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TMUA 2023 · Paper 1 · Question 19 of 20

TMUA 2023 Paper 1 Question 19

Differentiation and integration — Integrating a modulus. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Differentiation and integrationIntegrating a modulus8 options
The solution to the differential equation dydx = |−6x|   for all  x is y = f(x) + c, where c is a constant.

Which one of the following is a correct expression for f(x)?

  1. A−6x|x|
  2. B6x|x|
  3. C−3x|x|
  4. D3x|x|
  5. E−3x2
  6. F3x2
  7. G−x3
  8. Hx3
Show the answer and worked solution
answer · D
  1. A−6x|x|
  2. B6x|x|
  3. C−3x|x|
  4. D3x|x|
  5. E−3x2
  6. F3x2
  7. G−x3
  8. Hx3
Note |−6x| = 6|x|. For x ≥ 0 this is 6x, integrating to 3x2; for x < 0 it is −6x, integrating to −3x2. A single expression covering both is 3x|x|, since it equals 3x2 when x ≥ 0 and −3x2 when x < 0. Differentiating 3x|x| indeed returns 6|x|.