Algebra y funciones [219 marks]

Transcription

1 Algebra y funciones [219 marks] Let f() = 3 ln and g() = ln5 3. 1a. Epress g() in the form f() + lna, where a Z +. 1b. The graph of g is a transformation of the graph of f. Give a full geometric description of this transformation. In the epansion of (3 2) 12, the term in 5 12 can be epressed as ( ) (3) p ( 2). r (a) 2a. Write down the value of p, of and of r. (b) Find the coefficient of the term in 5. 2b. Write down the value of p, of and of r. 2c. Find the coefficient of the term in 5. Let log 3 p = 6 and log 3 = 7. (a) Find log. 3a. 3 p 2 p (b) Find log 3 ( ). [7 marks] (c) Find log 3 (9p). 3b. Find log 3p 2. p 3c. Find log 3 ( ). 3d. Find log3(9p).. The constant term in the epansion of ( + ), where a R is Find a. a a 2 6 [7 marks] Write down the value of 5a. (i) log 3 27; 5b. (ii) log 8 1; 8 5c. (iii) log 16. 5d. Hence, solve log log = 8 log log

2 Consider the epansion of ( + 3) 10. 6a. Write down the number of terms in this epansion. 6b. Find the term containing Consider the epansion of 2 (3 2 + ). The constant term is Find k. k 8 [7 marks] Let f() = p 3 + p 2 +. Find f (). 8a. Given that, show that. 8b. f () 0 p 2 3p Let f() = cos( ) + sin( ), for. 9a. Sketch the graph of f. 9b. Find the values of where the function is decreasing. 9c. The function f can also be written in the form f() = asin( ( + c)), where a R, and 0 c 2. Find the value of a; 9d. The function f can also be written in the form f() = asin( ( + c)), where a R, and 0 c 2. Find the value of c. Let f() = 3, where. 10a. Write down the euations of the vertical and horizontal asymptotes of the graph of f. 10b. The vertical and horizontal asymptotes to the graph of f intersect at the point Q(1, 3). Find the value of. 10c. The vertical and horizontal asymptotes to the graph of f intersect at the point Q(1, 3). The point P(, y) lies on the graph of f. Show that PQ = ( 1) ( ). 1 10d. The vertical and horizontal asymptotes to the graph of f intersect at the point Q(1, 3). Hence find the coordinates of the points on the graph of f that are closest to (1, 3). [6 marks]

3 Let f() = 5 2. Part of the graph of fis shown in the following diagram. The graph crosses the -ais at the points A and B. 11a. Find the -coordinate of A and of B. 11b. The region enclosed by the graph of f and the -ais is revolved 360 about the -ais. Find the volume of the solid formed. Let f() = 3 2 and g() = 5, for a. Find f 1 (). 12b. Show that (g )() =. f Let h() = 5, for 0. The graph of h has a horizontal asymptote at y = c. Find the y-intercept of the graph of h. 12d. Hence, sketch the graph of h. 12e. For the graph of h 1, write down the -intercept; 12f. For the graph of h 1, write down the euation of the vertical asymptote. 12g. Given that h 1 (a) = 3, find the value of a. Let f() = 5, for 5. Find f. 13a. 1 (2) 13b. Let g be a function such that g 1 eists for all real numbers. Given that g(30) = 3, find (f g 1 )(3).

4 Consider f() = ln( + 1). Find the value of f(0). 1a. 1b. Find the set of values of for which f is increasing. f 2 (3 ) ( 2 +1) The second derivative is given by () =. The euation f () = 0 has only three solutions, when = 0, ± 3 (±1.316 ). 1c. (i) Find f (1). (ii) Hence, show that there is no point of infleion on the graph of f at = 0. There is a point of infleion on the graph of f at = 3 ( = ). 1d. Sketch the graph of f, for 0. 1 The velocity of a particle in ms is given by v = e sin t 1, for 0 t 5. On the grid below, sketch the graph of v. 15a. Find the total distance travelled by the particle in the first five seconds. 15b. 15c. Write down the positive t -intercept. Let f and g be functions such that g() = 2f( + 1) + 5. (a) The graph of f is mapped to the graph of g under the following transformations: 16a. [6 marks] vertical stretch by a factor of k, followed by a translation ( p ). Write down the value of (i) k ; (ii) p ; (iii). (b) Let h() = g(3). The point A( 6, 5) on the graph of g is mapped to the point A on the graph of h. Find A.

5 The graph of f is mapped to the graph of g under the following transformations: 16b. vertical stretch by a factor of k, followed by a translation ( p ). Write down the value of (i) k ; (ii) p ; (iii). 16c. Let h() = g(3). The point A( 6, 5 ) on the graph of g is mapped to the point A on the graph of h. Find A. 100 (1+50 e 0.2 ) Let f() =. Part of the graph of f is shown below. Write down f(0). 17a. 17b. Solve f() = c. Find the range of f. 17d. Show that f () = 1000e 0.2. (1+50 e 0.2 ) 2 17e. Find the maimum rate of change of f. 1 Let f() = sin + 2, for Find f (). 18a. Let g be a uadratic function such that g(0) = 5. The line = 2 is the ais of symmetry of the graph of g. Find g(). 18b. The function g can be epressed in the form g() = a( h ) (i) 18c. Write down the value of h. (ii) Find the value of a.

6 18d. Find the value of for which the tangent to the graph of f is parallel to the tangent to the graph of g. [6 marks] (ln ) 2 Let f() =, for > a. Show that f ln () =. 19b. There is a minimum on the graph of f. Find the -coordinate of this minimum. Let g() = 1. The following diagram shows parts of the graphs of f and g. The graph of f has an -intercept at = p. 19c. Write down the value of p. 19d. The graph of g intersects the graph of f when =. Find the value of. 19e. The graph of g intersects the graph of f when =. Let R be the region enclosed by the graph of f, the graph of g and the line = p. Show that the area of is. R 1 2 Let f() = ( 1)( ). 20a. Find the -intercepts of the graph of f. 20b. The region enclosed by the graph of f and the -ais is rotated 360 about the -ais. Find the volume of the solid formed.

7 A particle moves along a straight line such that its velocity, v ms 1, is given by v(t) = 10te 1.7t, for t 0. 21a. On the grid below, sketch the graph of v, for 0 t. 21b. Find the distance travelled by the particle in the first three seconds. 21c. Find the velocity of the particle when its acceleration is zero. Let f() = 3 2. The following diagram shows part of the curve of f. The curve crosses the -ais at the point P. 22a. Write down the -coordinate of P. Write down the gradient of the curve at P. 22b. 22c. Find the euation of the normal to the curve at P, giving your euation in the form y = a + b. International Baccalaureate Organization 2016 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for Colegio Aleman de Barranuilla