Differentiating Transcendental Functions

11 Show that \(\mathrm { e } ^ { x }\) is an increasing function for all values of \(x\), as stated in line 39 .

11 Show that \(\mathrm { e } ^ { x }\) is an increasing function for all values of \(x\), as stated in line 39 .

10. Given that \(\theta\) is measured in radians, prove, from first principles, that the derivative of \(\sin \theta\) is \(\cos \theta\) You may assume the formula for \(\sin ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)

1 Find the gradient of the curve \(y = \ln ( 5 x + 1 )\) at the point where \(x = 4\).

1 At the point ( 1,0 ) on the curve \(y = \ln x\), which statement below is correct? Tick ( \(\checkmark\) ) one box. The gradient is negative and decreasing □ The gradient is negative and increasing
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-02_109_109_995_1306} The gradient is positive and decreasing □ The gradient is positive and increasing □

4 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 1 + \cos x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
P is the point on the curve with \(x\)-coordinate \(\frac { 1 } { 3 } \pi\). \begin{figure}[h] \end{figure}

1. Find the \(y\)-coordinate of P .
2. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the gradient of the curve at the point P .
3. Show that the derivative of \(\frac { \sin x } { 1 + \cos x }\) is \(\frac { 1 } { 1 + \cos x }\). Hence find the exact area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 3 } \pi\).
4. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arccos \left( \frac { 1 } { x } - 1 \right)\). State the domain of this inverse function, and add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.

2 The curve \(y = \frac { \ln x } { x ^ { 3 } }\) has one stationary point. Find the \(x\)-coordinate of this point.

6 The curve with equation \(y = x \ln x\) has one stationary point.

1. Find the exact coordinates of this point, giving your answers in terms of e .
2. Determine whether this point is a maximum or a minimum point.

1. $$\mathrm { f } ( x ) = x ^ { \left( x ^ { 2 } \right) } \quad x > 0$$ Use logarithms to find the \(x\) coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\) ．

2 Find the exact coordinates of the stationary point on the curve with equation \(y = 5 x \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

3 The curve with equation \(y = 6 \mathrm { e } ^ { x } - \mathrm { e } ^ { 3 x }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

3 The equation of a curve is \(y = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 4 x\). Find the exact \(x\)-coordinate of each of the stationary points of the curve and determine the nature of each stationary point.

5 A curve has equation \(y = 3 \mathrm { e } ^ { 2 x }\) Find the gradient of the curve at the point where \(y = 10\)

9 A curve has equation \(y = \mathrm { e } ^ { 2 x }\)
Find the coordinates of the point on the curve where the gradient of the curve is \(\frac { 1 } { 2 }\) Give your answer in an exact form.
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David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.

4 Fig. 9 shows the curve \(y = x \mathrm { e } ^ { - 2 x }\) together with the straight line \(y = m x\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P . The dashed line is the tangent at P . \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinate of P is \(- \frac { 1 } { 2 } \ln m\).
2. Find, in terms of \(m\), the gradient of the tangent to the curve at P . You are given that OP and this tangent are equally inclined to the \(x\)-axis.
3. Show that \(m = \mathrm { e } ^ { - 2 }\), and find the exact coordinates of P .
4. Find the exact area of the shaded region between the line OP and the curve.

8
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-10_643_414_260_863} The diagram shows the curve with equation $$y = 3 x ^ { 2 } \ln \left( \frac { 1 } { 6 } x \right) .$$ The curve crosses the \(x\)-axis at the point \(P\) and has a minimum point \(M\).

1. Find the gradient of the curve at the point \(P\).
2. Find the exact coordinates of the point \(M\).

6
\includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-08_451_1086_260_525} The diagram shows the curve with equation $$y = ( \ln x ) ^ { 2 } - 2 \ln x$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a minimum point \(M\).

1. Find the exact value of the gradient of the curve at each of the points \(A\) and \(B\).
2. Find the exact \(x\)-coordinate of \(M\).

3 Fig. 9 shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The function \(y = \mathrm { f } ( x )\) is given by $$f ( x ) = \ln \left( \frac { 2 x } { 1 + x } \right) , x > 0$$ The curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at P , and the line \(x = 2\) at Q . \begin{figure}[h] \end{figure}

1. Verify that the \(x\)-coordinate of P is 1 . Find the exact \(y\)-coordinate of Q .
2. Find the gradient of the curve at P. [Hint: use \(\ln \frac { a } { b } = \ln a - \ln b\).] The function \(\mathrm { g } ( x )\) is given by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } } { 2 - \mathrm { e } ^ { x } } , \quad x < \ln 2 .$$ The curve \(y = \mathrm { g } ( x )\) crosses the \(y\)-axis at the point R .
3. Show that \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\). Write down the gradient of \(y = \mathrm { g } ( x )\) at R .
4. Show, using the substitution \(u = 2 - \mathrm { e } ^ { x }\) or otherwise, that \(\int _ { 0 } ^ { \ln \frac { 4 } { 3 } } \mathrm {~g} ( x ) \mathrm { d } x = \ln \frac { 3 } { 2 }\). Using this result, show that the exact area of the shaded region shown in Fig. 9 is \(\ln \frac { 32 } { 27 }\). [Hint: consider its reflection in \(y = x\).]

4 The equation of a curve is \(y = \cos 2 x + 2 \sin x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the stationary points on the curve for \(0 < x < \pi\).

3 The equation of a curve is \(y = x + 2 \cos x\). Find the \(x\)-coordinates of the stationary points of the curve for \(0 \leqslant x \leqslant 2 \pi\), and determine the nature of each of these stationary points.

9
\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-3_307_601_1553_772} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Using the substitution \(u = \sin x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

4 The curve with equation \(y = \mathrm { e } ^ { - x } \sin x\) has one stationary point for which \(0 \leqslant x \leqslant \pi\).

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

4 The curve with equation \(y = \mathrm { e } ^ { - x } \sin x\) has one stationary point for which \(0 \leqslant x \leqslant \pi\).

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

5 The curve with equation \(y = \mathrm { e } ^ { - a x } \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac { 1 } { 2 } \pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.

3 The length, \(x\) metres, of a Green Anaconda snake which is \(t\) years old is given approximately by the formula $$x = 0.7 \sqrt { } ( 2 t - 1 ) ,$$ where \(1 \leqslant t \leqslant 10\). Using this formula, find

1. \(\frac { \mathrm { d } x } { \mathrm {~d} t }\),
2. the rate of growth of a Green Anaconda snake which is 5 years old.

6 A curve has equation \(y = \frac { 9 \mathrm { e } ^ { 2 x } + 16 } { \mathrm { e } ^ { x } - 1 }\).

1. Show that the \(x\)-coordinate of any stationary point on the curve satisfies the equation $$\mathrm { e } ^ { x } \left( 3 \mathrm { e } ^ { x } - 8 \right) \left( 3 \mathrm { e } ^ { x } + 2 \right) = 0$$
2. Hence show that the curve has only one stationary point and find its exact coordinates.

3 Find, in the form \(y = m x + c\), the equation of the tangent to the curve $$y = x ^ { 2 } \ln x$$ at the point with \(x\)-coordinate e.

1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).

4 A curve has equation \(y = x ^ { 4 } + 2 ^ { x }\) Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)

Differentiate, with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } + \ln 2 x\),
2. \(\left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\).

3. The point \(P\) lies on the curve with equation \(y = \ln \left( \frac { 1 } { 3 } x \right)\). The \(x\)-coordinate of \(P\) is 3 . Find an equation of the normal to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
(5)

2 Given that \(y = \ln ( 5 x )\)
find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$

6

1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
2. Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } ( \pi + \ln 4 )\).
   \(7 \quad\) Two lines \(l\) and \(m\) have equations \(\mathbf { r } = a \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) respectively, where \(a\) is a constant. It is given that the lines intersect.

7 The diagram shows part of the graph of \(y = \mathrm { e } ^ { - x ^ { 2 } }\)
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-10_376_940_607_392} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing. 7

1. Find the values of \(x\) for which the graph is concave. 7
2. The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded.
   \includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-11_372_937_584_355} Use the trapezium rule, with 4 strips, to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } e ^ { - x ^ { 2 } } d x\) Give your estimate to four decimal places.
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   7
3. Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate.
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4. By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place.
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   \section*{END OF SECTION A
   TURN OVER FOR SECTION B}

1. (a) Given that

$$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$ (b) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).

3 You are given that \(y = 4 x + \sin 8 x\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Find the smallest positive value of \(x\) for which \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0\), giving your answer in an exact form.

9. (i) Show that the two non-stationary points of inflection on the curve \(y = \ln \left( 1 + 4 x ^ { 2 } \right)\) are at \(x = \pm \frac { 1 } { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f9e0bca6-c2a3-4868-b38b-942ceabd4992-20_492_1064_237_513} The diagram shows the curve \(y = \ln \left( 1 + 4 x ^ { 2 } \right)\). The shaded region is bounded by the curve and a line parallel to the \(x\)-axis which meets the curve where \(x = \frac { 1 } { 2 }\) and \(x = - \frac { 1 } { 2 }\).
(ii) Show that the area of the shaded region is given by $$\int _ { 0 } ^ { \ln 2 } \sqrt { \mathrm { e } ^ { y } - 1 } \mathrm {~d} y$$ (iii) Show that the substitution \(\mathrm { e } ^ { y } = \sec ^ { 2 } \theta\) transforms the integral in part (ii) to \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } 2 \tan ^ { 2 } \theta \mathrm {~d} \theta\).
(iv) Hence find the exact area of the shaded region.
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6. A curve has equation \(y = x \mathrm { e } ^ { \frac { x } { 2 } }\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection.
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