1.07q Product and quotient rules: differentiation

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$

1. Deduce the range of values of \(x\) for which \(\mathrm { f } ( x ) \geqslant 0\)
2. Find \(\mathrm { f } ^ { \prime } ( x )\) giving your answer in simplest form. The point \(R ( - 4,84 )\) lies on \(C\).
   Given that the tangent to \(C\) at the point \(P\) is parallel to the tangent to \(C\) at the point \(R\) (c) find the \(x\) coordinate of \(P\).
   (d) Find the point to which \(R\) is transformed when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation,
   1. \(y = \mathrm { f } ( x - 3 )\)
   2. \(y = 4 \mathrm { f } ( x )\)

7. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geqslant - \frac { 7 } { 4 }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
2. Find the coordinates of \(P\).
3. Hence find the range of the function g defined by $$g ( x ) = - 4 f ( x ) \quad x \geqslant - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 \mathrm { f } \left( x - \frac { 3 } { 2 } \right) - 8$$

9. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
   (Solutions based entirely on calculator technology are not acceptable.)

11 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\). \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-4_631_901_532_571} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed.
2. In the first model the equation is $$y = \mathrm { e } ^ { - x } \cos 12 x$$ Show that this model has a maximum point close to \(A\) and a minimum point close to \(B\), and state the coordinates of these maximum and minimum points and also the \(y\) value when \(x = 0.3\).
3. In an alternative model the equation is $$y = f \cos ( \lambda x ) + g$$ where the constants \(f , \lambda\) and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). Find suitable values for \(f , \lambda\) and \(g\).
4. Using the alternative model, state the value of \(y\) when \(x = 0.3\) and hence comment on how accurate each model is in fitting the three given points.

12 Given \(y = x \mathrm { e } ^ { 2 x }\),

1. find the first four derivatives of \(y\) with respect to \(x\),
2. conjecture an expression for \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) in the form \(( a x + b ) \mathrm { e } ^ { 2 x }\), where \(a\) and \(b\) are functions of \(n\),
3. prove by induction that your result holds for all positive integers \(n\).

7 It is given that \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { - x } ( 2 - x )\).
2. Hence find the exact coordinates of the stationary points on the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

8 The function f is given by \(\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 3 x ^ { 2 } - 1 }\), for \(x > 1\). Show that f is a decreasing function.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-4_622_896_957_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

5

1. Differentiate \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) with respect to \(x\).
2. Hence show that \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) is increasing for all \(x\).

8

1. Using the quotient rule, show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tan 3 \theta ) = 3 + 3 \tan ^ { 2 } 3 \theta\) for \(- \frac { 1 } { 6 } \pi < \theta < \frac { 1 } { 6 } \pi\).
2. Hence find the value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 9 } \pi } \tan ^ { 2 } 3 \theta \mathrm {~d} \theta\), giving your answer in the simplest exact form.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-4_620_894_1064_342} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled \(C\) has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\). In the second model the equation is \(y = f \cos ( \lambda x ) + g\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-4_624_897_1062_342} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled \(C\) has coordinates \(( 0.3,0.04 )\). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + g\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

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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1
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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1
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.

Differentiate the following functions with respect to \(x\). a) \(x ^ { 3 } \ln ( 5 x )\) b) \(( x + \cos 3 x ) ^ { 4 }\)

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]

\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]

Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]

The curve \(y = \frac{\ln x}{x + 1}\) has one stationary point.

1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac{x + 1}{\ln x},$$ and that this \(x\)-coordinate lies between 3 and 4. [5]
2. Use the iterative formula $$x_{n+1} = \frac{x_n + 1}{\ln x_n}$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

\includegraphics{figure_8} The diagram shows the curve \(y = x\cos\frac{1}{2}x\) for \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac{dy}{dx}\) and show that \(4\frac{d^2y}{dx^2} + y + 4\sin\frac{1}{2}x = 0\). [5]
2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis. [5]

\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 \(y = \frac{1+x}{1+2x}\) for \(x > -\frac{1}{2}\). Show that the gradient of the curve is always negative. [3]

\includegraphics{figure_9} The diagram shows the curve \(y = (1 + x^2)\text{e}^{-\frac{3x}{4}}\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).

1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve. [4]
2. Show that the exact value of the area of \(R\) is \(18 - \frac{42}{\text{e}}\). [5]

Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that $$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3] Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]