IB Math Standard Level Calculus Practice 06-07

Transcription

1 Calculus Practice Problems #-22 are from Paper and should be done without a calculator. Problems 23-3 are from paper 2. A calculator may be used. An asterisk by the problem number indicates that a calculator is acceptable.. Let f (x) = e 5x. Write down f (x). Let g (x) = sin 2x. Write down g (x). (c) Let h (x) = e 5x sin 2x. Find h (x). 2. 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 maximum point. Complete the following table, noting whether ƒ (x) is positive, negative or zero at the given points. A B E f (x) Complete the following table, noting whether ƒ (x) is positive, negative or zero at the given points. A C E ƒ (x) 3. The velocity, v m s, of a moving object at time t seconds is given by v = 4t 3 2t. When t = 2, the displacement, s, of the object is 8 metres. Find an expression for s in terms of t. 4. The graph of a function g is given in the diagram below. The gradient of the curve has its maximum 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 < x < b e < x < ƒ Complete the table below by noting the points on the graph described by the following conditions. Conditions Point g (x) = 0, g (x) < 0 g (x) < 0, g (x) = 0 Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page of 0

2 5. A part of the graph of y = 2x x 2 is given in the diagram below. The shaded region is revolved through 360 about the x-axis. Write down an expression for this volume of revolution. Calculate this volume. 6. Consider the function ƒ : x α 3x 2 5x + k. Write down ƒ (x). The equation of the tangent to the graph of ƒ at x = p is y = 7x 9. Find the value of p; (c) k. 7. The diagram below shows the graph of ƒ (x) = x 2 e x for 0 x 6. There are points of inflexion at A and C and there is a maximum at B. Using the product rule for differentiation, find ƒ (x). Find the exact value of the y-coordinate of B. (c) The second derivative of ƒ is ƒ (x) = (x 2 4x + 2) e x. Use this result to find the exact value of the x- coordinate of C. 8. The displacement s metres at time t seconds is given by s = 5 cos 3t + t 2 + 0, for t 0. Write down the minimum value of s. Find the acceleration, a, at time t. (c) Find the value of t when the maximum value of a first occurs. Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 2 of 0

3 9. The following diagram shows the graph of a function f. Consider the following diagrams. Complete the table below, noting which one of the diagrams above represents the graph of f (x); f (x). Graph Diagram f (x) f " (x) 0. The velocity v in m s of a moving body at time t seconds is given by v = e 2t. When t = 0 5. the displacement of the body is 0 m. Find the displacement when t =.. The shaded region in the diagram below is bounded by f (x) = x, x = a, and the x-axis. The shaded region is revolved around the x-axis through 360. The volume of the solid formed is 0.845π. Find the value of a. 2. The velocity v of a particle at time t is given by v = e 2t + 2t. The displacement of the particle at time t is s. Given that s = 2 when t = 0, express s in terms of t. Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 3 of 0

4 3. The graph of the function y = f (x), 0 x 4, is shown below. Write down the value of (i) f (); (ii) f. On the diagram below, draw the graph of y = 3 f ( x). (c) On the diagram below, draw the graph of y = f (2x). 4. Let f (x) = x 3 3x 2 24x +. The tangents to the curve of f at the points P and Q are parallel to the x-axis, where P is to the left of Q. Calculate the coordinates of P and of Q. Let N and N2 be the normals to the curve at P and Q respectively. Write down the coordinates of the points where (i) the tangent at P intersects N2; (ii) the tangent at Q intersects N. Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 4 of 0

5 5. It is given that 3 Write down 3 f (x)dx = 5. Find the value of 3 2 f (x)dx. 6. Let f (x) = 2x 2 2. Given that f ( ) =, find f (x). (3x 2 + f (x))dx. 7. The velocity, v, in m s of a particle moving in a straight line is given by v = e 3t 2, where t is the time in seconds. Find the acceleration of the particle at t =. At what value of t does the particle have a velocity of 22.3 m s? (c) Find the distance travelled in the first second. 8. Let f (x) = 3 cos 2x + sin 2 x. Show that f (x) = 5 sin 2x. π 3π In the interval x, one normal to the graph of f has equation x = k. 4 4 Find the value of k. 9. The following diagram shows part of the graph of y = cos x for 0 x 2π. Regions A and B are shaded. Write down an expression for the area of A. Calculate the area of A. (c) Find the total area of the shaded regions. () () 20. Consider the function f (x) = 4x 3 + 2x. Find the equation of the normal to the curve of f at the point where x =. 2. Differentiate each of the following with respect to x. y = sin 3x () y = x tan x (c) y = ln x x Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 5 of 0

6 22. On the axes below, sketch a curve y = f (x) which satisfies the following conditions. x f (x) f (x) f (x) 2 x < 0 negative positive 0 0 positive 0 < x < positive positive 2 positive 0 < x 2 positive negative You may use a calculator on the remaining problems. 23. * Let f (x) = 4 3 x 2 + x + 4. (i) Write down f (x). (ii) Find the equation of the normal to the curve of f at (2, 3). (iii) This normal intersects the curve of f at (2, 3) and at one other point P. Find the x-coordinate of P. Part of the graph of f is given below. (9) Let R be the region under the curve of f from x = to x = 2. (i) Write down an expression for the area of R. (ii) Calculate this area. (iii) The region R is revolved through 360 about the x-axis. Write down an expression for the volume of the solid formed. (c) Find k f ( x) dx, giving your answer in terms of k. Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 6 of 0 (6) (6) (Total 2 marks)

7 24. * Consider the functions f and g where f (x) = 3x 5 and g (x) = x 2. Find the inverse function, f. (c) Let h (x) = Given that g (x) = x + 2, find (g f) (x). Given also that (f g) (x) f ( x), x 2. g ( x) x+3, solve (f g) (x) = (g f) (x). 3 (d) (i) Sketch the graph of h for 3 x 7 and 2 y 8, including any asymptotes. (ii) Write down the equations of the asymptotes. (e) 3x 5 The expression may also be written as 3 + x 3 (i) Find h(x) dx. (ii) Hence, calculate the exact value of 5 3 Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 7 of 0 x 2 h (x)dx. (f) On your sketch, shade the region whose area is represented by 5. Use this to answer the following. 3 h (x)dx. 25. * The function f is defined as f (x) = (2x +) e x, 0 x 3. The point P(0, ) lies on the graph of f (x), and there is a maximum point at Q. Sketch the graph of y = f (x), labelling the points P and Q. (i) Show that f (x) = ( 2x) e x. (ii) Find the exact coordinates of Q. (c) The equation f (x) = k, where k, has two solutions. Write down the range of values of k. (5) (5) () (Total 8 marks) (d) Given that f (x) = e x ( 3 + 2x), show that the curve of f has only one point of inflexion. (e) Let R be the point on the curve of f with x-coordinate 3. Find the area of the region enclosed by the curve and the line (PR). (7) (Total 2 marks) 26. * The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x p)(x q), where p, q. Write down (i) the value of p and of q; (ii) the equation of the axis of symmetry of the curve. Find the equation of the function in the form y = (x h) 2 + k, where h, k. d y (c) Find. dx (d) Let T be the tangent to the curve at the point (0, 5). Find the equation of T. (Total 0 marks) (7)

8 27. * The function f is defined as f (x) = e x sin x, where x is in radians. Part of the curve of f is shown below. There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects the x-axis at the point C. Write down the x-coordinate of the point C. (i) Find f (x). (ii) Write down the value of f (x) at the point B. (c) Show that f (x) = 2e x cos x. (d) (i) Write down the value of f (x) at A, the point of inflexion. (ii) Hence, calculate the coordinates of A. (e) Let R be the region enclosed by the curve and the x-axis, between the origin and C. (i) Write down an expression for the area of R. (ii) Find the area of R. 3x 28. * Let f (x) = p +, where p, q 2 2. x q Part of the graph of f, including the asymptotes, is shown below. () (Total 5 marks) The equations of the asymptotes are x =, x =, y = 2. Write down the value of (i) p; (ii) q. Let R be the region bounded by the graph of f, the x-axis, and the y-axis. (i) Find the negative x-intercept of f. (ii) Hence find the volume obtained when R is revolved through 360 about the x-axis. 2 (c) (i) Show that f (x) = 3( x + ) 2 ( x ) 2. (ii) Hence, show that there are no maximum or minimum points on the graph of f. (8) (d) Let g (x) = f (x). Let A be the area of the region enclosed by the graph of g and the x-axis, between x = 0 and x = a, where a > 0. Given that A = 2, find the value of a. (7) (Total 24 marks) (7) Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 8 of 0

9 29. * The function f (x) is defined as f (x) = 3 + 5, x. 2x 5 2 Sketch the curve of f for 5 x 5, showing the asymptotes. Using your sketch, write down (i) the equation of each asymptote; (ii) the value of the x-intercept; (iii) the value of the y-intercept. (c) The region enclosed by the curve of f, the x-axis, and the lines x = 3 and x = a, is revolved through 360 about the x-axis. Let V be the volume of the solid formed. (i) Find ( ) dx. 2 2x 5 2x 5 (ii) 28 Hence, given that V = π + 3ln3, find the value of a. 3 (0) (Total 7 marks) 30. * Consider the function f (x) e (2x ) 5 + ( 2x ), x. 2 Sketch the curve of f for 2 x 2, including any asymptotes. (i) Write down the equation of the vertical asymptote of f. (ii) Write down which one of the following expressions does not represent an area between the curve of f and the x-axis. (c) (iii) Justify your answer f (x)dx f (x)dx The region between the curve and the x-axis between x = and x =.5 is rotated through 360 about the x-axis. Let V be the volume formed. (i) Write down an expression to represent V. (ii) Hence write down the value of V. (d) Find f (x). (e) (i) Write down the value of x at the minimum point on the curve of f. (ii) The equation f (x) = k has no solutions for p k < q. Write down the value of p and of q. (Total 7 marks) Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 9 of 0

10 3. * A Ferris wheel with centre O and a radius of 5 metres is represented in the diagram below. Initially seat A is at ground level. The next seat is B, where π A ÔB =. 6 Find the length of the arc AB. Find the area of the sector AOB. (c) The wheel turns clockwise through an angle of 2π. Find the height of A above the ground. 3 The height, h metres, of seat C above the ground after t minutes, can be modelled by the function π h (t) = 5 5 cos 2t +. 4 (d) (i) Find the height of seat C when t = 4 π. (e) (ii) Find the initial height of seat C. (iii) Find the time at which seat C first reaches its highest point. Find h (t). (f) For 0 t π, (i) sketch the graph of h ; (ii) find the time at which the height is changing most rapidly. (8) (5) (Total 22 marks) Z:\Dropbox\Desert\SL\7Calculus\LP_SL2Calculus202-3.doc on 02/27/203 at :27 PM Page 0 of 0