A-Level Mathematics October/November 2024 Q5(b): Given that the curve passes through the point (4, 11), find the equation of the curve.

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  To find the equation of the curve, $y$, we must integrate the expression for the gradient, $\frac{dy}{dx}$. From part (a), we have $\frac{dy}{dx} = 4x - 3\sqrt{x} + 1$.

  $y = \int (4x - 3x^{\frac{1}{2}} + 1) \,dx$

  Integrating term by term: $y = \frac{4x^2}{2} - \frac{3x^{\frac{3}{2}}}{\frac{3}{2}} + x + c$ $y = 2x^2 - 3 \left(\frac{2}{3}\right) x^{\frac{3}{2}} + x + c$ $y = 2x^2 - 2x^{\frac{3}{2}} + x + c$

  We are given that the curve passes through the point $(4, 11)$. We substitute $x=4$ and $y=11$ into the equation to find the constant of integration, $c$.

  $11 = 2(4)^2 - 2(4)^{\frac{3}{2}} + (4) + c$ $11 = 2(16) - 2(\sqrt{4})^3 + 4 + c$ $11 = 32 - 2(2)^3 + 4 + c$ $11 = 32 - 2(8) + 4 + c$ $11 = 32 - 16 + 4 + c$ $11 = 20 + c$ $c = 11 - 20$ $c = -9$

  Therefore, the equation of the curve is: $y = 2x^2 - 2x^{\frac{3}{2}} + x - 9$

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  How the marks are awarded

  • M1 — Awarded for attempting to integrate the expression for dy/dx. This is shown by an increase in the power of at least one term, leading to the form $k_1x^2 + k_2x^{3/2} + k_3x$.
  • A1 — Awarded for the correct, simplified integrated expression $2x^2 - 2x^{3/2} + x$. The constant of integration, $c$, may or may not be present at this stage.
  • M1 — Awarded for substituting the coordinates $x=4$ and $y=11$ into their integrated equation, which must include a constant of integration ('+c'), and attempting to solve for $c$.
  • A1 — Awarded for the final, completely correct equation $y = 2x^2 - 2x^{3/2} + x - 9$, which must include the correctly determined value of $c=-9$.

  Common mistakes

  • Forgetting to include the constant of integration, '+c', after integrating. This makes it impossible to find the specific curve and loses the final two marks.
  • Errors in integrating the term with a fractional power, for example, incorrectly calculating $\int -3x^{1/2} \,dx$ as $-3x^{-1/2} / (-1/2)$.
  • Arithmetic errors when substituting $x=4$, particularly in evaluating $4^{3/2}$. A common mistake is to calculate this as 6 instead of $(\sqrt{4})^3 = 8$.
  • Incorrectly substituting the coordinates, for example using $x=11$ and $y=4$, which leads to an incorrect value for $c$.

  Examiner tip: Always remember to include the constant of integration, '+c', when finding an indefinite integral, as it is essential for finding the specific solution that passes through a given point.

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