Finding a curve from its gradientEdexcel International A Level Maths: Subtopic test

1. 1
   A curve passes through the point $(2,9)$ and has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=3x^2-4x+1$.

   (a)

   Integrate the gradient function to find a general expression for $y$.

   [1 mark]

   • A$x^3-2x^2+x$
   • B$x^3-2x^2+x+c$
   • C$6x-4+c$
   • D$x^3-4x^2+x+c$

   (b)

   Find the value of the constant $c$.

   [1 mark]

   • A$7$
   • B$9$
   • C$2$
   • D$-7$

   (c)

   Find the value of $y$ on the curve when $x=-1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The gradient of a curve at the point $(x,y)$ is given by $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{8}{x^3}-3\sqrt{x}$ for $x>0$. The curve passes through the point $(1,4)$.

   (a)

   Integrate the gradient function to find a general expression for $y$.

   [1 mark]

   • A$4x^{-2}-2x^{\frac32}+c$
   • B$-4x^{-2}-\frac92x^{\frac32}+c$
   • C$-24x^{-4}-\frac32x^{-\frac12}+c$
   • D$-4x^{-2}-2x^{\frac32}+c$

   (b)

   Find the value of the constant $c$.

   [1 mark]

   • A$4$
   • B$-10$
   • C$10$
   • D$-6$

   (c)

   Find the value of $y$ on the curve when $x=4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $y=f(x)$ has gradient function $f'(x)=(2x+3)(x-1)$ and passes through the point $(3,10)$.

   (a)

   Show that $f(x)=\frac23x^3+\frac12x^2-3x+c$, where $c$ is a constant.

   [3 marks]

   (b)

   Find $f(x)$, and hence find the $y$-coordinate of the point on the curve where $x=-3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=6\sqrt{x}-\dfrac{3}{x^2}+2$ for $x>0$, and passes through the point $(1,10)$.

   (a)

   Find an equation of $C$.

   [6 marks]

   (b)

   A second curve $E$ has the same gradient function as $C$ and passes through $(4,40)$. Find an equation of $E$, and find the vertical distance between $C$ and $E$. Justify why this distance is the same for every value of $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions