1.07n Stationary points: find maxima, minima using derivatives

9 A curve has parametric equations \(x = 1 - \cos t , y = \sin t \sin 2 t\), for \(0 \leqslant t \leqslant \pi\).

1. Find the coordinates of the points where the curve meets the \(x\)-axis.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos 2 t + 2 \cos ^ { 2 } t\). Hence find, in an exact form, the coordinates of the stationary points.
3. Find the cartesian equation of the curve. Give your answer in the form \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a polynomial.
4. Sketch the curve.

3 The function f is defined by \(\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }\), for \(x \in \mathbb { R }\) and \(a > 0\).

1. For the curve with equation \(y = \mathrm { f } ( x )\),
   1. write down the equation of the asymptote,
   2. find the range of values that \(y\) can take.
   3. For the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\), write down
      (a) the equation of the line of symmetry,
      (b) the maximum and minimum values of \(y\),
   4. the set of values of \(x\) for which the curve is defined.

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

1. Find the equations of the asymptotes of the curve.
2. Write down the coordinates of the points where the curve cuts the axes.
3. Show that the curve has no stationary points.
4. Sketch the curve and the asymptotes.

2 Given that \(y = \frac { x ^ { 2 } + x + 1 } { ( x - 1 ) ^ { 2 } }\), prove that \(y \geqslant \frac { 1 } { 4 }\) for all \(x \neq 1\).

8

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
   1. \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
   2. \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
   3. Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
   4. Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that $$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
   5. Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).

5 A function is defined by \(\mathrm { f } ( x ) = \sinh ^ { - 1 } x + \sinh ^ { - 1 } \left( \frac { 1 } { x } \right)\), for \(x \neq 0\).

1. When \(x > 0\), show that the value of \(\mathrm { f } ( x )\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\) is \(2 \ln ( 1 + \sqrt { 2 } )\).
2. \includegraphics[max width=\textwidth, alt={}, center]{72a1330a-c6dc-4f3a-9b0e-333b099f4509-3_497_659_520_708} The diagram shows the graph of \(y = \mathrm { f } ( x )\) for \(x > 0\). Sketch the graph of \(y = \mathrm { f } ( x )\) for \(x < 0\) and state the range of values that \(\mathrm { f } ( x )\) can take for \(x \neq 0\).

7 The equation of a curve is \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ( x - 7 ) }\).

1. Write down the equations of the asymptotes.
2. Find the coordinates of the stationary points on the curve.
3. Find the coordinates of the point where the curve meets one of its asymptotes.
4. Sketch the curve.

5 A curve has equation \(y = \frac { x ^ { 2 } - 8 } { x - 3 }\).

1. Find the equations of the asymptotes of the curve.
2. Prove that there are no points on the curve for which \(4 < y < 8\).
3. Sketch the curve. Indicate the asymptotes in your sketch.

10 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant. Obtain the equation of each of the asymptotes of \(C\). In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\). In both cases the coordinates of the turning points must be indicated.

The curve \(C\) has equation $$y = \frac { x ( x + 1 ) } { ( x - 1 ) ^ { 2 } }$$

1. Obtain the equations of the asymptotes of \(C\).
2. Show that there is exactly one point of intersection of \(C\) with the asymptotes and find its coordinates.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence
   1. find the coordinates of any stationary points of \(C\),
   2. state the set of values of \(x\) for which the gradient of \(C\) is negative.
   3. Draw a sketch of \(C\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + \lambda x - 6 \lambda ^ { 2 } } { x + 3 }$$ where \(\lambda\) is a constant such that \(\lambda \neq 1\) and \(\lambda \neq - \frac { 3 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and deduce that if \(C\) has two stationary points then \(- \frac { 3 } { 2 } < \lambda < 1\).
2. Find the equations of the asymptotes of \(C\).
3. Draw a sketch of \(C\) for the case \(0 < \lambda < 1\).
4. Draw a sketch of \(C\) for the case \(\lambda > 3\).

9 The curve \(C\) with equation $$y = \frac { a x ^ { 2 } + b x + c } { x - 1 }$$ where \(a , b\) and \(c\) are constants, has two asymptotes. It is given that \(y = 2 x - 5\) is one of these asymptotes.

1. State the equation of the other asymptote.
2. Find the value of \(a\) and show that \(b = - 7\).
3. Given also that \(C\) has a turning point when \(x = 2\), find the value of \(c\).
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\).

9 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 2 x + 3 } { x ^ { 2 } + 2 }$$ Show that, for all \(x , 1 \leqslant y \leqslant \frac { 5 } { 2 }\). Find the coordinates of the turning points on \(C\). Find the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of any intersections with the \(y\)-axis and the asymptote.

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Find the coordinates of the turning points on \(C\). Draw a sketch of \(C\).

10 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\). State the equations of the asymptotes of \(C\). Show that \(y \leqslant \frac { 25 } { 12 }\) at all points of \(C\). Find the coordinates of any stationary points of \(C\). Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.

7 The curve \(C\) has parametric equations $$x = \sin t , \quad y = \sin 2 t , \quad \text { for } 0 \leqslant t \leqslant \pi .$$ Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\). Hence, or otherwise, find the coordinates of the stationary points on \(C\) and determine their nature.

5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \sin ^ { 2 } x \mathrm {~d} x\), for \(n \geqslant 0\). By differentiating \(\cos ^ { n - 1 } x \sin ^ { 3 } x\) with respect to \(x\), prove that $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 } \quad \text { for } n \geqslant 2$$ Hence find the exact value of \(I _ { 4 }\).

7 A curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(0 < y < 8\). Sketch \(C\), giving the coordinates of the turning points.

9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 6 } { 1 - x }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the turning points of \(C\).
3. Find the coordinates of any intersections with the coordinate axes.
4. Sketch \(C\).

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + b } { x + b }$$ where \(b\) is a positive constant.

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) does not intersect the \(x\)-axis.
3. Justifying your answer, find the number of stationary points on \(C\).
4. Sketch C. Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.

10 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where \(a\) is a constant and \(a > 2\).

1. Find the equations of the asymptotes of \(C _ { 1 }\).
2. Find the equation of the oblique asymptote of \(C _ { 2 }\).
3. Show that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
4. Find the coordinates of the stationary points of \(C _ { 2 }\).
5. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C _ { 2 }\) crosses the axes.]

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { k x - 1 }$$ where \(k\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points of \(C\).
3. Sketch \(C\).

The curve \(C\) has equation $$y = \frac { ( x - a ) ( x - b ) } { x - c }$$ where \(a , b , c\) are constants, and it is given that \(0 < a < b < c\).

1. Express \(y\) in the form $$x + P + \frac { Q } { x - c }$$ giving the constants \(P\) and \(Q\) in terms of \(a , b\) and \(c\).
2. Find the equations of the asymptotes of \(C\).
3. Show that \(C\) has two stationary points.
4. Given also that \(a + b > c\), sketch \(C\), showing the asymptotes and the coordinates of the points of intersection of \(C\) with the axes.

The curve \(C\) has equation $$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$ where \(a\) is a constant not equal to 1,2 or 3 .

1. Write down the equations of the asymptotes of \(C\).
2. Show that \(C\) meets the asymptote parallel to the \(x\)-axis at the point where \(x = \frac { 2 a - 3 } { a - 2 }\).
3. Show that the \(x\)-coordinates of any stationary points on \(C\) satisfy $$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$ and hence find the set of values of \(a\) for which \(C\) has stationary points.
4. Sketch the graph of \(C\) for
   1. \(a > 3\),
   2. \(2 < a < 3\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 2 \lambda x } { x ^ { 2 } - 2 x + \lambda }$$ where \(\lambda\) is a constant and \(\lambda \neq - 1\).

1. Show that \(C\) has at most two stationary points.
2. Show that if \(C\) has exactly two stationary points then \(\lambda > - \frac { 5 } { 4 }\).
3. Find the set of values of \(\lambda\) such that \(C\) has two vertical asymptotes.
4. Find the \(x\)-coordinates of the points of intersection of \(C\) with
   1. the \(x\)-axis,
   2. the horizontal asymptote.
   3. Sketch \(C\) in each of the cases
      (a) \(\lambda < - 2\),
      (b) \(\lambda > 2\).