k + ln x 2, para x 0, k, en el punto en que x = 2, tiene la 1. La normal a la curva y = x

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. Sea f la function,tal que, f ( ) d = 8.

1 1 1. La normal a la curva = k + ln, para 0, k, en el punto en que =, tiene la ecuación 3 + = b,donde b. halle eactamente el valor de k. Answer: (Total 6 puntos).. Sea f la function,tal que, f ( ) d = (a) Deducir el valor de, (i) 3 0 f ( ) d ; 3 ( +. (ii) ( f ) ) d d 0 (b) f ( )d = 8, escriba el valor de c de d. c Answers: (a) (i) (ii)... (b) c =..., d =... 1

2 (Total 6 marks) 3. En un eperimento, el número de bacterias n en un líquido, está dado por la fórmula n = 650 e kt, donde t, el tiempo, está en minutos k es una constante. El número de bacterias se duplica cada veinte minutos. Halle, (a) eactamente k; (b) la razón a la que el número de bacterias se incrementa cuando t = 90. Answers: (a).. (b)... (Total 6 marks) 4. En la tabla se muestra algunos valores de dos funciones, f g, así como de sus derivadas, f g f() g() 1 5 f () g ()

3 3 Calcular: (a) d (f() + g()), cuando = 4; d 3 ( ) (b) g' ( ) + 6 d. 1 Answers: (a).. (b)... (Total 6 marks) 5. La ecuación de una curva puede escribise en la forma, = a( p)( q). intersecta al eje en A(, 0) B(4, 0). La curva = f() se muestra abajo. 4 A B (a) (i) Escriba el valor de p de q. (ii) El punto (6, 8) está en la curva, halle el valor de a. 6 (iii) Escriba la ecuación de la curva en la forma = a + b + c. (5) 3

4 4 (b) (i) Halle d. d (ii) Se dibuja una tangente en el punto P. El gradiente de esta tangente es 7. halle las coordenadas de P. (4) (c) La línea L pasa a través de B(4, 0), es perpendicular a la tangente de la curva en el punto B. (i) Halle la ecuación de L. (ii) Halle la coordenada (abscisa) del punto en el que L intersecta a la curva otra vez.. (6) (Total 15 marks) 6. Sea f() = (3 + 4) 5.Hallar (a) (b) f (); f()d. Answers: (a).. (b)... (Total 6 marks) 4

5 5 7. Use the substitution u = + to find ( + ) 3 d (Total 6 marks) 8. Consider the differential equation d dθ = e θ. + 1 (a) Use the substitution = e θ to show that d d = ( +. 1) (b) d Find. ( + 1) (c) Hence find in terms of θ, if = when θ = 0. (4) (4) (Total 11 marks) 5

6 6 9. Find the total area of the two regions enclosed b the curve = and the line = + 3. Answer:... (Total 6 marks) 10. Let h() = ( )sin( 1) for 5 5. The curve of h() is shown below. There is a minimum point at R and a maimum point at S. The curve intersects the -ais at the points (a, 0) (1, 0) (, 0) and (b, 0). (, a 0) R S (, b 0) (a) Find the eact value of (i) a; (ii) b. () The regions between the curve and the -ais are shaded for a as shown. 6

7 7 (b) (i) Write down an epression which represents the total area of the shaded regions. (ii) Calculate this total area. (c) (i) The -coordinate of R is Find the -coordinate of S. (5) (ii) Hence or otherwise, find the range of values of k for which the equation ( )sin( 1) = k has four distinct solutions. (4) (Total 11 marks) 11. The diagram shows part of the graph of = e. = e P ln (a) Find the coordinates of the point P, where the graph meets the -ais. () The shaded region between the graph and the -ais, bounded b = 0 and = ln, is rotated through 360 about the -ais. (b) Write down an integral which represents the volume of the solid obtained. (4) (c) Show that this volume is π. (5) (Total 11 marks) 1. The diagram below shows part of the graph of f() = sin( + π) and the shaded region A. 7

8 8 A P Q 0 1 This graph crosses the -ais at P and Q. The point P has coordinates ( 3 π, 0). (a) Find the -coordinate of Q. (b) Use the substitution u = 3 + π to find f () d. (c) Hence, using our answer to (b), find the area of the region A. () (4) (Total 9 marks) 8

9 9 13. (a) Find (1 + 3 sin( + ))d. (b) The diagram shows part of the graph of the function f() = sin( + ). The area of the shaded region is given b f ( )d. 0 a Find the value of a. Answers: (a).. (b).. (Total 6 marks) 9

10 Calculate the area enclosed b the curves = ln and = e e, > 0. Answer:.. (Total 6 marks) 15. Using the substitution =, or otherwise, find d. Answer:..... (Total 6 marks) 16. Consider the function f() = cos + sin. (a) (i) Show that f( 4 π ) = 0. (ii) Find in terms of π, the smallest positive value of which satisfies f() = 0. 10

11 11 The diagram shows the graph of = e (cos + sin ), 3. The graph has a maimum turning point at C(a, b) and a point of infleion at D. 6 C(a, b) 4 D (b) Find d. d (c) Find the eact value of a and of b. π (d) Show that at D, = e 4. (4) (5) (e) Find the area of the shaded region. () (Total 17 marks) 11

12 Given that g( )d = 10, deduce the value of 3 1 (a) g ( )d; 1 3 ( (b) g( ) + 4)d. 1 1 Answers: (a).. (b)... (Total 6 marks) 18. The diagram below shows the shaded region R enclosed b the graph of = -ais, and the vertical line = k. 1+, the = 1+ R k (a) Find d. d 1

13 13 (b) Using the substitution u = 1 + or otherwise, show that 3 1+ d = (1 + ) 3 + c. (c) Given that the area of R equals 1, find the value of k. (Total 9 marks) 19. In this question, s represents displacement in metres, and t represents time in seconds. (a) The velocit v ms 1 d s of a moving bod ma be written as v = = 30 at, where a is a dt constant. Given that s = 0 when t = 0, find an epression for s in terms of a and t. (5) Trains approaching a station start to slow down when the pass a signal which is 00 m from the station. (b) The velocit of Train 1 t seconds after passing the signal is given b v = 30 5t. (i) (ii) Write down its velocit as it passes the signal. Show that it will stop before reaching the station. (5) (c) Train slows down so that it stops at the station. Its velocit is given b d s v = = 30 at, where a is a constant. dt (i) Find, in terms of a, the time taken to stop. (ii) Use our solutions to parts (a) and (c)(i) to find the value of a. (5) (Total 15 marks) 13

14 14 0. The diagram below shows a sketch of the graph of the function = sin(e ) where 1, and is in radians. The graph cuts the -ais at A, and the -ais at C and D. It has a maimum point at B. A B C D (a) Find the coordinates of A. (b) The coordinates of C ma be written as (ln k, 0). Find the eact value of k. () () (c) (i) Write down the -coordinate of B. (ii) Find d. d (iii) Hence, show that at B, = ln π. (6) (d) (i) Write down the integral which represents the shaded area. (ii) Evaluate this integral. (5) (e) (i) Cop the above diagram into our answer booklet. (There is no need to cop the shading.) On our diagram, sketch the graph of = 3. (ii) The two graphs intersect at the point P. Find the -coordinate of P. (Total 18 marks) 14

15 15 1. The diagram below shows the graph of 1 = f(), On the aes below, sketch the graph of = 0 f ( t)dt, marking clearl the points of infleion (Total 6 marks) 15

16 16. The diagram shows part of the curve = sin. The shaded region is bounded b the curve and the lines = 0 and = 4 π 3. 3π 4 π 3π Given that sin 4 = 3π and cos 4 =, calculate the eact area of the shaded region. Answer:.. (Total 6 marks) 16

17 17 3. The figure below shows part of the curve = The curve crosses the -ais at the points A, B and C. 0 A B C (a) Find the -coordinate of A. (b) Find the -coordinate of B. (c) Find the area of the shaded region. Answers: (a).. (b)... (c).. (Total 6 marks) 17

18 Consider the function f ( ) =, where n (a) Show that the derivative f ( ) = f ( ). (b) Sketch the function f(), showing clearl the local maimum of the function and its horizontal asmptote. You ma use the fact that 1n lim = 0. (5) (c) Find the Talor epansion of f() about = e, up to the second degree term, and show that this polnomial has the same maimum value as f() itself. (5) (Total 13 marks) 5. Let = sin(k) k cos(k), where k is a constant. d Show that = k sin( k). d (Total 3 marks) 6. Find the area enclosed b the curves = and = e 3, given that Answers:.. (Total 3 marks) 18

19 19 7. The diagram below shows part of the graph of the function 3 f : a A P 15 0 Q B The graph intercepts the -ais at A( 3,0), B(5,0) and the origin, O. There is a minimum point at P and a maimum point at Q. (a) The function ma also be written in the form where a < b. Write down the value of (i) a; f : a ( a) ( b), (ii) b. () (b) Find (i) f (); (ii) the eact values of at which f '() = 0; (iii) the value of the function at Q. (7) (c) (i) Find the equation of the tangent to the graph of f at O. (ii) This tangent cuts the graph of f at another point. Give the -coordinate of this point. (4) (d) Determine the area of the shaded region. () (Total 15 marks) 19

20 Let f(t) = t 3 1. Find f ( t) dt. 5 t 3 Answers:.. (Total 3 marks) 9. Find (a) sin (3 + 7) d; 4 d (b) e. Answers:.. (Total 4 marks) 0

21 1 1n 30. The function f is given b f ( ) =, > n (a) (i) Show that f ( ) =. Hence (ii) prove that the graph of f can have onl one local maimum or minimum point; (iii) find the coordinates of the maimum point on the graph of f. 1n 3 (b) B showing that the second derivative f ( ) = or otherwise, 3 find the coordinates of the point of infleion on the graph of f. (6) (6) (c) The region S is enclosed b the graph of f, the -ais, and the vertical line through the maimum point of f, as shown in the diagram below. =( f ) 0 (i) Would the trapezium rule overestimate or underestimate the area of S? Justif our answer b drawing a diagram or otherwise. (ii) Find f ( ) d, b using the substitution u = ln, or otherwise. (iii) Using f ( ) d, find the area of S. (4) (4) (d) The Newton-Raphson method is to be used to solve the equation f () = 0. (i) Show that it is not possible to find a solution using a starting value of 1 = 1. (ii) Starting with 1 = 0.4, calculate successive approimations, 3,... for the root of the equation until the absolute error is less than Give all answers correct to five decimal places. (4) (Total 30 marks) 1

23 3 31. In this part of the question, radians are used throughout. The function f is given b f() = (sin ) cos. The following diagram shows part of the graph of = f(). A C O B The point A is a maimum point, the point B lies on the -ais, and the point C is a point of infleion. (a) Give the period of f. (1) (b) From consideration of the graph of = f(), find to an accurac of one significant figure the range of f. (1) (c) (i) Find f (). (ii) Hence show that at the point A, cos = 1. 3 (iii) Find the eact maimum value. (9) (d) Find the eact value of the -coordinate at the point B. (1) 3

24 4 (e) (i) Find f () d. (ii) Find the area of the shaded region in the diagram. (4) (f) Given that f () = 9(cos ) 3 7 cos, find the -coordinate at the point C. (4) (Total 0 marks) 3. The diagram shows a sketch of the graph of = f () for a b. = f ( ) a b 4

25 5 On the grid below, which has the same scale on the -ais, draw a sketch of the graph of = f() for a b, given that f(0) = 0 and f() 0 for all. On our graph ou should clearl indicate an minimum or maimum points, or points of infleion. a b (Total 3 marks) 33. (a) Sketch and label the graphs of f ( ) = e and g ( ) = e 1 for 0 1, and shade the region A which is bounded b the graphs and the -ais. (b) Let the -coordinate of the point of intersection of the curves = f() and = g() be p. Without finding the value of p, show that (c) (d) p < area of region A < p. Find the value of p correct to four decimal places. Epress the area of region A as a definite integral and calculate its value. (4) () (Total 1 marks) 5

26 6 6

27 7 34. The area of the enclosed region shown in the diagram is defined b +, a +, where a > 0. 0 a This region is rotated 360 about the -ais to form a solid of revolution. Find, in terms of a, the volume of this solid of revolution. Answers:.. (Total 4 marks) 35. A rock-climber slips off a rock-face and falls verticall. At first he falls freel, but after seconds a safet rope slows him down. The height h metres of the rock-climber after t seconds of the fall is given b: H = 50 5t, 0 t H = 90 40t + 5t, t 5 (a) Find the height of the rock-climber when t =. (b) Sketch a graph of h against t for 0 t 5. (1) (4) (c) Find dh dt for: (i) 0 t 7

28 8 (ii) t 5 () 8

29 9 (d) Find the velocit of the rock-climber when t =. () (e) Find the times when the velocit of the rock-climber is zero. (f) Find the minimum height of the rock-climber for 0 t 5. (Total 15 marks) 36. Given that f() = ( + 5) 3 find (a) f (); (b) f ( )d. Answers: (a).. (b)... (Total 4 marks) 9

30 A curve with equation =f() passes through the point (1, 1). Its gradient function is f () = + 3. Find the equation of the curve. (Total 4 marks) 38. The main runwa at Concordville airport is km long. An aeroplane, landing at Concordville, touches down at point T, and immediatel starts to slow down. The point A is at the southern end of the runwa. A marker is located at point P on the runwa. A T P B km Not to scale As the aeroplane slows down, its distance, s, from A, is given b s = c + 100t 4t, where t is the time in seconds after touchdown, and c metres is the distance of T from A. (a) The aeroplane touches down 800 m from A, (i.e. c = 800). (i) (ii) Find the distance travelled b the aeroplane in the first 5 seconds after touchdown. Write down an epression for the velocit of the aeroplane at time t seconds after touchdown, and hence find the velocit after 5 seconds. () 30

31 31 The aeroplane passes the marker at P with a velocit of 36 m s 1. Find (iii) how man seconds after touchdown it passes the marker; () (iv) the distance from P to A. (b) Show that if the aeroplane touches down before reaching the point P, it can stop before reaching the northern end, B, of the runwa. (5) (Total 15 marks) 39. Let f() = ln 5 3, 0. 5 < <, a, b ; (a, b are values of for which f() is not defined). (a) (i) Sketch the graph of f(), indicating on our sketch the number of zeros of f(). Show also the position of an asmptotes. (ii) Find all the zeros of f(), (that is, solve f() = 0). (b) Find the eact values of a and b. () (c) Find f(), and indicate clearl where f() is not defined. (d) Find the eact value of the -coordinate of the local maimum of f(), for 0 < < 1.5. (You ma assume that there is no point of infleion.) (e) Write down the definite integral that represents the area of the region enclosed b f() and the -ais. (Do not evaluate the integral.) () (Total 16 marks) 31

32 3 40. The diagram shows the graph of = f (). = f ( ) Indicate, and label clearl, on the graph (a) (b) (c) the points where = f() has minimum points; the points where = f() has maimum points; the points where = f() has points of infleion. (Total 3 marks) 3

33 Let f () = 3. (a) (b) f ( 5 + h) f (5) Evaluate for h = 0.1. h f ( 5 + h) f (5) What number does approach as h approaches zero? h Answers: (a).. (b)... (Total 4 marks) 4. Using the substitution u = 1 + 1, or otherwise, find the integral d. Answers:.. (Total 4 marks) 33

34 The diagram shows part of the graph of = 1. The area of the shaded region is units. 0 1 a Find the eact value of a. Answers: (Total 4 marks) 34

35 (a) Find the equation of the tangent line to the curve = ln at the point (e, 1), and verif that the origin is on this line. (4) (b) Show that d ( ln ) = ln. d () (c) The diagram shows the region enclosed b the curve = ln, the tangent line in part (a), and the line = 0. 1 (e, 1) Use the result of part (b) to show that the area of this region is 1 e 1. (4) (Total 10 marks) 45. The area between the graph of = e and the -ais from = 0 to = k (k > 0) is rotated through 360 about the -ais. Find, in terms of k and e, the volume of the solid generated. Answers: (Total 4 marks) 35

36 In the diagram, PTQ is an arc of the parabola = a, where a is a positive constant, and PQRS is a rectangle. The area of the rectangle PQRS is equal to the area between the arc PTQ of the parabola and the -ais. T S R P O Q =a Find, in terms of a, the dimensions of the rectangle. Answers: (Total 4 marks) 36

Topic 6: Calculus Differentiation. 6.1 Product Quotient Chain Rules Paper 2

Topic 6: Calculus Differentiation Standard Level 6.1 Product Quotient Chain Rules Paper 1. Let f(x) = x 3 4x + 1. Expand (x + h) 3. Use the formula f (x) = lim h 0 f ( x + h) h f ( x) to show that the