5.4 Tangents and normalsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The curve $C$ has equation $y=x^3-4x+1$ and passes through the point $P(2,1)$.

   (a)

   Find the gradient of the tangent to $C$ at $P$.

   [1 mark]

   • A$12$
   • B$1$
   • C$8$
   • D$-\frac{1}{8}$

   (b)

   Find the gradient of the normal to $C$ at $P$.

   [1 mark]

   • A$-\frac{1}{8}$
   • B$8$
   • C$\frac{1}{8}$
   • D$-8$

   (c)

   Find the equation of the tangent to $C$ at $P$, giving your answer in the form $y=mx+c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=2x^2-5x+3$.

   (a)

   At which point on the curve is the tangent parallel to the line $y=3x+1$?

   [1 mark]

   • A$(1,0)$
   • B$(3,6)$
   • C$(2,7)$
   • D$(2,1)$

   (b)

   Find the equation of the normal to the curve at the point where $x=1$.

   [1 mark]

   • A$y=-x+1$
   • B$y=x-1$
   • C$y=x+1$
   • D$y=-x-1$

   (c)

   Find the equation of the tangent to the curve at the point where $x=3$. Give your answer in the form $ax+by+d=0$, where $a$, $b$ and $d$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=x^3-3x^2+2x+4$. The point $A$ on the curve has $x$-coordinate $3$.

   (a)

   Find the equation of the tangent to the curve at $A$.

   [3 marks]

   (b)

   Find the equation of the normal to the curve at $A$, and the coordinates of the point where this normal meets the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The profile of a hill is modelled by $h(x)=-0.01x^3+0.3x^2$, for $0\le x\le30$, where $x$ metres is the horizontal distance from the start of the slope and $h$ metres is the height above ground level.

   (a)

   Find the equation of the tangent to the profile at the point where $x=5$.

   [6 marks]

   (b)

   A straight tunnel is drilled into the hill along the normal to the profile at the point where $x=5$, going down to ground level ($y=0$).
   (i) Find the equation of the normal.
   (ii) Find the value of $x$ at which the tunnel reaches ground level.
   (iii) Find the length of the tunnel.

   [6 marks]

   Total for question 4: 12 marks

End of questions