1.07i Differentiate x^n: for rational n and sums

The diagram shows part of a curve with a maximum point \(M\). \includegraphics{figure_5} The equation of the curve is $$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$

1. Find \(\frac{dy}{dx}\). [3]
2. Hence find the coordinates of the maximum point \(M\). [4]
3. The point \(P(1, 14)\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20x - 6\). [3]
4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(RM\). [3]

\includegraphics{figure_2} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, x \geq 0.$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.

1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
2. Use integration to find the area of \(R\). [5]

Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),

1. find, to 3 significant figures, the value of x for which f(x) = 5. [3]
2. Show that f(x) may be written in the form \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]

\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

\includegraphics{figure_2} Figure 2 shows part of the curve C with equation y = f(x), where $$f(x) = x^3 - 6x^2 + 5x.$$ The curve crosses the x-axis at the origin O and at the points A and B.

1. Factorise f(x) completely [3 marks]
2. Write down the x-coordinates of the points A and B. [1 marks]
3. Find the gradient of C at A. [3 marks] The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
4. Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]

For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),

1. find \(\frac{dy}{dx}\), [2]
2. find the coordinates of each of the stationary points, [5]
3. determine the nature of each stationary point. [3]

The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).

1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

$$f(x) = \frac{(x^2 - 3)^2}{x^3}, \quad x \neq 0.$$

1. Show that \(f(x) = x - 6x^{-1} + 9x^{-3}\). [2]
2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]
3. Verify that the graph of \(y = f(x)\) has stationary points at \(x = \pm\sqrt{3}\). [2]
4. Determine whether the stationary value at \(x = \sqrt{3}\) is a maximum or a minimum. [3]

\includegraphics{figure_1} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac{1}{3}x^2 - \frac{1}{4}x^3\). The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

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

Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]

Given that \(y = 6x^3 + \sqrt{x} + 3\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [5]

\includegraphics{figure_11} Fig. 11 shows a sketch of the cubic curve \(y = \text{f}(x)\). The values of \(x\) where it crosses the \(x\)-axis are \(-5\), \(-2\) and \(2\), and it crosses the \(y\)-axis at \((0, -20)\).

1. Express f(\(x\)) in factorised form. [2]
2. Show that the equation of the curve may be written as \(y = x^3 + 5x^2 - 4x - 20\). [2]
3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4. Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place. [6]
4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \text{f}(2x)\). [2]

Find \(\frac{dy}{dx}\) when

1. \(y = 2x^{-5}\). [2]
2. \(y = ^4\sqrt{x}\). [3]