OCR MEI C2 (Core Mathematics 2)

The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).

1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve. [7]
3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\). Write down the coordinates of the minimum point of the transformed curve. [3]

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

1. Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
2. Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
3. 1. Expand \((2 + h)^4\). [3]
   2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
   3. Show how your result in part (iii) (B) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]

1. Calculate the gradient of the chord joining the points on the curve \(y = x^2 - 7\) for which \(x = 3\) and \(x = 3.1\). [2]
2. Given that \(f(x) = x^2 - 7\), find and simplify \(\frac{f(3 + h) - f(3)}{h}\). [3]
3. Use your result in part (ii) to find the gradient of \(y = x^2 - 7\) at the point where \(x = 3\), showing your reasoning. [2]
4. Find the equation of the tangent to the curve \(y = x^2 - 7\) at the point where \(x = 3\). [2]
5. This tangent crosses the \(x\)-axis at the point P. The curve crosses the positive \(x\)-axis at the point Q. Find the distance PQ, giving your answer correct to 3 decimal places. [3]

In Fig. 5, A and B are the points on the curve \(y = 2^x\) with \(x\)-coordinates 3 and 3.1 respectively. \includegraphics{figure_5}

1. Find the gradient of the chord AB. Give your answer correct to 2 decimal places. [2]
2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2^x\) at A. [2]