Show derivative equals expression

5

1. Given that \(y = \tan ^ { 2 } x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \tan x + 2 \tan ^ { 3 } x\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( \tan x + \tan ^ { 2 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).

5 The equation of a curve is \(y = x ^ { 3 } \mathrm { e } ^ { - x }\).

1. Show that the curve has a stationary point where \(x = 3\).
2. Find the equation of the tangent to the curve at the point where \(x = 1\).

4 The equation of a curve is \(y = \frac { 1 + \mathrm { e } ^ { - x } } { 1 - \mathrm { e } ^ { - x } }\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is always negative.
2. The gradient of the curve is equal to - 1 when \(x = a\). Show that \(a\) satisfies the equation \(\mathrm { e } ^ { 2 a } - 4 \mathrm { e } ^ { a } + 1 = 0\). Hence find the exact value of \(a\).

7 The curve \(y = \sin \left( x + \frac { 1 } { 3 } \pi \right) \cos x\) has two stationary points in the interval \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. By considering the formula for \(\cos ( A + B )\), show that, at the stationary points on the curve, \(\cos \left( 2 x + \frac { 1 } { 3 } \pi \right) = 0\).
3. Hence find the exact \(x\)-coordinates of the stationary points.

8. A scientist is studying a population of fish in a lake. The number of fish, \(N\), in the population, \(t\) years after the start of the study, is modelled by the equation $$N = \frac { 600 \mathrm { e } ^ { 0.3 t } } { 2 + \mathrm { e } ^ { 0.3 t } } \quad t \geqslant 0$$ Use the equation of the model to answer parts (a), (b), (c), (d) and (e).

1. Find the number of fish in the lake at the start of the study.
2. Find the upper limit to the number of fish in the lake.
3. Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.
4. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { A \mathrm { e } ^ { 0.3 t } } { \left( 2 + \mathrm { e } ^ { 0.3 t } \right) ^ { 2 } }$$ where \(A\) is a constant to be found. Given that when \(t = T , \frac { \mathrm {~d} N } { \mathrm {~d} t } = 8\)
5. find the value of \(T\) to one decimal place.
   (Solutions relying entirely on calculator technology are not acceptable.)
   \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-27_2644_1840_118_111}

6. (i) The curve \(C _ { 1 }\) has equation $$y = 3 \ln \left( x ^ { 2 } - 5 \right) - 4 x ^ { 2 } + 15 \quad x > \sqrt { 5 }$$ Show that \(C _ { 1 }\) has a stationary point at \(x = \frac { \sqrt { p } } { 2 }\) where \(p\) is a constant to be found.
(ii) A different curve \(C _ { 2 }\) has equation $$y = 4 x - 12 \sin ^ { 2 } x$$

1. Show that, for this curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = A + B \sin 2 x$$ where \(A\) and \(B\) are constants to be found.
2. Hence, state the maximum gradient of this curve.

7. A curve has equation $$y = \ln ( 1 - \cos 2 x ) , \quad x \in \mathbb { R } , 0 < x < \pi$$ Show that

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k \cot x\), where \(k\) is a constant to be found. Hence find the exact coordinates of the point on the curve where
2. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { 3 }\)

1. (a) By writing \(\sec \theta = \frac { 1 } { \cos \theta }\), show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \sec \theta ) = \sec \theta \tan \theta\)
   (b) Given that

$$x = \mathrm { e } ^ { \sec y } \quad x > \mathrm { e } , \quad 0 < y < \frac { \pi } { 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \sqrt { \mathrm {~g} ( x ) } } , \quad x > \mathrm { e }$$ where \(\mathrm { g } ( x )\) is a function of \(\ln x\).

5 Show that the curve \(y = x ^ { 2 } \ln x\) has a stationary point when \(x = \frac { 1 } { \sqrt { \mathrm { e } } }\).

5 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \cos ^ { 2 } x } , - \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), together with its asymptotes \(x = \frac { 1 } { 2 } \pi\) and \(x = - \frac { 1 } { 2 } \pi\). \begin{figure}[h] \end{figure}

1. Use the quotient rule to show that the derivative of \(\frac { \sin x } { \cos x }\) is \(\frac { 1 } { \cos ^ { 2 } x }\).
2. Find the area bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \frac { 1 } { 2 } \mathrm { f } \left( x + \frac { 1 } { 4 } \pi \right)\).
3. Verify that the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) cross at \(( 0,1 )\).
4. State a sequence of two transformations such that the curve \(y = \mathrm { f } ( x )\) is mapped to the curve \(y = \mathrm { g } ( x )\). On the copy of Fig. 9, sketch the curve \(y = \mathrm { g } ( x )\), indicating clearly the coordinates of the minimum point and the equations of the asymptotes to the curve.
5. Use your result from part (ii) to write down the area bounded by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = - \frac { 1 } { 4 } \pi\).

9

1. The function f is defined for all real values of \(x\) by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$ (a) Show that \(\mathrm { f } ^ { \prime } ( x ) > 0\) for all \(x\).
   (b) Show that the set of values of \(x\) for which \(\mathrm { f } ^ { \prime \prime } ( x ) > 0\) is the same as the set of values of \(x\) for which \(\mathrm { f } ( x ) > 0\), and state what this set of values is.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-04_634_830_641_699} The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x } ,$$ where \(k\) is a constant greater than 1 . The graph of \(y = \mathrm { g } ( x )\) is shown above. Find the range of g , giving your answer in simplified form.

8 Fig. 8 shows the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\). P is the point on this curve with \(x\)-coordinate 1 , and R is the point \(\left( 0 , - \frac { 7 } { 8 } \right)\). \begin{figure}[h] \end{figure}

1. Find the gradient of PR.
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that PR is a tangent to the curve.
3. Find the exact coordinates of the turning point Q .
4. Differentiate \(x \ln x - x\). Hence, or otherwise, show that the area of the region enclosed by the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is \(\frac { 59 } { 24 } - \frac { 1 } { 4 } \ln 2\).

1

1. Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence show that the curve \(y = \mathrm { e } ^ { - x } \sin 2 x\) has a stationary point when \(x = \frac { 1 } { 2 } \arctan 2\).

5 Given that \(y = \ln \left( \sqrt { \frac { 2 x - 1 } { 2 x + 1 } } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 x - 1 } - \frac { 1 } { 2 x + 1 }\).

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x \quad 0 < x < \pi$$ a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin ^ { 2 } x\)
b. Find the coordinates of the points of inflection of the curve.

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 \(\frac { \cos h - 1 } { h } \rightarrow 0\)

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

3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when $$y = \mathrm { e } ^ { 3 x } + \ln x$$
2. 1. Given that \(u = \frac { \sin x } { 1 + \cos x }\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 1 } { 1 + \cos x }\).
   2. Hence show that if \(y = \ln \left( \frac { \sin x } { 1 + \cos x } \right)\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } x\).

7

1. By writing \(\sec x = ( \cos x ) ^ { - 1 }\), use the chain rule to show that, if \(y = \sec x\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
2. The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the \(y\)-coordinate of the stationary point of the graph of \(y = \mathrm { f } ( x )\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
   [0pt] [6 marks]

8. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   [0pt]
2. Show that the curve has a stationary point in the interval [1.3,1.4]. The point \(A\) on the curve has \(x\)-coordinate 2 .
3. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
   The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
4. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.

1. The curve \(C\) has equation

$$y = \arccos \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2$$

1. Show that \(C\) has no stationary points. The normal to \(C\), at the point where \(x = 1\), crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\). Given that \(O\) is the origin,
2. show that the area of the triangle \(O A B\) is \(\frac { 1 } { 54 } \left( p \sqrt { 3 } + q \pi + r \sqrt { 3 } \pi ^ { 2 } \right)\) where \(p\), \(q\) and \(r\) are integers to be determined.
   (5)