justif maimum, e.g. nature table proof of l = 5 4 a 2 b = 3 5 (8 a) 3 complete proof leading to A = da da =... = a 6 a = 4 7 e.g. nature table, comp. the square hsn.uk.net Page 40

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49 A ball is thrown verticall upwards. After t seconds its height is h metres, where h = t 4 9t 2. (a) Find the speed of the ball after 1 second. 3 (b) For how man seconds is the ball travelling upwards? 2 hsn.uk.net Page 49

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52 A curve is such that d d = 2 1. Show that no tangent to this curve is parallel to the line with equation = 0. 4 Part Marks Level Calc. Content Answer U1 C3 4 C CN G2, C4 Proof B ss: epress line in standard form 2 ic: interpret gradient 3 ss: consider solving d d = grad. from 2 4 ic: conclusion 1 = so m line = m tang. = m line = 0 4 impossible, since 2 0 hsn.uk.net Page 52