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Higher Mathematics

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Higher Mathematics

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  1. Higher Mathematics or Everything you wanted to know about Higher Maths but … Higher Mathematics: HappySmile Productions

  2. Pick a number Higher Mathematics: HappySmile Productions

  3. answer Question 1 • How do you draw the graph of –f(x) given the graph of f(x) ? Higher Mathematics: HappySmile Productions

  4. Return to question Answer to Question 1 • Reflect the graph in the x-axis Higher Mathematics: HappySmile Productions

  5. answer Question 2 • For what values is the function f(x) = √(x-a) undefined? Higher Mathematics: HappySmile Productions

  6. Return to question Answer to Question 2 • x < a Higher Mathematics: HappySmile Productions

  7. answer Question 3 • If f(x) = xn then what is f’(x) ? Higher Mathematics: HappySmile Productions

  8. Return to question Answer to Question 3 • nxn-1 Higher Mathematics: HappySmile Productions

  9. y (a,b) x answer Question 4 • What is the • general equation • of a straight • line passing • through (a,b) • with gradient m ? Higher Mathematics: HappySmile Productions

  10. Return to question y (a,b) x Answer to Question 4 • y – b = m(x – a) Higher Mathematics: HappySmile Productions

  11. answer Question 5 • If f(t) is the distance travelled in a certain time t seconds, then what does f’(t) represent ? Higher Mathematics: HappySmile Productions

  12. Return to question Answer to Question 5 • Speed (velocity) Higher Mathematics: HappySmile Productions

  13. answer Question 6 • What is an altitude of a triangle ? Higher Mathematics: HappySmile Productions

  14. Return to question Answer to Question 6 • A line from a vertex, perpendicular to the opposite side Higher Mathematics: HappySmile Productions

  15. answer Question 7 • What is logaa equal to ? Higher Mathematics: HappySmile Productions

  16. Return to question Answer to Question 7 • 1 Higher Mathematics: HappySmile Productions

  17. answer Question 8 • Name the steps you take in order to differentiate functions like f(x) = x2+ 3x + 1 √x Higher Mathematics: HappySmile Productions

  18. Return to question Answer to Question 8 • (i) Change roots to powers (ii) split up into 3 fractions (iii) simplify each term (iv) differentiate Higher Mathematics: HappySmile Productions

  19. answer Question 9 • How do you find where a curve crosses the y-axis ? Higher Mathematics: HappySmile Productions

  20. Return to question Answer to Question 9 • Let x = 0 and solve Higher Mathematics: HappySmile Productions

  21. answer Question 10 • How do you draw the graph of 2f(x) given the graph of f(x) ? Higher Mathematics: HappySmile Productions

  22. Return to question Answer to Question 10 • Double the y-coordinates Higher Mathematics: HappySmile Productions

  23. answer Question 11 • How do you find the gradient of a line, given the angle the line makes with the positive direction of the x-axis ? Higher Mathematics: HappySmile Productions

  24. Return to question y Θ x Answer to Question 11 • m = tan Θ Higher Mathematics: HappySmile Productions

  25. answer Question 12 • What is the equation of a horizontal line passing through the point (a,b) ? Higher Mathematics: HappySmile Productions

  26. Return to question y (a,b) x Answer to Question 12 • y = b Higher Mathematics: HappySmile Productions

  27. answer Question 13 • What do you know about the gradient of two perpendicular lines ? Higher Mathematics: HappySmile Productions

  28. Return to question y m1 x m2 Answer to Question 13 • They multiply together to give -1 i.e. m1xm2 = -1 Higher Mathematics: HappySmile Productions

  29. answer Question 14 • What is a median of a triangle ? Higher Mathematics: HappySmile Productions

  30. Return to question Answer to Question 14 • A line from a vertex to the middle of the opposite side Higher Mathematics: HappySmile Productions

  31. answer Question 15 • What is the gradient of a horizontal line ? Higher Mathematics: HappySmile Productions

  32. Return to question y x Answer to Question 15 • 0 Higher Mathematics: HappySmile Productions

  33. answer Question 16 • How do you find the rate of change of a function at a particular point ? Higher Mathematics: HappySmile Productions

  34. Return to question Answer to Question 16 • (i) differentiate (ii) fit in given x value Higher Mathematics: HappySmile Productions

  35. answer Question 17 • How do you find the limit of a recurrence relation from its formula ? Higher Mathematics: HappySmile Productions

  36. Return to question Answer to Question 17 • (i) let un+1 = un = L • (ii) solve equation Remember: For un+1=aun+b, a limit exists if -1<a<1 Higher Mathematics: HappySmile Productions

  37. answer Question 18 • How do you know when a recurrence relation of the form un+1 = aun + bconverges ? Higher Mathematics: HappySmile Productions

  38. Return to question Answer to Question 18 • -1 < a < 1 Higher Mathematics: HappySmile Productions

  39. answer Question 19 • Given a rec. relation in the form un+1 = aun + b and 3 consecutive terms, how do you find the values of a and b? Higher Mathematics: HappySmile Productions

  40. Return to question Answer to Question 19 • (i) fit 1st term into un and 2nd term into un+1 (ii) fit 2nd term into un and the 3rd term into un+1 (iii) solve simultaneous equations Higher Mathematics: HappySmile Productions

  41. answer Question 20 • How do • you show • that a curve • is always decreasing? Higher Mathematics: HappySmile Productions

  42. Return to question Answer to Question 20 • (i) differentiate (ii) show that f’(x) is negative and a square e.g. –(x+a)2 Higher Mathematics: HappySmile Productions

  43. answer Question 21 • How would you find the maximum or minimum value of a function given its equation? Higher Mathematics: HappySmile Productions

  44. Return to question Answer to Question 21 • (i) differentiate (ii) let f’(x) = 0 (iii) solve to find the stationary points (iv) draw the nature table (v) read off max or min Higher Mathematics: HappySmile Productions

  45. answer Question 22 • Which two points does the graph of y = logax always pass through ? Higher Mathematics: HappySmile Productions

  46. Return to question Answer to Question 22 • (1,0) and (a,1) Higher Mathematics: HappySmile Productions

  47. answer Question 23 • How do you find where a curve is decreasing ? Higher Mathematics: HappySmile Productions

  48. Return to question Answer to Question 23 • (i) differentiate (ii) let f’(x) = 0 (iii) solve to find st pts (iv) draw nature table (v) read values where graph is decreasing Higher Mathematics: HappySmile Productions

  49. answer Question 24 • If y is the equation of a curve, what is represented by dy/dx ? Higher Mathematics: HappySmile Productions

  50. Return to question Answer to Question 24 • The gradient Higher Mathematics: HappySmile Productions