1.07s Parametric and implicit differentiation

10 The diagram shows the curve with equation $$x = ( y - 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the \(y\)-axis at \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{afc8561d-94ae-42c0-bc6c-e9b091938368-3_588_780_1087_680}

1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Find the exact gradient of the curve at each of the points \(A\) and \(B\).

8 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 3 t .$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point when \(t = 0\), giving your answer in the form \(a y + b x + c = 0\) where \(a , b\) and \(c\) are integers.

8 The curve \(C\) has equation \(y ^ { 3 } + 6 y ^ { 2 } - 2 y = 3 x ^ { 2 } + 2 x\). Show that the equation of the normal to \(C\) at the point \(( 1,1 )\) can be written in the form \(8 y + 13 x - 21 = 0\).

7 A curve, \(C\), is given parametrically by \(x = 2 \cos \theta , y = 3 \sin \theta , 0 < \theta < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 2 } \cot \theta\). A tangent to \(C\) intersects the \(x\)-axis and \(y\)-axis at \(P\) and \(Q\) respectively.
2. Show that the midpoint of \(P Q\) has coordinates \(\left( \sec \theta , \frac { 3 } { 2 } \operatorname { cosec } \theta \right)\).
3. Hence show that the midpoint of \(P Q\) lies on the curve \(\frac { 4 } { x ^ { 2 } } + \frac { 9 } { y ^ { 2 } } = 4\).

7 Find the coordinates of the two stationary points of the curve $$9 x ^ { 2 } + 4 y ^ { 2 } - 6 x - 4 y = 34$$ showing that one is a maximum and one is a minimum.

7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.

1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
2. Show that this normal does not meet the curve again.

8 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 3 t .$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point when \(t = 0\), giving your answer in the form \(a y + b x + c = 0\) where \(a , b\) and \(c\) are integers.

12 The curve \(C\) is defined parametrically by $$x = t + \ln ( \cosh t ) , \quad y = \sinh t$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \cosh ^ { 2 } t\).
2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \cosh ^ { 2 } t ( 2 \sinh t - \cosh t )\).
3. Find the exact value of \(t\) at the point on \(C\) where \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

A curve \(C\) has parametric equations \(x = \sin \theta , y = \cos 2 \theta\). a) The equation of the tangent to the curve \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 4 }\) is \(y = m x + c\). Find the exact values of \(m\) and \(c\).
b) Find the coordinates of the points of intersection of the curve \(C\) and the straight line \(x + y = 1\). The diagram below shows a sketch of the graph of \(y = f ( x )\). The graph crosses the \(y\)-axis at the point \(( 0 , - 2 )\), and the \(x\)-axis at the point \(( 8,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-3_784_1080_1407_513}
a) Sketch the graph of \(y = - 4 f ( x + 3 )\). Indicate the coordinates of the point where the graph crosses the \(x\)-axis and the \(y\)-coordinate of the point where \(x = - 3\).
b) Sketch the graph of \(y = 3 + f ( 2 x )\). Indicate the \(y\)-coordinate of the point where \(x = 4\).

a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\) ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\) iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\) b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
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The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
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A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.

The parametric equations of the curve \(C\) are $$x = 3 - 4 t + t ^ { 2 } , \quad y = ( 4 - t ) ^ { 2 }$$ a) Find the coordinates of the points where \(C\) meets the \(y\)-axis.
b) Show that the \(x\)-axis is a tangent to the curve \(C\). a) Prove that $$\cos ( \alpha - \beta ) + \sin ( \alpha + \beta ) \equiv ( \cos \alpha + \sin \alpha ) ( \cos \beta + \sin \beta )$$ b) i) Hence show that \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta }\) can be expressed as \(\cos \theta + \sin \theta\).
ii) Explain why \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta } \neq \cos \theta + \sin \theta\) when \(\theta = \frac { 3 \pi } { 16 }\).

A curve \(C\) has the equation $$x^3 + 2xy - x - y^3 - 20 = 0$$

1. [(a)] Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). \hfill [5]
2. [(b)] Find an equation of the tangent to \(C\) at the point \((3, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. \hfill [2]

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point \(P\) lies on \(C\) and has coordinates \((3, 2)\). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).

1. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]

The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.

1. [(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p\pi + q\pi^2\), where \(p\) and \(q\) are rational numbers to be determined. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}

\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]

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_7} The diagram shows the curve with parametric equations $$x = k \tan t, \quad y = 3 \sin 2t - 4 \sin t,$$ for \(0 < t < \frac{1}{2}\pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).

1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction. [3]
2. Express \(\frac{dy}{dx}\) in terms of \(k\) and \(\cos t\). [4]
3. Given that the normal to the curve at \(P\) has gradient \(\frac{9}{10}\), find the value of \(k\), giving your answer as an exact fraction. [3]

\includegraphics{figure_6} The diagram shows the curve with parametric equations $$x = 1 + \sqrt{t}, \quad y = (\ln t + 2)(\ln t - 3),$$ for \(0 < t < 25\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\) and has a minimum point \(M\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4\ln t - 2}{\sqrt{t}}\). [4]
2. Find the exact gradient of the curve at \(B\). [2]
3. Find the exact coordinates of \(M\). [3]

A curve has parametric equations $$x = \frac{e^{2t} - 2}{e^{2t} + 1}, \quad y = e^{3t} + 1.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [4]
2. Find the exact gradient of the curve at the point where the curve crosses the \(y\)-axis. [3]

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]

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]

The parametric equations of a curve are $$x = 2\theta + \sin 2\theta, \quad y = 1 - \cos 2\theta.$$ Show that \(\frac{dy}{dx} = \tan \theta\). [5]

\includegraphics{figure_5} The diagram shows the curve with equation $$x^3 + xy^2 + ay^2 - 3ax^2 = 0,$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\). [6]

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

1. Show that \(\frac{dy}{dx} = \cot t\). [5]
2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2. Give your answer correct to 3 significant figures. [2]

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]