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DSE M2 Applications of Differentiation Practice

DSE M2 · 60 practice questions with instant marking and worked solutions.

Applications of Differentiation is an important part of the DSE M2 mathematics curriculum. This topic develops tangents and normals, extrema, points of inflexion, asymptotes, rates of change and optimisation. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Applications of Differentiation has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Applications of Differentiation; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Applications of Differentiation set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Applications of Differentiation is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    Consider the normal to the curve $y=2x^{2}+3x-8$ at $x=4$. Find the gradient of the normal (as a fraction in lowest terms). Choose the correct answer.
  2. 2 Basic
    Consider the tangent to the curve $y=2x^2$ at $x=-5$. Find the $y$-intercept of that tangent. Give the answer only.

  3. 3 Basic
    It is given that $f(x)=x^3-75x+4$. Find the local minimum value of $f(x)$. Show the essential working and give the answer.

  4. 4 Basic
    The edge of a cube, $x$ cm, increases at $1$ cm/s. The rate of increase of its volume (in cm³/s) when $x=7$ is 343. Is this true?
  5. 5 Basic
    Consider the normal to the curve $y=2x^{2}+2x+4$ at $x=-1$. Find the gradient of the normal (as a fraction in lowest terms). Choose the correct answer.
  6. 6 Basic
    Consider the tangent to the curve $y=2x^2$ at $x=2$. Find the $y$-intercept of that tangent. Give the answer only.

  7. 7 Basic
    It is given that $f(x)=x^3-12x+1$. Find the local minimum value of $f(x)$. Show the essential working and give the answer.

  8. 8 Basic
    The edge of a cube, $x$ cm, increases at $1$ cm/s. The rate of increase of its volume (in cm³/s) when $x=3$ is 9. Is this true?
  9. 9 Basic
    Consider the normal to the curve $y=2x^{2}-x+1$ at $x=-1$. Find the gradient of the normal (as a fraction in lowest terms). Choose the correct answer.
  10. 10 Basic
    Consider the tangent to the curve $y=2x^2$ at $x=-4$. Find the $y$-intercept of that tangent. Give the answer only.

  11. 11 Basic
    It is given that $f(x)=x^3-75x-4$. Find the local minimum value of $f(x)$. Show the essential working and give the answer.

  12. 12 Basic
    The edge of a cube, $x$ cm, increases at $1$ cm/s. The rate of increase of its volume (in cm³/s) when $x=5$ is 125. Is this true?
  13. 13 Standard
    Squares of side $x$ cm are cut from the corners of a 18 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Choose the correct answer.
  14. 14 Standard
    The curve $y=x^{3}-3x^{2}-4x-4$ has one point of inflexion. Find the $x$-coordinate of the point of inflexion. Give the answer only.

  15. 15 Standard
    Consider the curve $y=\frac{-2x+4}{x-1}$. Find the value of $q$ in its horizontal asymptote $y=q$. Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Let $f(x)=x^3-12x+4$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the local maximum.
    2. (b) Find the local maximum value.
    3. (c) Find the local minimum value.
  17. 17 Standard
    Squares of side $x$ cm are cut from the corners of a 6 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Choose the correct answer.
  18. 18 Standard
    The curve $y=-x^{3}-9x^{2}+4$ has one point of inflexion. Find the $y$-coordinate of the point of inflexion. Give the answer only.

  19. 19 Standard
    Consider the curve $y=\frac{x-5}{x+5}$. Find the value of $q$ in its horizontal asymptote $y=q$. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    Let $f(x)=x^3-27x-4$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the local maximum.
    2. (b) Find the local maximum value.
    3. (c) Find the local minimum value.
  21. 21 Standard
    Squares of side $x$ cm are cut from the corners of a 30 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Choose the correct answer.
  22. 22 Standard
    The curve $y=-x^{3}+3x^{2}+x+2$ has one point of inflexion. Find the $x$-coordinate of the point of inflexion. Give the answer only.

  23. 23 Standard
    Consider the curve $y=\frac{3x-5}{x-3}$. Find the value of $q$ in its horizontal asymptote $y=q$. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    Let $f(x)=x^3-3x+1$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the local maximum.
    2. (b) Find the local maximum value.
    3. (c) Find the local minimum value.
  25. 25 Advanced
    Consider the normal to the curve $y=x^{2}+4x$ at $x=-1$. Find the gradient of the normal (as a fraction in lowest terms). Choose the correct answer.
  26. 26 Advanced
    Consider the tangent to the curve $y=x^2$ at $x=4$. Find the $y$-intercept of that tangent. Give the answer only.

  27. 27 Advanced
    It is given that $f(x)=x^3-27x-5$. The local minimum value of $f(x)$ is -59. Is this true?
  28. 28 Advanced DSE long question
    Daily profit is $P(x)=-x^2+6x-31$ (hundred dollars) for output $x$ (hundred items). Answer each part; give the answer only.
    1. (a) Find $P'(3)$.
    2. (b) Find the $x$ that maximises profit.
    3. (c) Find the maximum profit (hundred dollars).
  29. 29 Advanced
    Consider the normal to the curve $y=-x^{2}+7$ at $x=4$. Find the gradient of the normal (as a fraction in lowest terms). Choose the correct answer.
  30. 30 Advanced
    Consider the tangent to the curve $y=x^2$ at $x=-1$. Find the $y$-intercept of that tangent. Give the answer only.

  31. 31 Advanced
    It is given that $f(x)=x^3-27x$. The local minimum value of $f(x)$ is -54. Is this true?
  32. 32 Advanced DSE long question
    Daily profit is $P(x)=-3x^2+48x-21$ (hundred dollars) for output $x$ (hundred items). Answer each part; give the answer only.
    1. (a) Find $P'(1)$.
    2. (b) Find the $x$ that maximises profit.
    3. (c) Find the maximum profit (hundred dollars).
  33. 33 Advanced
    Consider the normal to the curve $y=2x^{2}-3x+2$ at $x=-2$. Find the gradient of the normal (as a fraction in lowest terms). Choose the correct answer.
  34. 34 Advanced
    Consider the tangent to the curve $y=x^2$ at $x=-4$. Find the $y$-intercept of that tangent. Give the answer only.

  35. 35 Advanced
    It is given that $f(x)=x^3-75x+1$. The local minimum value of $f(x)$ is -249. Is this true?
  36. 36 Advanced DSE long question
    Daily profit is $P(x)=-3x^2+36x-38$ (hundred dollars) for output $x$ (hundred items). Answer each part; give the answer only.
    1. (a) Find $P'(2)$.
    2. (b) Find the $x$ that maximises profit.
    3. (c) Find the maximum profit (hundred dollars).
  37. 37 Challenging
    Squares of side $x$ cm are cut from the corners of a 36 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Choose the correct answer.
  38. 38 Challenging
    Squares of side $x$ cm are cut from the corners of a 12 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Show the essential working and give the answer.

  39. 39 Challenging
    The curve $y=2x^{3}-18x^{2}+6x-5$ has one point of inflexion. The $x$-coordinate of the point of inflexion is -3. Is this true?
  40. 40 Challenging DSE long question
    Let $f(x)=x^3-3x+3$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the local maximum.
    2. (b) Find the local maximum value.
    3. (c) Find the local minimum value.
  41. 41 Challenging
    The curve $y=-x^{3}+4x+6$ has one point of inflexion. Find the $x$-coordinate of the point of inflexion. Choose the correct answer.
  42. 42 Challenging
    Squares of side $x$ cm are cut from the corners of a 36 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Show the essential working and give the answer.

  43. 43 Challenging
    The curve $y=3x^{3}-18x^{2}+5x-3$ has one point of inflexion. The $x$-coordinate of the point of inflexion is -2. Is this true?
  44. 44 Challenging DSE long question
    Let $f(x)=x^3-3x-6$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the local maximum.
    2. (b) Find the local maximum value.
    3. (c) Find the local minimum value.
  45. 45 Challenging
    The edge of a cube, $x$ cm, increases at $3$ cm/s. Find the rate of increase of its volume (in cm³/s) when $x=5$. Choose the correct answer.
  46. 46 Challenging
    Squares of side $x$ cm are cut from the corners of a 24 cm square sheet, which is folded into an open box. Find the maximum volume of the box in cm³. Show the essential working and give the answer.

  47. 47 Challenging
    The curve $y=-x^{3}+6x^{2}+2x-1$ has one point of inflexion. The $x$-coordinate of the point of inflexion is -2. Is this true?
  48. 48 Challenging DSE long question
    Let $f(x)=x^3-27x-2$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the local maximum.
    2. (b) Find the local maximum value.
    3. (c) Find the local minimum value.
  49. 49 Elite
    Consider the curve $y=\frac{-2x}{x+1}$. Find the value of $p$ in its vertical asymptote $x=p$. Give the answer only.

  50. 50 Elite
    Consider the normal to the curve $y=x^{2}+2x-4$ at $x=-3$. Find the gradient of the normal (as a fraction in lowest terms). Show the essential working and give the answer.

  51. 51 Elite
    Consider the tangent to the curve $y=3x^2$ at $x=-3$. The $y$-intercept of that tangent is -27. Is this true?
  52. 52 Elite DSE long question
    Daily profit is $P(x)=-2x^2+16x-23$ (hundred dollars) for output $x$ (hundred items). Answer each part; give the answer only.
    1. (a) Find $P'(2)$.
    2. (b) Find the $x$ that maximises profit.
    3. (c) Find the maximum profit (hundred dollars).
  53. 53 Elite
    Consider the curve $y=\frac{3x+7}{x+2}$. Find the value of $p$ in its vertical asymptote $x=p$. Give the answer only.

  54. 54 Elite
    Consider the normal to the curve $y=x^{2}+5x+7$ at $x=2$. Find the gradient of the normal (as a fraction in lowest terms). Show the essential working and give the answer.

  55. 55 Elite
    Consider the tangent to the curve $y=x^2$ at $x=-5$. The $y$-intercept of that tangent is -25. Is this true?
  56. 56 Elite DSE long question
    Daily profit is $P(x)=-3x^2+24x-12$ (hundred dollars) for output $x$ (hundred items). Answer each part; give the answer only.
    1. (a) Find $P'(1)$.
    2. (b) Find the $x$ that maximises profit.
    3. (c) Find the maximum profit (hundred dollars).
  57. 57 Elite
    Consider the curve $y=\frac{3x-3}{x+5}$. Find the value of $q$ in its horizontal asymptote $y=q$. Give the answer only.

  58. 58 Elite
    Consider the normal to the curve $y=-x^{2}+5x$ at $x=0$. Find the gradient of the normal (as a fraction in lowest terms). Show the essential working and give the answer.

  59. 59 Elite
    Consider the tangent to the curve $y=2x^2$ at $x=-3$. The $y$-intercept of that tangent is -18. Is this true?
  60. 60 Elite DSE long question
    Daily profit is $P(x)=-2x^2+24x-38$ (hundred dollars) for output $x$ (hundred items). Answer each part; give the answer only.
    1. (a) Find $P'(2)$.
    2. (b) Find the $x$ that maximises profit.
    3. (c) Find the maximum profit (hundred dollars).

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Applications of Differentiation

This topic needs particular care with tangents and normals, extrema, points of inflexion, asymptotes, rates of change and optimisation. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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