1.07a Derivative as gradient: of tangent to curve

A curve has equation $$y = 4x^2 - 5x$$ The curve passes through the point \(P(2, 6)\) Use differentiation from first principles to find the value of the gradient of the curve at \(P\). [4]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\text{m s}^{-1}\), where \(v = 2t^4 + kt^2 - 4\). The acceleration of \(P\) when \(t = 2\) is \(28\text{m s}^{-2}\).

1. Show that \(k = -9\). [3]
2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). [3]

When \(t = 1\), \(P\) is at the point \((-6.4125, 0)\).

1. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity. [4]

% Figure 2 shows curve with vertical asymptotes at x = -2 and x = 2, horizontal asymptote at y = 1, with U-shaped region between asymptotes \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{x^2 - 2}{x^2 - 4}\) and \(x \neq \pm 2\). The curve cuts the \(y\)-axis at \(U\).

1. Write down the coordinates of the point \(U\). [1]

The point \(P\) with \(x\)-coordinate \(a\) (\(a \neq 0\)) lies on \(C\).

1. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left(0, \frac{a^2 - 2}{a^2 - 4} - \frac{(a^2 - 4)^2}{4}\right)$$ [6]

The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).

1. 1. Show that $$\frac{a^2}{2(a^2-4)} - \frac{(a^2-4)^2}{4} = a^2 + \frac{(a^2-4)^4}{16}$$
   2. Hence, show that $$(a^2 - 4)^2 = 1$$
   3. Find the centre and radius of \(E\).
   [10]

[Total 17 marks]

4 A particle \(P\) moves on a straight line, starting from rest at a point \(O\) of the line. The time after \(P\) starts to move is \(t \mathrm {~s}\), and the particle moves along the line with constant acceleration \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it passes through a point \(A\) at time \(t = 8\). After passing through \(A\) the velocity of \(P\) is \(\frac { 1 } { 2 } t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) immediately after it passes through \(A\). Hence show that the acceleration of \(P\) decreases by \(\frac { 1 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it passes through \(A\).
2. Find the distance moved by \(P\) from \(t = 0\) to \(t = 27\).