IB SL REVIEW and PRACTICE

Transcription

1 IB SL REVIEW and PRACTICE Topic: CALCULUS Here are sample problems that deal with calculus. You ma use the formula sheet for all problems. Chapters 16 in our Tet can help ou review. NO CALCULATOR Problems (1-5) 1. Find the equation of the normal to the curve with equation Working: = + 1 at the point (1,). Answers:.. (Total 4 marks). The graph represents the function f: p cos, p. Find the value of p; the area of the shaded region. 1

2 . Differentiate with respect to 4 e sin 4. The function f is such that f () =. When the graph of f is drawn, it has a minimum point at (, 7). (c) Show that f () = and hence find f(). Find f (0), f ( 1) and f( 1). Hence sketch the graph of f labeling it with the information obtained in part. (6) (4) (Note: It is not necessar to find the coordinates of the points where the graph cuts the -ais.) (Total 1 marks) 5. The diagram shows part of the graph of = e. = e P ln 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 60 about the -ais. Write down an integral which represents the volume of the solid obtained. (4) (c) Show that this volume is. (5) (Total 11 marks)

3 6. The diagram shows part of the graph of = 1 (1 ). 0 Write down an integral which represents the area of the shaded region. Find the area of the shaded region. Working: Answers:..... (Total 4 marks) 7. A curve has equation = ( 4). For this curve find (i) (iii) the -intercepts; the coordinates of the maimum point; the coordinates of the point of infleion. (9) Use our answers to part to sketch a graph of the curve for 0 4, clearl indicating the features ou have found in part. (c) (i) On our sketch indicate b shading the region whose area is given b the following integral: 0 ( 4) d. Eplain, using our answer to part, wh the value of this integral is greater than 0 but less than 40. (Total 15 marks)

4 8. If f () = cos, and f =, find f(). 9. The diagram shows part of the graph of the curve with equation = e cos. P( a, b) 0 Show that d d = e ( cos sin ). d Find. d (4) There is an infleion point at P(a, b). (c) Use the results from parts and to prove that: (i) tan a = 4 ; the gradient of the curve at P is e. (5) (Total 14 marks) 4

5 10. Given that f() = ( + 5) find f (); f ( )d. 11. The diagram shows the graph of the function = 1 + 1, 0 < 4. Find the eact value of the area of the shaded region. 4 = The point P ( 1, 0 ) lies on the graph of the curve of = sin( 1). Find the gradient of the tangent to the curve at P. Working: Answers:.. (Total 4 marks) 1. Find sin ( 7)d; 4 d e. 5

6 14. Consider the function f() = k sin +, where k is a constant. Find f (). When =, the gradient of the curve of f() is 8. Find the value of k. Working: Answers:..... (Total 4 marks) 15. A ball is dropped verticall from a great height. Its velocit v is given b v = 50 50e 0.t, t 0 where v is in metres per second and t is in seconds. Find the value of v when (i) t = 0; t = 10. (i) Find an epression for the acceleration, a, as a function of t. What is the value of a when t = 0? (c) (i) As t becomes large, what value does v approach? As t becomes large, what value does a approach? (d) (iii) Eplain the relationship between the answers to parts (i) and. Let metres be the distance fallen after t seconds. (i) Show that = 50t + 50e 0.t + k, where k is a constant. Given that = 0 when t = 0, find the value of k. (iii) Find the time required to fall 50 m, giving our answer correct to four significant figures. (7) (Total 15 marks) 6

7 16. The derivative of the function f is given b f() = sin, for 1. The graph of f passes through the point (0, ). Find an epression for f(). NO # Let f() =. Find f (); f ( )d. 19. The graph of = has a maimum point between = 1 and =. Find the coordinates of this maimum point. 0. The diagram below shows the shaded region R enclosed b the graph of = -ais, and the vertical line = k. 1, the = 1+ R k Find d. d Using the substitution u = 1 + or otherwise, show that 1 d = (1 + ) + c. (c) Given that the area of R equals 1, find the value of k. 1. Let f() = e + 5cos. Find f (). (Total 9 marks) 7

8 . Consider the function f() = 1 + e. (i) Find f(). Eplain briefl how this shows that f() is a decreasing function for all values of (i.e. that f() alwas decreases in value as increases). Let P be the point on the graph of f where = 1. Find an epression in terms of e for (i) the -coordinate of P; the gradient of the tangent to the curve at P. (c) Find the equation of the tangent to the curve at P, giving our answer in the form = a + b. (d) (i) Sketch the curve of f for 1. Draw the tangent at = 1. (iii) (iv) Shade the area enclosed b the curve, the tangent and the -ais. Find this area. (7) (Total 14 marks). Let f be a function such that f ( ) d 8. 0 Deduce the value of (i) 0 f ( ) d ; f ) d d 0 (. f ( )d 8, write down the value of c and of d. c 8

9 4. Consider the function h:, 1. ( 1) A sketch of part of the graph of h is given below. A P Not to scale The line (AB) is a vertical asmptote. The point P is a point of infleion. B Write down the equation of the vertical asmptote. Find h'(), writing our answer in the form a ( 1) n (1) where a and n are constants to be determined. (4) (c) Given that 8 h ( ), calculate the coordinates of P. 4 ( 1) (Total 8 marks) 9

10 CALCULATOR ALLOWED Problems 6-6. 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) Find the eact value of (i) a; 7 b. The regions between the curve and the -ais are shaded for a as shown. (i) Write down an epression which represents the total area of the shaded regions. Calculate this total area. (5) (c) (i) The -coordinate of R is Find the -coordinate of S. 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) 7. Let f() = 1 + cos() for 0 π, and is in radians. (i) Find f(). Find the values for for which f() = 0, giving our answers in terms of. (6) The function g() is defined as g() = f() 1, 0 π. (i) The graph of f ma be transformed to the graph of g b a stretch 1in the -direction with scale factor followed b another transformation. Describe full this other transformation. Find the solution to the equation g() = f() 10

11 8. The diagram below shows part of the graph of f() = sin( + ) and the shaded region A. A P Q 0 1 This graph crosses the -ais at P and Q. The point P has coordinates ( π, 0). Find the -coordinate of Q. Use the substitution u = + to find f () d. (c) Hence, using our answer to, find the area of the region A. (4) (Total 9 marks) 9. Consider the function f() = cos + sin. (i) Show that f( 4 π ) = 0. Find in terms of, the smallest positive value of which satisfies f() = 0. The diagram shows the graph of = e (cos + sin ),. The graph has a maimum turning point at C(a, b) and a point of infleion at D. 11

12 6 C(a, b) 4 D 1 1 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) 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 Find the coordinates of A. 1

13 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. Find d. d (iii) Hence, show that at B, = ln π. (6) (d) (i) Write down the integral which represents the shaded area. 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 =. The two graphs intersect at the point P. Find the -coordinate of P. (Total 18 marks) 1. Note: Radians are used throughout this question. Draw the graph of = + cos, 0 5, on millimetre square graph paper, using a scale of cm per unit. Make clear (i) the integer values of and on each ais; (c) (d) the approimate positions of the -intercepts and the turning points. Without the use of a calculator, show that is a solution of the equation + cos = 0. Find another solution of the equation + cos = 0 for 0 5, giving our answer to si significant figures. Let R be the region enclosed b the graph and the aes for 0. Shade R on our diagram, and write down an integral which represents the area of R. (5) (e) Evaluate the integral in part (d) to an accurac of si significant figures. (If ou consider it d necessar, ou can make use of the result ( sin cos ) cos.) d (Total 15 marks) 1

14 . In this question ou should note that radians are used throughout. (i) Sketch the graph of = cos, for 0 making clear the approimate positions of the positive intercept, the maimum point and the end-points. Write down the approimate coordinates of the positive -intercept, the maimum point and the end-points. (7) Find the eact value of the positive -intercept for 0. Let R be the region in the first quadrant enclosed b the graph and the -ais. (c) (i) Shade R on our diagram. Write down an integral which represents the area of R. (d) Evaluate the integral in part (c), either b using a graphic displa calculator, or b using the following information. d ( sin + cos sin ) = cos. d (Total 15 marks) 14