2. Find the equation of the normal to the curve with equation y = x at the point (1, 2). (Total 4 marks)

Size: px

Start display at page:

Download "2. Find the equation of the normal to the curve with equation y = x at the point (1, 2). (Total 4 marks)"

Mary Wilkinson
5 years ago
Views:

1 CHAPTER 3 REVIEW FOR SLs ONLY 1. Find the coordinates of the point on the graph of = 2 at which the tangent is parallel to the line = 5. (Total 4 marks) 2. Find the equation of the normal to the curve with equation = at the point (1, 2). (Total 4 marks) 3. Let f () = (a) There are two points of infleion on the graph of f. Write down the -coordinates of these points. Let g() = f (). Eplain wh the graph of g has no points of infleion. 4. The following diagram shows a rectangular area ABCD enclosed on three sides b 60 m of fencing, and on the fourth b a wall AB. (Total 5 marks) Find the width of the rectangle that gives its maimum area. IB Questionbank Maths SL 1

2 5. The diagram shows the graph of = f (). 0 On the grid below sketch the graph of = f () The graph of = has a maimum point between = 1 and = 3. Find the coordinates of this maimum point. IB Questionbank Maths SL 2

3 7. 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. (a) Calculate the coordinates of P and of Q. Let N 1 and N 2 be the normals to the curve at P and Q respectivel. Write down the coordinates of the points where (i) the tangent at P intersects N 2 ; (ii) the tangent at Q intersects N Consider the function f () = Find the equation of the normal to the curve of f at the point where =1. 9. The diagram shows part of the graph of = f (). The -intercepts are at points A and C. There is a minimum at B, and a maimum at D. (a) (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. 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. 10. A farmer wishes to create a rectangular enclosure, ABCD, of area 525 m 2, as shown below. (Total 7 marks) The fencing used for side AB costs $11 per metre. The fencing for the other three sides costs $3 per metre. The farmer creates an enclosure so that the cost is a minimum. Find this minimum cost. (Total 7 marks) IB Questionbank Maths SL 3

4 11. The function f is defined b f : Let N be the normal to the curve at the point where the graph intercepts the -ais. (a) Show that the equation of N ma be written as = Find the coordinates of the other point of intersection of the normal and the curve. (c) Let R be the region enclosed between the curve and N. Find the area of R. (Total 13 marks) 12. The function f () is defined as f () = ( h) 2 + k. The diagram below shows part of the graph of f (). The maimum point on the curve is P (3, 2). 4 2 P(3, 2) (a) Write down the value of (i) h; (ii) k Show that f () can be written as f () = (c) Find f (). The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q. (d) (i) Calculate the gradient of L. (ii) Find the equation of L. (iii) The line L intersects the curve again at R. Find the -coordinate of R. (8) (Total 13 marks) IB Questionbank Maths SL 4

5 13. The parabola shown has equation 2 = 9. P 2 = 9 M Q (a) Verif that the point P (4, 6) is on the parabola. The line (PQ) is the normal to the parabola at the point P, and cuts the -ais at Q. (i) Find the equation of (PQ) in the form a + b + c = 0. (ii) Find the coordinates of Q. S is the point 9, 0. 4 (c) Verif that SP = SQ. (d) The line (PM) is parallel to the -ais. From part (c), eplain wh (QP) bisects the angle S Pˆ M. 14. The function f is given b f () = 2 1,, 3. 3 (a) (i) Show that = 2 is an asmptote of the graph of = f (). (c) (ii) (iii) Find the vertical asmptote of the graph. Write down the coordinates of the point P at which the asmptotes intersect. Find the points of intersection of the graph and the aes. Hence sketch the graph of = f (), showing the asmptotes b dotted lines. (d) Show that f () = 7 ( 3) the point S where = 4. 2 and hence find the equation of the tangent at (e) The tangent at the point T on the graph is parallel to the tangent at S. Find the coordinates of T. (f) Show that P is the midpoint of [ST]. (Total 16 marks) (6) (l) (Total 24 marks) IB Questionbank Maths SL 5

IB SL REVIEW and PRACTICE

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