The curve \(y =k{x^2} - 8x - 6\) has a gradient of 4 at the point on it where \(x = 3\). The value of \(k\) is?

The normal to the curve \(y = 2{x^3} - 12{x^2} + 19x + 3\) at the point \(\left( {1,12} \right)\) cuts the \(y\)-axis at \(\left( {0,b} \right)\). The value of \(b\) is? 

At the point \(\left( {1,3} \right)\) on the curve \(y = a{x^2} + bx + 2\), the tangent is parallel to the \(x\)-axis. The values of \(a\) and \(b\) are?

The tangent to the curve \(y =2\sqrt x + 1\) at the point \(\left( {4,5} \right)\) on it, meets the \(x\) - axis at \(Q\). The coordinates of \(Q\) are?

At the point (1,4) on the parabola \(y = a{x^2} + bx + 2\) the tangent is parallel to the \(x\) axis. Find the values of \(a\) and \(b\).

The tangent to the curve \(y = {x^3} - 4{x^2} + bx - 7\) at \(x=2\), is inclined at \({45^o}\) to the \(x\) axis. Find b.

The diagram shows a graph of the cubic function \(y = x^3\) and the tangent line to this curve at the point where \(x = p\).

(i) Find the gradient of the tangent line at \(x = p\).
(ii) Find the equation of the tangent line at \(x = p\).
(iii) Find the value of \(p\) if the tangent line crosses the \(y\)-axis at \((0,\, -2)\)

The line \(ax + by + c = 0\) is a tangent line to the curve \(y = x^4 - 2x + 1\) at the point \((-1, 4)\). Find \(a\), \(b\) and \(c\).

The parabolas \(y = x^2 + bx + c\) and \(y = 4x − x^2\) have a common tangent at the point \((3,3)\). Find the values of \(b\) and \(c\).

The tangent to the curve \(y = x^3 − 4x^2 + bx − 7\) at \(x = 2\), is inclined at \(45^\circ\) to the \(x\) axis. Find \(b\).

For the parabola \(y = x^2 − 3x + 5\) it is given that \(\dfrac{dy}{dx}= 2x − 3\). Find the coordinates of the point on the parabola at which the tangent is inclined at \(135^\circ\) to the \(x\)-axis.

The tangent to the curve \(y = k\sqrt {x}\) at the point where \(x = 4\) is inclined at \(60^\circ\) to the \(x\)-axis. Find the value of \(k\).

For the parabola \(f (x) = ax^2 + bx + c\) ,

(i) Show by first principals that \(f '(x) = 2ax + b\)
(ii) Hence show that the axis of symmetry of the parabola is \(x =− \dfrac{b}{2a}\). 

An object moves in a straight line with position given by \(s(t) = {t^3} - 11{t^2} + 12t\) m from O, where \(t\) is in seconds, \(t \ge 0\) . 

i) Find the velocity function 
ii) Find the instaneous velocity at 2 seconds

A projectile is fired into the air and its height in metres is given by \(h = 40t − 5t^2 +10\), where \(t\) is in seconds. Find:

(i) The initial height.
(ii) The initial velocity.

An object is travelling along a straight line over time t seconds, with displacement according to the formula \(s(t) = t^3 + 6t^2 − 2t + 5\) \(m\). Find

i) The equation of its velocity
ii) The equation of its acceleration.

The diagram shows the graph of a certain function \(f(x)\). Copy this graph and on the same set of axes, draw a sketch of the derivative \(f'(x)\) of the function.

The diagram shows the graph of a certain function \(f(x)\). Copy this graph and on the same set of axes, draw a sketch of the derivative \(f'(x)\) of the function.

If the tangent to the curve \(f(x) = \dfrac{k}{{x + 1}}\) is parallel to the line \(y = 2x + 1\), at \(x = 1\), then the value of the constant \(k\) is ?

If the tangent to the curve \(f(x) = k\sqrt {x + 1} \), at the point where \(x = 1\) has a gradient of \(\dfrac{1}{{\sqrt 2 }}\), then the value of the constant \(k\) is? 

The tangent line to the curve \(y = 2{x^2} + x\) touches the curve at \((1,3)\). The equation of the tangent line is?

Find the value(s) of \(b\) such that \(y=2 x+b\) is a tangent to the parabola \(y=2 x^2+6 x-5\)