National Certificate of Educational Achievement · Mathematics and Statistics

Calculus Methods, Derivatives, Kinematics and Coordinate Geometry

40 practice questions, each with the answer and an explanation.

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All 40 questions

  1. What is the derivative of the function f(x) = 4x^3 - 5x^2 + 7x - 9?
  2. What is the gradient of the curve y = x^2 - 6x + 8 at the point where x = 5?
  3. At what x-coordinate does the curve f(x) = x^3 - 12x have a local minimum?
  4. What is the equation of the tangent line to the curve y = x^2 + 3x at the point (1, 4)?
  5. The displacement of a toy rocket in metres is given by s(t) = 30t - 5t^2, where t is in seconds. What is its velocity at t = 2 seconds?
  6. What is the acceleration of an object whose displacement is given by s(t) = 2t^3 - 9t^2 + 12t at t = 3 seconds?
  7. If f'(x) = 6x^2 - 4x + 1 and f(1) = 5, what is the constant of integration C?
  8. Over what interval is the function f(x) = x^2 - 8x + 12 strictly increasing?
  9. What is the gradient of the normal to the curve y = 2x^2 - 3x at the point (2, 2)?
  10. A farmer has 100 metres of fencing to enclose a rectangular sheep pen against an existing stone wall (so only three sides need fencing). What width x perpendicular to the wall maximises the area?
  11. Find the distance between the points A(-2, 3) and B(4, -5) in the Cartesian plane.
  12. What are the coordinates of the midpoint of the line segment joining (3, -7) and (9, 5)?
  13. What is the anti-derivative (indefinite integral) of g(x) = 6x^2 - 10x + 4?
  14. At what point on the curve y = x^3 - 3x^2 + 2 does the inflection point occur?
  15. What is the gradient of a straight line that is parallel to 4x - 2y = 7?
  16. An object is thrown vertically upwards such that its height in metres is h(t) = 40t - 5t^2. What is the maximum height reached?
  17. Evaluate the definite integral of f(x) = 3x^2 from x = 1 to x = 4.
  18. What is the rate of change of the area of a circle with respect to its radius r when r = 5 cm?
  19. If the gradient of a curve is given by dy/dx = 3/sqrt(x) and it passes through (4, 15), what is the equation of the curve?
  20. For the cubic curve y = 2x^3 - 9x^2 + 12x - 4, what are the coordinates of the local maximum turning point?
  21. A cylinder is to be constructed with a fixed total surface area of 150*pi cm^2. What radius r maximises the volume of the cylinder?
  22. For what values of the constant c is the cubic function f(x) = x^3 + 3x^2 + cx - 1 strictly increasing for all real numbers x?
  23. A particle moves in a straight line with velocity v(t) = 3t^2 - 18t + 24 for t >= 0. What is the total distance travelled by the particle between t = 0 and t = 5 seconds?
  24. Find the points on the curve y = 2x^3 - 3x^2 - 10x + 1 where the tangent is parallel to the line y = 2x + 7.
  25. A curve has a gradient given by dy/dx = k*x - 4. If the curve has a stationary point at x = 2 and passes through (1, 3), find the equation of the curve.
  26. What is the maximum area of a rectangle that can be inscribed under the parabola y = 12 - x^2 and above the x-axis, with its base on the x-axis and symmetric about the y-axis?
  27. The cost in thousands of dollars to manufacture x thousand items is C(x) = x^3 - 6x^2 + 15x + 10. For what production level x is the marginal cost a minimum?
  28. Differentiate y = (3x - 2)^4 with respect to x using the chain rule.
  29. What is the gradient of the chord connecting x = 1 and x = 3 on the curve y = 2x^2 + 1?
  30. If a curve has dy/dx = 4 - 2x, over what domain is the curve concave down?
  31. Find the equation of the line passing through (4, -1) with a gradient of -3.
  32. What is the derivative of f(x) = 5 / x^2 with respect to x?
  33. At what x-coordinate does the tangent to y = x^2 - 4x + 7 have a gradient of 6?
  34. If s(t) = t^3 - 6t^2 + 9t describes displacement, at what times is the object stationary?
  35. Evaluate the anti-derivative integral( (4x^3 + 6x) dx ).
  36. Differentiate y = 3x^4 - 2x with respect to x.
  37. Find the gradient of the curve y = x^2 - 4x at the point where x = 3.
  38. A particle moves so that its displacement is s = t^3 - 6t^2 + 9t metres after t seconds. At what times is the particle at rest?
  39. Find the coordinates of the turning point of the curve y = x^2 - 6x + 5.
  40. Find the equation of the tangent to the curve y = x^2 at the point where x = 2.

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