1.07m Tangents and normals: gradient and equations

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1. Sketch the graph of \(y = \sqrt { 2 } \sin x\) for \(0 \leqslant x \leqslant 2 \pi\). The points \(P\) and \(Q\) on the graph have \(x\)-coordinates \(\frac { 1 } { 4 } \pi\) and \(\frac { 3 } { 4 } \pi\), respectively.
2. Determine the equation of the tangent to the curve at \(P\). The normals to the curve at \(P\) and \(Q\) intersect at the point \(R\).
3. Determine the exact coordinates of \(R\).

The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.

1. Find \(\frac{dy}{dx}\). [2]
2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan^{-1}(2)\) with the positive \(x\)-axis. [4]

\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.

1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]

The equation of a curve is \(y = (5-2x)^{\frac{1}{2}} + 5\) for \(x < \frac{5}{2}\).

1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\). [4]
2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is \(-3\). Find the equation of the perpendicular bisector of \(AB\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]

The equation of a curve is such that \(\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}\). The curve passes through the point \(P(2, 8)\).

1. Find the equation of the normal to the curve at \(P\). [2]
2. Find the equation of the curve. [4]

\includegraphics{figure_7} The diagram shows part of the curve with equation \(y = \frac{12}{\sqrt{2x+1}}\). The point \(A\) on the curve has coordinates \(\left(\frac{7}{2}, 6\right)\).

1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = mx + c\). [4]
2. Find the area of the region bounded by the curve and the lines \(x = 0\), \(x = \frac{7}{2}\) and \(y = 0\). [4]

The equation of a curve is \(y = 4 + 5x + 6x^2 - 3x^3\).

1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases. [4]
2. It is given that \(y = 9x + k\) is a tangent to the curve. Find the value of the constant \(k\). [4]

A function f with domain \(x > 0\) is such that \(\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}\). It is given that the curve with equation \(y = \mathrm{f}(x)\) passes through the point \((1, 0)\).

1. Find the equation of the normal to the curve at the point \((1, 0)\). [3]
2. Find f\((x)\). [4]

It is given that the equation \(\mathrm{f}'(x) = 0\) can be expressed in the form $$125x^2 - 128x + 192 = 0.$$

1. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]

The equation of a curve is \(y = kx^{\frac{1}{2}} - 4x^2 + 2\), where \(k\) is a constant.

1. Find \(\frac{\text{d}y}{\text{d}x}\) and \(\frac{\text{d}^2y}{\text{d}x^2}\) in terms of \(k\). [2]
2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature. [4]
3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6. Find this value of \(k\). [6]

The equation of a curve is \(y = \frac{1}{6}(2x - 3)^3 - 4x\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis. [3]
3. Find the set of values of \(x\) for which \(\frac{1}{6}(2x - 3)^3 - 4x\) is an increasing function of \(x\). [3]

A curve has equation \(y = \frac{4}{3x - 4}\) and \(P(2, 2)\) is a point on the curve.

1. Find the equation of the tangent to the curve at \(P\). [4]
2. Find the angle that this tangent makes with the \(x\)-axis. [2]

The curve \(y = \frac{10}{2x + 1} - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).

1. Show that the equation of \(AC\) is \(5y + 4x = 8\). [5]
2. Find the distance \(AC\). [2]

\includegraphics{figure_10} The diagram shows part of the curve \(y = \frac{8}{\sqrt{(3x + 4)}}\). The curve intersects the \(y\)-axis at \(A(0, 4)\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).

1. Find the coordinates of \(B\). [5]
2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal. [6]

The equation of a curve is \(y = \frac{A}{2x - 1}\).

1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360°\) about the \(x\)-axis. [4]
2. Given that the line \(2y = x + c\) is a normal to the curve, find the possible values of the constant \(c\). [6]

The line \(3y + x = 25\) is a normal to the curve \(y = x^2 - 5x + k\). Find the value of the constant \(k\). [6]

\includegraphics{figure_10} The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{3}}\) and the tangent to the curve at the point \(A\). The \(x\)-coordinate of \(A\) is 4.

1. Find the equation of the tangent to the curve at \(A\). [5]
2. Find, showing all necessary working, the area of the shaded region. [5]
3. A point is moving along the curve. At the point \(P\) the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of \(P\). [3]

\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \sqrt{x^3 + x^2}\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [4]
2. \(P\) is the point on the curve with \(x\)-coordinate \(3\). Find the \(y\)-coordinate of the point where the normal to the curve at \(P\) crosses the \(y\)-axis. [6]

\includegraphics{figure_10} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}}\).

1. The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(AB\). [6]
2. The tangent to the curve at a point \(T\) is parallel to \(AB\). Find the coordinates of \(T\). [3]
3. Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(AB\). [3]

A curve is such that \(\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-\frac{3}{2}}\), where \(a\) is a positive constant. The point \(A(a^2, 3)\) lies on the curve. Find, in terms of \(a\),

1. the equation of the tangent to the curve at \(A\), simplifying your answer, [3]
2. the equation of the curve. [4]

It is now given that \(B(16, 8)\) also lies on the curve.

1. Find the value of \(a\) and, using this value, find the distance \(AB\). [5]

A curve is defined by the parametric equations $$x = 4\cos^2 t, \quad y = \sqrt{3}\sin 2t,$$ for values of \(t\) such that \(0 < t < \frac{1}{2}\pi\). Find the equation of the normal to the curve at the point for which \(t = \frac{1}{6}\pi\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [7]

\includegraphics{figure_6} The diagram shows the curve \(y = (4 - x)e^x\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Write down the coordinates of \(A\) and \(B\). [2]
2. Find the \(x\)-coordinate of \(M\). [4]
3. The point \(P\) on the curve has \(x\)-coordinate \(p\). The tangent to the curve at \(P\) passes through the origin \(O\). Calculate the value of \(p\). [5]

The equation of a curve is \(3x^2 + 4xy + y^2 = 24\). Find the equation of the normal to the curve at the point \((1, 3)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [8]

A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$

1. Find the equation of the tangent to the curve at the origin. [5]
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

The parametric equations of a curve are $$x = \tan^2 2t, \quad y = \cos 2t,$$ for \(0 < t < \frac{1}{4}\pi\).

1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\cos^3 2t\). [4]
2. Hence find the equation of the normal to the curve at the point where \(t = \frac{1}{8}\pi\). Give your answer in the form \(y = mx + c\). [4]