Sketch then find derivative/gradient/tangent

Questions that ask to sketch the curve and then find the derivative, gradient at a point, or equation of a tangent line.

3 questions

Edexcel C1 2012 January Q8
8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
  3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
  4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.
Edexcel C1 2011 June Q10
10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$
  1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
  3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
  4. Find the \(x\)-coordinate of \(B\).
Edexcel C1 Q5
5. The curve \(C\) with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).
  1. Sketch the curve \(C\), showing the coordinates of \(A\) and \(B\).
  2. Show that the tangent to \(C\) at \(A\) has the equation $$x + y = 2 .$$