1. Given the function f (x) = x 2 3bx + (c + 2), determine the values of b and c such that f (1) = 0 and f (3) = 0.

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Aileen Hines
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1 Chapter Review IB Questions 1. Given the function f () = 3b + (c + ), determine the values of b and c such that f = 0 and f = 0. (Total 4 marks). Consider the function ƒ : k. (a) Write down ƒ (). The equation of the tangent to the graph of ƒ at = p is y = 7 9. Find the value of p; k. 3. Let f () = 6 3. Find f (). 4. Let f() = k 4. The point P(1, k) lies on the curve of f. At P, the normal to the curve is parallel to 1 y =. Find the value of k Let f () = 3 1. (a) Find f (). Find the gradient of the curve of f () at the point (, 1). 6. Consider f() = + (a) Find f (). p, 0, where p is a constant. There is a minimum value of f() when =. Find the value of p. 7. Consider the curve with equation f() = p + q, where p and q are constants. The point A(1, 3) lies on the curve. The tangent to the curve at A has gradient 8. Find the value of p and of q. (Total 7 marks) IB Questionbank Maths SL 1

2 8. The following diagram shows part of the graph of a quadratic function, with equation in the form y = ( p)( q), where p, q. (a) Write down (i) the value of p and of q; (ii) the equation of the ais of symmetry of the curve. Find the equation of the function in the form y = ( h) + k, where h, k. Find d y. d (d) Let T be the tangent to the curve at the point (0, 5). Find the equation of T. (Total 10 marks) 9. Consider f () = Part of the graph of f is shown below. There is a maimum point at M, and a point of infleion at N. (a) Find f (). Find the -coordinate of M. Find the -coordinate of N. (d) The line L is the tangent to the curve of f at (3, 1). Find the equation of L in the form y = a + b. (Total 14 marks) IB Questionbank Maths SL

3 10. Let g() = (a) 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 14 marks) 11. Let f() = (a) Epand ( + h) 3. Use the formula f () = lim h 0 f ( h) f ( ) h to show that the derivative of f() is 3 4. The tangent to the curve of f at the point P(1, ) is parallel to the tangent at a point Q. Find the coordinates of Q. (d) The graph of f is decreasing for p < < q. Find the value of p and of q. (e) Write down the range of values for the gradient of f. (Total 15 marks) 1. The equation of a curve may be written in the form y = a( p)( q). The curve intersects the -ais at A(, 0) and B(4, 0). The curve of y = f () is shown in the diagram below. y 4 A B (a) (i) Write down the value of p and of q. (ii) Given that the point (6, 8) is on the curve, find the value of a. (iii) Write the equation of the curve in the form y = a + b + c. 6 (i) d y Find. d (ii) A tangent is drawn to the curve at a point P. The gradient of this tangent is 7. Find the coordinates of P. The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B. (i) Find the equation of L. (ii) Find the -coordinate of the point where L intersects the curve again. IB Questionbank Maths SL 3 (5) (6) (Total 15 marks)

4 13. Let f () = (a) Write down the equation of the vertical asymptote of y = f (). Find f (). Give your answer in the form a b ( 5 1) where a and b. (Total 5 marks) 14. Radian measure is used, where appropriate, throughout the question. 3 Consider the function y. 5 The graph of this function has a vertical and a horizontal asymptote. (a) Write down the equation of (i) the vertical asymptote; (ii) the horizontal asymptote. d Find, simplifying the answer as much as possible. d y How many points of infleion does the graph of this function have? IB Questionbank Maths SL 4

5 15. Consider the function h:, 1. A sketch of part of the graph of h is given below. A y P Not to scale B The line (AB) is a vertical asymptote. The point P is a point of infleion. (a) Write down the equation of the vertical asymptote. Find h (), writing your answer in the form a where a and n are constants to be determined. 8 Given that h ( ), calculate the coordinates of P. 4 n (Total 8 marks) IB Questionbank Maths SL 5

6 16. Consider the function f given by f () = A part of the graph of f is given below. y 13 0, 1. 0 The graph has a vertical asymptote and a horizontal asymptote, as shown. (a) Write down the equation of the vertical asymptote. f (100) = 1.91 f ( 100) =.09 f (1000) = 1.99 (i) Evaluate f ( 1000). (ii) Write down the equation of the horizontal asymptote. 9 7 Show that f () =, The second derivative is given by f () = 4, 1. (d) Using values of f () and f () eplain why a minimum must occur at = 3. (e) There is a point of infleion on the graph of f. Write down the coordinates of this point. (Total 10 marks) IB Questionbank Maths SL 6

7 17. Let f() = 3 + 0, for ±. The graph of f is given below. 4 The y-intercept is at the point A. (a) (i) Find the coordinates of A. (ii) Show that f () = 0 at A. 40(3 4) 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. Describe the behaviour of the graph of f for large. (d) Write down the range of f. diagram not to scale (7) (6) (Total 16 marks) IB Questionbank Maths SL 7

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.

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. 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.