1.07m Tangents and normals: gradient and equations

The parametric equations of a curve are $$x = t^2 + 1, \quad y = 4t + \ln(2t - 1).$$

1. Express \(\frac{dy}{dx}\) in terms of \(t\). [3]
2. Find the equation of the normal to the curve at the point where \(t = 1\). Give your answer in the form \(ax + by + c = 0\). [3]

\includegraphics{figure_10} The diagram shows the curve \(y = x^2 \cos 2x\) for \(0 \leq x \leq \frac{1}{4}\pi\). The curve has a maximum point at \(M\) where \(x = p\).

1. Show that \(p\) satisfies the equation \(p = \frac{1}{2} \tan^{-1} \left(\frac{1}{p}\right)\). [3]
2. Use the iterative formula \(p_{n+1} = \frac{1}{2} \tan^{-1} \left(\frac{1}{p_n}\right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis. [5]

The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis. [2]
2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]

\includegraphics{figure_11} The diagram shows part of the curve \(y = \frac{x}{2} + \frac{6}{x}\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).

1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\). [6]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360° about the \(x\)-axis. Give your answer in terms of \(\pi\). [6]

The equation of a curve is \(y = x \ln(8 - x)\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).

1. Show that \(a\) satisfies the equation \(x = 8 - \frac{8}{\ln(8 - x)}\). [3]
2. Verify by calculation that \(a\) lies between 2.9 and 3.1. [2]
3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

The curve \(C\) has equation \(y = \frac{2x^2 + kx}{x + 1}\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. [5] For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes. [6]

The curve \(C\) has equation $$y = \frac{x^2 + ax - 1}{x + 1},$$ where \(a\) is constant and \(a > 1\).

1. Find the equations of the asymptotes of \(C\). [3]
2. Show that \(C\) intersects the \(x\)-axis twice. [1]
3. Justifying your answer, find the number of stationary points on \(C\). [2]
4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. [3]

The straight line with equation \(y = 4x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = 2x^2 + 8x + 3\) Calculate the value of \(c\) [5]

\includegraphics{figure_3} A sketch of part of the curve \(C\) with equation $$y = 20 - 4x - \frac{18}{x}, \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = -2x + 7\). [6]

The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of \(B\). [5]

The curve \(C\) has equation \(y = 4x^2 + \frac{5-x}{x}\), \(x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate \(1\).

1. Show that the value of \(\frac{dy}{dx}\) at \(P\) is \(3\). [5]
2. Find an equation of the tangent to \(C\) at \(P\). [3]

This tangent meets the \(x\)-axis at the point \((k, 0)\).

1. Find the value of \(k\). [2]

The gradient of the curve \(C\) is given by $$\frac{dy}{dx} = (3x - 1)^2.$$ The point \(P(1, 4)\) lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\). [4]
2. Find an equation for the curve \(C\) in the form \(y = f(x)\). [5]
3. Using \(\frac{dy}{dx} = (3x - 1)^2\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2x\). [2]

The curve \(C\) has equation \(y = \frac{1}{3}x^3 - 4x^2 + 8x + 3\). The point \(P\) has coordinates \((3, 0)\).

1. Show that \(P\) lies on \(C\). [1]
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [5]

Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).

1. Find the coordinates of \(Q\). [5]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = (x - 1)(x^2 - 4).$$ The curve cuts the \(x\)-axis at the points \(P\), \((1, 0)\) and \(Q\), as shown in Figure 2.

1. Write down the \(x\)-coordinate of \(P\) and the \(x\)-coordinate of \(Q\). [2]
2. Show that \(\frac{dy}{dx} = 3x^2 - 2x - 4\). [3]
3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point \((-1, 6)\). [2]

The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point \((-1, 6)\).

1. Find the exact coordinates of \(R\). [5]

The curve \(C\) with equation \(y = f(x)\), \(x \neq 0\), passes through the point \((3, 7\frac{1}{2})\). Given that \(f'(x) = 2x + \frac{3}{x^2}\),

1. find \(f(x)\). [5]
2. Verify that \(f(-2) = 5\). [1]
3. Find an equation for the tangent to \(C\) at the point \((-2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The curve \(C\) has equation \(y = f(x)\), \(x \neq 0\), and the point \(P(2, 1)\) lies on \(C\). Given that $$f'(x) = 3x^2 - 6 - \frac{8}{x^3},$$

1. find \(f(x)\). [5]
2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers. [4]

The curve \(C\) has equation \(y = 4x + 3x^{-1} - 2x^2\), \(x > 0\).

1. Find an expression for \(\frac{dy}{dx}\). [3]
2. Show that the point \(P(4, 8)\) lies on \(C\). [1]
3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3y - x + 20.$$ [4]

The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).

1. Find the length \(PQ\), giving your answer in a simplified surd form. [3]

A curve \(C\) has equation \(y = x^3 - 5x^2 + 5x + 2\).

1. Find \(\frac{dy}{dx}\) in terms of \(x\). [2]

The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2. The \(x\)-coordinate of \(P\) is 3.

1. Find the \(x\)-coordinate of \(Q\). [2]
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [3]

This tangent intersects the coordinate axes at the points \(R\) and \(S\).

1. Find the length of \(RS\), giving your answer as a surd. [4]

The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),

1. find \(y\) in terms of \(x\). [4]
2. Verify that \(C\) passes through the point \((4, 0)\). [2]
3. Find an equation of the tangent to \(C\) at \(P\). [3]

The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).

1. Calculate the exact \(x\)-coordinate of \(Q\). [5]

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

1. find \(\frac{dy}{dx}\). [2]

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]

The curve \(C\) has equation \(y = \text{f}(x)\) and the point \(P(3, 5)\) lies on \(C\). Given that $$\text{f}(x) = 3x^2 - 8x + 6,$$

1. find \(\text{f}'(x)\). [4]
2. Verify that the point \((2, 0)\) lies on \(C\). [2]

The point \(Q\) also lies on \(C\), and the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).

1. Find the \(x\)-coordinate of \(Q\). [5]

The curve \(C\) has equation \(y = x^3 - 5x + \frac{2}{x}\), \(x \neq 0\). The points \(A\) and \(B\) both lie on \(C\) and have coordinates \((1, -2)\) and \((-1, 2)\) respectively.

1. Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\). [5]
2. Show that an equation for the normal to \(C\) at \(A\) is \(4y = x - 9\). [4]

The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).

1. Find the length of \(PQ\). [4]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2x - \frac{2}{\sqrt{x}}, \quad x > 0.$$ The point \(A(1, 5)\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B(b, 0)\), where \(b\) is a constant and \(b > 0\).

1. Verify that \(b = 4\). [1]

The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2.

1. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\). [4]
2. Find the coordinates of the point \(D\). [1]

The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(AD\) and the \(x\)-axis.

1. Use integration to find the area of \(R\). [6]

\includegraphics{figure_11} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, \quad x \geq 0.$$ 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]

The curve \(C\) has equation \(y = 2e^x + 3x^2 + 2\). The point \(A\) with coordinates \((0, 4)\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\). [5]