Differentiation Applications

Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).

1 Differentiate \(10 x ^ { 4 } + 12\).

7 Given that \(y \in \mathbb { R }\), prove that $$( 2 + 3 y ) ^ { 4 } + ( 2 - 3 y ) ^ { 4 } \geq 32$$ Fully justify your answer.
[0pt] [6 marks]

5 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).

5 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).

1 Given that \(\mathrm { f } ( x ) = 6 x ^ { 3 } - 5 x\), find

1. \(\mathrm { f } ^ { \prime } ( x )\),
2. \(f ^ { \prime \prime } ( 2 )\).

5 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.

1

1. Show that \(4 x ^ { 2 } - 12 x + 3 = 4 \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - 6\).
2. State the coordinates of the minimum point of the curve \(y = 4 x ^ { 2 } - 12 x + 3\).

10 Find the coordinates of the points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 9 } { x }\) at which the tangent is parallel to the line \(y = 8 x + 3\).

9 The equation of a curve is \(y = 24 \sqrt { x } - 8 x ^ { \frac { 3 } { 2 } } + 16\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Find the coordinates of the turning point.
3. Determine the nature of the turning point.

4 A curve has equation \(y = x ^ { 2 } - 2 x - 3\). A point is moving along the curve in such a way that at \(P\) the \(y\)-coordinate is increasing at 4 units per second and the \(x\)-coordinate is increasing at 6 units per second. Find the \(x\)-coordinate of \(P\).

6 Given that \(y = \frac { 5 } { x ^ { 2 } } - \frac { 1 } { 4 x } + x\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the displacement \(s \mathrm {~m}\) from \(O\) is given by \(s = t ^ { 3 } - 4 t ^ { 2 } + 4 t\) and the velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression for \(v\) in terms of \(t\).
2. Find the two values of \(t\) for which \(P\) is at instantaneous rest.
3. Find the minimum velocity of \(P\).

7. Differentiate with respect to \(x\), giving each answer in its simplest form.

1. \(( 1 - 2 x ) ^ { 2 }\)
2. \(\frac { x ^ { 5 } + 6 \sqrt { } x } { 2 x ^ { 2 } }\)

10 A curve has equation \(y = 2 x ^ { 2 } - 8 x \sqrt { x } + 8 x + 1\) for \(x \geq 0\) 10

1. Prove that the curve has a maximum point at ( 1,3 )
   Fully justify your answer.
   10
2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]