(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.

Transcription

1 Calculus Review Packet 1. Consider the function f() = Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point, M, and a local minimum point, N. (d) (i) Use f () to find the -coordinate of M and of N. (ii) Hence or otherwise write down the coordinates of M and of N. (5) (e) Sketch the graph of f() for 5 7 and 60 y 60. Mark clearly M and N on your graph. Lines L 1 and L 2 are parallel, and they are tangents to the graph of f() at points A and B respectively. L 1 has equation y = (f) (i) Find the -coordinate of A and of B. (ii) Find the y-coordinate of B. (6) (Total 21 marks) 2. The curve y = p 2 + q 4 passes through the point (2, 10). Use the above information to write down an equation in p and q. The gradient of the curve y = p 2 + q 4 at the point (2, 10) is 1. dy (i) Find. d (ii) Hence, find a second equation in p and q. Solve the equations to find the value of p and of q. (Total 8 marks) IB Questionbank Mathematical Studies 3rd edition 1

2 3. Sketch the graph of y = 2 for 2 3. Indicate clearly where the curve intersects the y-ais. Write down the equation of the asymptote of the graph of y = 2. On the same aes sketch the graph of y = Indicate clearly where this curve intersects the and y aes. (d) Using your graphic display calculator, solve the equation = 2. (e) Write down the maimum value of the function f() = (f) Use Differential Calculus to verify that your answer to (e) is correct. (5) (Total 16 marks) 4. Let f() = Find f (). Find the value of f ( 3). Find the value of for which f () = 0. (Total 6 marks) IB Questionbank Mathematical Studies 3rd edition 2

3 5. Consider the curve y = (i) Write down the value of y when is 2. (ii) Write down the coordinates of the point where the curve intercepts the y-ais. Sketch the curve for 4 3 and 10 y 10. Indicate clearly the information found in. dy Find. d (d) Let L 1 be the tangent to the curve at = 2. Let L 2 be a tangent to the curve, parallel to L 1. (i) Show that the gradient of L 1 is 12. (ii) (iii) Find the -coordinate of the point at which L 2 and the curve meet. Sketch and label L 1 and L 2 on the diagram drawn in. (8) (e) It is known that dy d > 0 for < 2 and > b where b is positive. (i) Using your graphic display calculator, or otherwise, find the value of b. (ii) Describe the behaviour of the curve in the interval 2 < < b. (iii) Write down the equation of the tangent to the curve at = 2. (5) (Total 23 marks) IB Questionbank Mathematical Studies 3rd edition 3

4 6. It is not necessary to use graph paper for this question. Sketch the curve of the function f () = for values of from 2 to 4, giving the intercepts with both aes. On the same diagram, sketch the line y = 7 2 and find the coordinates of the point of intersection of the line with the curve. Find the value of the gradient of the curve where =1.7. (Total 8 marks) 8. A function is represented by the equation f () = a Find f (). The function f () has a local maimum at the point where = 1. Find the value of a. (Total 6 marks) 15. Consider the function f () = Differentiate f () with respect to. Calculate f () when = 1. Calculate the values of when f () = 0. (d) Calculate the coordinates of the local maimum and the local minimum points. (e) On graph paper, taking aes 6 3 and 0 y 80, draw the graph of f () indicating clearly the local maimum, local minimum and y-intercept. (Total 14 marks) IB Questionbank Mathematical Studies 3rd edition 4

5 7. The diagram below shows the graph of a line L passing through (1, 1) and (2, 3) and the graph P of the function f () = y L P 0 Find the gradient of the line L. Differentiate f (). Find the coordinates of the point where the tangent to P is parallel to the line L. (d) (e) Find the coordinates of the point where the tangent to P is perpendicular to the line L. Find (i) (ii) the gradient of the tangent to P at the point with coordinates (2, 6); the equation of the tangent to P at this point. (f) State the equation of the ais of symmetry of P. (e) Find the coordinates of the verte of P and state the gradient of the curve at this point. (Total 18 marks) IB Questionbank Mathematical Studies 3rd edition 5

6 9. A farmer has a rectangular enclosure with a straight hedge running down one side. The area of the enclosure is 162 m 2. He encloses this area using metres of the hedge on one side as shown on the diagram below. diagram not to scale If he uses y metres of fencing to complete the enclosure, 324 show that y = +. The farmer wishes to use the least amount of fencing. Find dy. d Find the value of which makes y a minimum. (d) Calculate this minimum value of y. (e) 324 Using y = + find the values of a and b in the following table y a 37.5 b 45 (f) Draw an accurate graph of this function using a horizontal scale starting at 0 and taking 2 cm to represent 10 metres, and a vertical scale starting at 30 with 4 cm to represent 10 metres. (5) (g) Write down the values of for which y increases. (Total 20 marks) IB Questionbank Mathematical Studies 3rd edition 6

7 10. Epand the epression 2( 2 1). Hence differentiate f () = 2( 2 1) with respect to. Find the gradient of the tangent to the curve y = f () at the point where = 1. (d) If the angle between the -ais and the tangent in part is, write down the value of tan. (Total 6 marks) Consider the function f () = Calculate the value of f () when = 1. (d) Differentiate f (). Find f (l). Eplain what f (l) represents. (e) Find the equation of the tangent to the curve f () at the point where = 1. (f) Determine the -coordinate of the point where the gradient of the curve is zero. (Total 16 marks) 12. Differentiate the following function with respect to : f () = Calculate the -coordinates of the points on the curve where the gradient of the tangent to the curve is equal to 6. (Total 6 marks) IB Questionbank Mathematical Studies 3rd edition 7

8 13. Differentiate the function y = At a certain point (, y) on this curve the gradient is 5. Find the co-ordinates of this point. (Total 6 marks) 14. On the same graph sketch the curves y = 2 1 and y = 3 for values of from 0 to 4 and values of y from 0 to 4. Show your scales on your aes. Find the points of intersection of these two curves. 1 (i) Find the gradient of the curve y = 3 in terms of. (ii) Find the value of this gradient at the point (1, 2). 1 (d) Find the equation of the tangent to the curve y = 3 at the point (1, 2). (Total 15 marks) Write in the form 3 a where a. 2 3 b Hence differentiate y = giving your answer in the form where c 2 c +. (Total 6 marks) IB Questionbank Mathematical Studies 3rd edition 8

9 16. A closed bo has a square base of side and height h. Write down an epression for the volume, V, of the bo. Write down an epression for the total surface area, A, of the bo. The volume of the bo is 1000 cm 3 Epress h in terms of. (d) Hence show that A = 4000 l da (e) Find. d (f) Calculate the value of that gives a minimum surface area. (g) Find the surface area for this value of. (Total 15 marks) 20. A function is given as y = a 2 + b + 6. dy Find. d If the gradient of this function is 2 when is 6 write an equation in terms of a and b. If the point (3, 15) lies on the graph of the function find a second equation in terms of a and b. (Total 6 marks) IB Questionbank Mathematical Studies 3rd edition 9

10 18. At the circus a clown is swinging from an elastic rope. A student decides to investigate the motion of the clown. The results can be shown on the graph of the function f () = (0.8 )(5 sin 100), where is the horizontal distance in metres. Sketch the graph of f () for 0 10 and 3 f () 5. (5) Find the coordinates of the first local maimum point. Find the coordinates of one point where the curve cuts the y-ais. Another clown is fired from a cannon. The clown passes through the points given in the table below: Horizontal distance () Vertical distance (y) (d) Find the correlation coefficient, r, and comment on the value for r. (e) Write down the equation of the regression line of y on. (f) Sketch this line on the graph of f () in part. (g) Find the coordinates of one of the points where this line cuts the curve. (Total 16 marks) IB Questionbank Mathematical Studies 3rd edition 10

11 19. The cost of producing a mathematics tetbook is $15 (US dollars) and it is then sold for $. Find an epression for the profit made on each book sold. A total of ( ) books is sold. Show that the profit made on all the books sold is P = dp (i) Find. d (ii) Hence calculate the value of to make a maimum profit (d) Calculate the number of books sold to make this maimum profit. (Total 10 marks) 22. Consider the function g () = Find g () g (l) (Total 5 marks) IB Questionbank Mathematical Studies 3rd edition 11

12 21. f () D C y = f( ) A B E K L F H G Given the graph of f () state (d) (e) the intervals from A to L in which f () is increasing. the intervals from A to L in which f () is decreasing. a point that is a maimum value. a point that is a minimum value. the name given to point K where the gradient is zero. (Total 5 marks) 23. A function g () = Find g (). Solve g () = 0. (i) Calculate the values of g () when = 3; = 0. (ii) Hence state whether the function is increasing or decreasing at = 3; = 0. (Total 9 marks) IB Questionbank Mathematical Studies 3rd edition 12

13 24. The diagram shows a part of the curve y = f (). y For what values of is f '() = 0? For what range of values of is f '() <0? (Total 5 marks) 25. Consider the function f () = (i) Find f ' (). (ii) Find the gradient of the curve f () when = 3. Find the -coordinates of the points on the curve where the gradient is equal to 12. (i) Calculate the -coordinates of the local maimum and minimum points. (ii) Hence find the coordinates of the local minimum. (6) (d) For what values of is the value of f () increasing? (Total 15 marks) IB Questionbank Mathematical Studies 3rd edition 13

14 26. The distance s metres run by an athlete in t minutes is given by the formula s(t) = 250t + 5t t 3, 0 t 70. Calculate the distance run after 50 minutes. 1 (i) Show that 50 minutes and 1 second may be written as 50 minutes (ii) Calculate the distance run after 50 minutes. 60 (iii) Calculate the value of 1 s 50 s (5) (Total 6 marks) 27. The curve y = f() has its only local minimum value at = a and its only local maimum value at = b. If a < 0 and b > 0, sketch a possible curve of y = f () indicating clearly the points (a, f ) and (b, f ). Given that 0 < h < 1 and b a > 1, are the following statements about the curve y = f () TRUE or FALSE? (i) (ii) (iii) f (a + h) < f f (b h) is positive The tangent to the curve at the point (a, f ) is parallel to the vertical ais. (iv) The gradient of the tangent to the curve at the point (a, f ) is equal to zero. (v) f (a h) < f < f (a + h) (5) (Total 7 marks) IB Questionbank Mathematical Studies 3rd edition 14

15 28. A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length cm are cut from each corner of the card as shown in the diagram below. What is left is then folded to make an open bo, of length l cm and width w cm. 24 cm 9 cm Write epressions, in terms of, for (i) the length, l; (ii) the width, w. Show that the volume (B m 3 ) of the bo is given by B = db Find. d (d) (i) Find the value of which gives the maimum volume of the bo. (ii) Calculate the maimum volume of the bo. (Total 8 marks) 29. The function g is defined as follows g: p 2 + q + c, p, q, c Find g () If g () = 2 + 6, find the values of p and q. g() has a minimum value of 12 at the point A. Find (i) the -coordinate of A; (ii) the value of c. (Total 7 marks) IB Questionbank Mathematical Studies 3rd edition 15

16 30. The letters A to E are placed at particular points on the curve y = f(). y C B D A E y = f( ) What is the gradient of the curve y =f() at the point marked C? dy In passing from point B, through point C, to point D what is happening to? Is it d decreasing or increasing? (Total 3 marks) IB Questionbank Mathematical Studies 3rd edition 16

17 31. A cylinder is cut from a solid wooden sphere of radius 8 cm as shown in the diagram. The height of the cylinder is 2h cm. D C h O 8 cm h 8 cm A E B Find AE (the radius of the cylinder), in terms of h. Show that the volume (V) of the cylinder may be written as V= 2 h (64 h 2 ) cm 3. (i) Determine, correct to three significant figures, the height of the cylinder with the greatest volume that can be produced in this way. (5) (ii) Calculate this greatest volume, giving your answer correct to the nearest cm 3. (Total 12 marks) IB Questionbank Mathematical Studies 3rd edition 17

18 32. The diagram below is a part of the graph of the function y = 2 2. P is the point (1, 2) on the curve and Q is a neighbouring point. y y = 2 2 Q (1, 2) P R O MN Complete the following table PR QN QR Gradient of PQ Show how you can use your table to (i) predict the gradient of the curve y = 2 2 at the point (1,2); (ii) deduce the gradient of the curve y = at the point (1,5). (Total 8 marks) IB Questionbank Mathematical Studies 3rd edition 18

19 33. The function f () is given by f () = , for 1 3. Differentiate f () with respect to. Complete the table below f () f '() Use the information in your table to sketch the graph of f (). (d) Write down the gradient of the tangent to the curve at the point (3, 9). (Total 8 marks) 34. The perimeter of a rectangle is 24 metres. The table shows some of the possible dimensions of the rectangle. Find the values of a, b, c, d and e. Length (m) Width (m) Area (m 2 ) a 10 b 3 c 27 4 d e If the length of the rectangle is m, and the area is A m 2, epress A in terms of only. What are the length and width of the rectangle if the area is to be a maimum? (Total 6 marks) (d) Write down the gradient of the tangent to the curve at the point (3, 9). (Total 8 marks) IB Questionbank Mathematical Studies 3rd edition 19