The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

Transcription

1 Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3. The graph of f() = 6 4, for, is shown below. The region enclosed by the curve of f and the -ais is rotated 360 about the -ais. Find the volume of the solid formed.

2 4. Let f() = k 4. The point P(, k) lies on the curve of f. At P, the normal to the curve is parallel to y = Find the value of k A function f is defined for 4 3. The graph of f is given below. The graph has a local maimum when = 0, and local minima when = 3, =. Write down the -intercepts of the graph of the derivative function, f. (b) Write down all values of for which f () is positive. (c) At point D on the graph of f, the -coordinate is 0.5. Eplain why f () < 0 at D. 6. Let g() = sin. Find g (). (b) Find the gradient of the graph of g at = π. (Total 7 marks)

3 3 7. The graph of the function y = f () passes through the point, 4. The gradient function of f is given as f () = sin ( 3). Find f (). 8. Let f() = e cos. Find the gradient of the normal to the curve of f at = π. 9. The following diagram shows the graphs of the displacement, velocity and acceleration of a moving object as functions of time, t. Complete the following table by noting which graph A, B or C corresponds to each function. Function displacement acceleration Graph (b) Write down the value of t when the velocity is greatest.

4 0. The graph of y = between = 0 and = a is rotated 360 about the -ais. The volume of the solid formed is 3π. Find the value of a. (Total 7 marks). A function f has its first derivative given by f () = ( 3) 3. Find the second derivative. (b) Find f and f. () (c) The point P on the graph of f has -coordinate 3. Eplain why P is not a point of infleion. (Total 5 marks). Let f() = e 3 π and g() = sin. 3 Write down (i) (ii) f (); g (). (b) Let h() = e 3 π π sin. Find the eact value of h Consider f() = p +, 0, where p is a constant. Find f (). (b) There is a minimum value of f() when =. Find the value of p. 4. Find d. 3 3 (b) Given that d = ln P, find the value of P A particle moves along a straight line so that its velocity, v m s at time t seconds is given by v = 6e 3t + 4. When t = 0, the displacement, s, of the particle is 7 metres. Find an epression for s in terms of t. (Total 7 marks)

5 6. Let 5 3 f ( )d. Show that 5 f ( ) d 4. (b) Find the value of 5 f ) d f ( ) ( d. (5) (Total 7 marks) 7. The diagram shows part of the graph of y = f (). The -intercepts are at points A and C. There is a minimum at B, and a maimum at D. (i) Write down the value of f () at C. (ii) Hence, show that C corresponds to a minimum on the graph of f, i.e. it has the same -coordinate. (b) Which of the points A, B, D corresponds to a maimum on the graph of f? () (c) Show that B corresponds to a point of infleion on the graph of f. (Total 7 marks)

6 8. Find the equation of the tangent to the curve y = e at the point where =. Give your answer in terms of e. 9. Find ( 3 ) d. 0 (b) Find e d. (Total 7 marks) 0. The velocity v m s of a moving body at time t seconds is given by v = 50 0t. Find its acceleration in m s. (b) The initial displacement s is 40 metres. Find an epression for s in terms of t.. The velocity v of a particle at time t is given by v = e t + t. The displacement of the particle at time t is s. Given that s = when t = 0, epress s in terms of t.. Let f () = The tangents to the curve of f at the points P and Q are parallel to the -ais, where P is to the left of Q. Calculate the coordinates of P and of Q. Let N and N be the normals to the curve at P and Q respectively. (b) Write down the coordinates of the points where (i) the tangent at P intersects N ; (ii) the tangent at Q intersects N.

7 3 3. It is given that f ()d = 5. 3 Write down f ()d. 3 (b) Find the value of (3 + f ())d. 4. Let f () =. Given that f ( ) =, find f (). 5. Let f () = 3 cos + sin. Show that f () = 5 sin. π 3π (b) In the interval, one normal to the graph of f has equation = k. 4 4 Find the value of k. 6. Consider the function f () = Find the equation of the normal to the curve of f at the point where =. 7. Differentiate each of the following with respect to. y = sin 3 () (b) (c) y = y = tan ln

8 8. On the aes below, sketch a curve y = f () which satisfies the following conditions. f () f () f () 0 negative positive 0 0 positive 0 positive positive positive 0 positive negative 9. Let f () = e 5. Write down f (). (b) (c) Let g () = sin. Write down g (). Let h () = e 5 sin. Find h ().

9 30. The following diagram shows part of the curve of a function ƒ. The points A, B, C, D and E lie on the curve, where B is a minimum point and D is a maimum point. Complete the following table, noting whether ƒ () is positive, negative or zero at the given points. A B E f () (b) Complete the following table, noting whether ƒ () is positive, negative or zero at the given points. A C E ƒ () 3. The velocity, v m s, of a moving object at time t seconds is given by v = 4t 3 t. When t =, the displacement, s, of the object is 8 metres. Find an epression for s in terms of t.

10 3. The graph of a function g is given in the diagram below. The gradient of the curve has its maimum value at point B and its minimum value at point D. The tangent is horizontal at points C and E. Complete the table below, by stating whether the first derivative g is positive or negative, and whether the second derivative g is positive or negative. Interval g g a b e ƒ (b) Complete the table below by noting the points on the graph described by the following conditions. Conditions Point g () = 0, g () 0 g () 0, g () = A part of the graph of y = is given in the diagram below. The shaded region is revolved through 360 about the -ais. (b) Write down an epression for this volume of revolution. Calculate this volume.

11 34. Consider the function ƒ : k. Write down ƒ (). The equation of the tangent to the graph of ƒ at = p is y = 7 9. Find the value of (b) p; (c) k. 35. The diagram below shows the graph of ƒ () = e for 0 6. There are points of infleion at A and C and there is a maimum at B. Using the product rule for differentiation, find ƒ (). (b) Find the eact value of the y-coordinate of B. (c) The second derivative of ƒ is ƒ () = ( 4 + ) e. Use this result to find the eact value of the -coordinate of C. 36. The shaded region in the diagram below is bounded by f () =, = a, and the -ais. The shaded region is revolved around the -ais through 360. The volume of the solid formed is Find the value of a.

12 37. Figure shows the graphs of the functions f, f, f 3, f 4. Figure includes the graphs of the derivatives of the functions shown in Figure, eg the derivative of f is shown in diagram (d). Figure Figure y y f O O y y f (b) O O y y f 3 (c) O O f 4 y (d) y O O y (e) O Complete the table below by matching each function with its derivative. Function f Derivative diagram (d) f f 3 f 4

13 38. The diagram shows part of the curve y = sin. The shaded region is bounded by the curve and the lines y = 0 and = 3π. 4 y 3 4 3π 3π Given that sin = and cos =, calculate the eact area of the shaded region. 4 4 Working: Answer: Let f () = 3. f ( 5 h) f (5) Evaluate for h = 0.. h f ( 5 h) f (5) (b) What number does approach as h approaches zero? h Working: Answers:... (b)... (Total 4 marks)

14 40. The diagram shows part of the graph of y =. The area of the shaded region is units. y 0 a Find the eact value of a. Working: Answer:... (Total 4 marks)

15 Section B 4. The velocity v m s of a particle at time t seconds, is given by v = t + cost, for 0 t. Write down the velocity of the particle when t = 0. () When t = k, the acceleration is zero. π (b) (i) Show that k =. 4 π (ii) Find the eact velocity when t =. 4 (8) π dv π dv (c) When t <, > 0 and when t >, > 0. 4 dt 4 dt Sketch a graph of v against t. (d) Let d be the distance travelled by the particle for 0 t. (i) Write down an epression for d. (ii) Represent d on your sketch. (Total 6 marks) 4. The following diagram shows part of the graph of the function f() =. diagram not to scale The line T is the tangent to the graph of f at =. Show that the equation of T is y = 4. (b) Find the -intercept of T. (5)

16 (c) The shaded region R is enclosed by the graph of f, the line T, and the -ais. (i) Write down an epression for the area of R. (ii) Find the area of R. (9) (Total 6 marks) 43. The following diagram shows part of the graph of a quadratic function f. The -intercepts are at ( 4, 0) and (6, 0) and the y-intercept is at (0, 40). Write down f() in the form f() = 0( p)( q). (b) Find another epression for f() in the form f() = 0( h) + k. (c) Show that f() can also be written in the form f() = A particle moves along a straight line so that its velocity, v m s, at time t seconds is given by v = t 0t, for 0 t 6. (d) (i) Find the value of t when the speed of the particle is greatest. (ii) Find the acceleration of the particle when its speed is zero. (7) (Total 5 marks)

17 3 44. Let f() = 3. Part of the graph of f is shown below. 3 There is a maimum point at A and a minimum point at B(3, 9). Find the coordinates of A. (8) (b) Write down the coordinates of (i) the image of B after reflection in the y-ais; (ii) the image of B after translation by the vector ; 5 (iii) the image of B after reflection in the -ais followed by a horizontal stretch with scale factor. (6) (Total 4 marks)

18 cos 45. Let f() =, for sin 0. sin Use the quotient rule to show that f () =. sin (5) (b) Find f (). π π In the following table, f = p and f = q. The table also gives approimate values of f () and f () π near =. π π π f ().0 p.0 f () 0.03 q 0.03 (c) Find the value of p and of q. (d) Use information from the table to eplain why there is a point of infleion on the graph of f where =. π (Total 3 marks)

19 46. Consider the function f with second derivative f () = 3. The graph of f has a minimum point at A(, 4) and a maimum point at B 4 358,. 3 7 Use the second derivative to justify that B is a maimum. 3 (b) Given that f = + p, show that p = 4. (c) Find f(). (7) (Total 4 marks) 47. Let f() = 6 + 6sin. Part of the graph of f is shown below. The shaded region is enclosed by the curve of f, the -ais, and the y-ais. Solve for 0 < π. (i) 6 + 6sin = 6; (ii) sin = 0. (b) Write down the eact value of the -intercept of f, for 0 <. (c) The area of the shaded region is k. Find the value of k, giving your answer in terms of π. (5) () (6) π Let g() = 6 + 6sin. The graph of f is transformed to the graph of g. (d) Give a full geometric description of this transformation. 3π p p (e) Given that g( )d = k and 0 p < π, write down the two values of p. (Total 7 marks)

20 48. Let f() = 3. The following diagram shows part of the graph of f. diagram not to scale The point P (a, f), where a > 0, lies on the graph of f. The tangent at P crosses the -ais at the point Q, 0. 3 This tangent intersects the graph of f at the point R(, 8). a (i) Show that the gradient of [PQ] is. a 3 3 (ii) Find f. (iii) Hence show that a =. (7) The equation of the tangent at P is y = 3. Let T be the region enclosed by the graph of f, the tangent [PR] and the line = k, between = and = k where < k <. This is shown in the diagram below.

21 diagram not to scale (b) Given that the area of T is k + 4, show that k satisfies the equation k 4 6k + 8 = 0. (9) (Total 6 marks)

22 49. A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below. The point P(, y) is a verte of the rectangle and also lies on the circle. The angle between (OP) and the π -ais is θ radians, where 0 θ. Write down an epression in terms of θ for (i) ; (ii) y. Let the area of the rectangle be A. (b) Show that A = 8 sin θ. da (c) (i) Find. d (ii) (iii) Hence, find the eact value of θ which maimizes the area of the rectangle. Use the second derivative to justify that this value of θ does give a maimum. (8) (Total 3 marks)

23 a 50. Let f() =, 8 8, a. The graph of f is shown below. The region between = 3 and = 7 is shaded. Show that f( ) = f(). a( 3) (b) Given that f () =, find the coordinates of all points of infleion. 3 ( ) (7) a (c) It is given that f ( )d ln( ) C. (i) Find the area of the shaded region, giving your answer in the form p ln q. (ii) Find the value of f ( )d. 8 4 (7) (Total 6 marks)

24 0 5. Let f() = 3 +, for ±. The graph of f is given below. 4 diagram not to scale The y-intercept is at the point A. (i) Find the coordinates of A. (ii) Show that f () = 0 at A. (7) 40(3 4) (b) The second derivative f () =. Use this to 3 ( 4) (i) justify that the graph of f has a local maimum at A; (ii) eplain why the graph of f does not have a point of infleion. (6) (c) Describe the behaviour of the graph of f for large. () (d) Write down the range of f. (Total 6 marks)

25 5. Let f() =. Line L is the normal to the graph of f at the point (4, ). Show that the equation of L is y = (b) Point A is the -intercept of L. Find the -coordinate of A. In the diagram below, the shaded region R is bounded by the -ais, the graph of f and the line L. (c) Find an epression for the area of R. (d) The region R is rotated 360 about the -ais. Find the volume of the solid formed, giving your answer in terms of π. (8) (Total 7 marks)

26 53. Consider f () = Part of the graph of f is shown below. There is a maimum point at M, and 3 a point of infleion at N. Find f (). (b) Find the -coordinate of M. (c) Find the -coordinate of N. (d) The line L is the tangent to the curve of f at (3, ). Find the equation of L in the form y = a + b. (Total 4 marks) 54. Let f : sin 3. (i) Write down the range of the function f. (ii) Consider f () =, 0. Write down the number of solutions to this equation. Justify your answer. (5) (b) Find f (), giving your answer in the form a sin p cos q where a, p, q. π (c) Let g () = 3 sin (cos ) for 0. Find the volume generated when the curve of g is revolved through about the -ais. (7) (Total 4 marks)

27 55. The acceleration, a m s, of a particle at time t seconds is given by a = t + cost. Find the acceleration of the particle at t = 0. (b) Find the velocity, v, at time t, given that the initial velocity of the particle is m s. (c) Find 3, giving your answer in the form p q cos 3. 0 (5) (7) (d) What information does the answer to part (c) give about the motion of the particle? (Total 6 marks) 56. Let g() = (b) Find the two values of at which the tangent to the graph of g is horizontal. For each of these values, determine whether it is a maimum or a minimum. (8) (6) (Total 4 marks) 57. The diagram below shows part of the graph of y = sin. The shaded region is between = 0 and = m. Write down the period of this function. (b) Hence or otherwise write down the value of m. (c) Find the area of the shaded region. (6) (Total 0 marks)