1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

7 The point \(P ( x , y )\) is moving along the curve \(y = x ^ { 2 } - \frac { 10 } { 3 } x ^ { \frac { 3 } { 2 } } + 5 x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).

5 A curve has equation \(y = 3 + \frac { 12 } { 2 - x }\).

1. Find the equation of the tangent to the curve at the point where the curve crosses the \(x\)-axis.
2. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.04 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\).

2 A point is moving along the curve \(y = 2 x + \frac { 5 } { x }\) in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 1\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 4 x + 1 )\) and \(( 2,5 )\) is a point on the curve.

1. Find the equation of the curve.
2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \(( 2,5 )\).
3. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \times \frac { \mathrm { d } y } { \mathrm {~d} x }\) is constant.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }\). The point \(P ( 2,9 )\) lies on the curve.

1. A point moves on the curve in such a way that the \(x\)-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the \(y\)-coordinate when the point is at \(P\). [2]
2. Find the equation of the curve.

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.

1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.

3 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-04_489_465_258_840} The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is \(h \mathrm {~cm}\) the surface of the water is a square of side \(\frac { 1 } { 2 } h \mathrm {~cm}\).

1. Express the volume of water in the container in terms of \(h\).
   [0pt] [The volume of a pyramid having a base area \(A\) and vertical height \(h\) is \(\frac { 1 } { 3 } A h\).]
   Water is steadily dripping into the container at a constant rate of \(20 \mathrm {~cm} ^ { 3 }\) per minute.
2. Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm . \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-06_403_773_258_685} In the diagram, \(A B = A C = 8 \mathrm {~cm}\) and angle \(C A B = \frac { 2 } { 7 } \pi\) radians. The circular \(\operatorname { arc } B C\) has centre \(A\), the circular arc \(C D\) has centre \(B\) and \(A B D\) is a straight line.

9 The equation of a curve is \(x y = 12\) and the equation of a line \(l\) is \(2 x + y = k\), where \(k\) is a constant.

1. In the case where \(k = 11\), find the coordinates of the points of intersection of \(l\) and the curve.
2. Find the set of values of \(k\) for which \(l\) does not intersect the curve.
3. In the case where \(k = 10\), one of the points of intersection is \(P ( 2,6 )\). Find the angle, in degrees correct to 1 decimal place, between \(l\) and the tangent to the curve at \(P\).

8 The equation of a curve is \(y = \frac { 6 } { 5 - 2 x }\).

1. Calculate the gradient of the curve at the point where \(x = 1\).
2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(y\) has a constant value of 0.02 units per second. Find the rate of increase of \(x\) when \(x = 1\).
3. The region between the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume obtained is \(\frac { 12 } { 5 } \pi\).

8 The equation of a curve is \(y = \sqrt { } \left( 8 x - x ^ { 2 } \right)\). Find

1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and the coordinates of the stationary point on the curve,
2. the volume obtained when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

3 An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing at midday.

3 The equation of a curve is \(y = \frac { 2 } { \sqrt { } ( 5 x - 6 ) }\).

1. Find the gradient of the curve at the point where \(x = 2\).
2. Find \(\int \frac { 2 } { \sqrt { } ( 5 x - 6 ) } \mathrm { d } x\) and hence evaluate \(\int _ { 2 } ^ { 3 } \frac { 2 } { \sqrt { } ( 5 x - 6 ) } \mathrm { d } x\).

9 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-4_584_670_881_740} The diagram shows part of the curve \(y = \frac { 8 } { x } + 2 x\) and three points \(A , B\) and \(C\) on the curve with \(x\)-coordinates 1, 2 and 5 respectively.

1. A point \(P\) moves along the curve in such a way that its \(x\)-coordinate increases at a constant rate of 0.04 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing as \(P\) passes through \(A\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-4_547_1057_255_543} The diagram shows the curve \(y = \sqrt { } \left( x ^ { 4 } + 4 x + 4 \right)\).

1. Find the equation of the tangent to the curve at the point ( 0,2 ).
2. Show that the \(x\)-coordinates of the points of intersection of the line \(y = x + 2\) and the curve are given by the equation \(( x + 2 ) ^ { 2 } = x ^ { 4 } + 4 x + 4\). Hence find these \(x\)-coordinates.
3. The region shaded in the diagram is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of revolution.

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends \(A B E\) and \(D C F\) are identical isosceles triangles. Angle \(A B E =\) angle \(B A E = 30 ^ { \circ }\). The length of \(A D\) is 40 cm . The tank is fixed in position with the open top \(A B C D\) horizontal. Water is poured into the tank at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). The depth of water, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\) (see Fig. 2).

1. Show that, when the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by \(V = ( 40 \sqrt { } 3 ) h ^ { 2 }\).
2. Find the rate at which \(h\) is increasing when \(h = 5\).

9 A curve passes through the point \(A ( 4,6 )\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + 2 x ^ { - \frac { 1 } { 2 } }\). A point \(P\) is moving along the curve in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 3 units per minute.

1. Find the rate at which the \(y\)-coordinate of \(P\) is increasing when \(P\) is at \(A\).
2. Find the equation of the curve.
3. The tangent to the curve at \(A\) crosses the \(x\)-axis at \(B\) and the normal to the curve at \(A\) crosses the \(x\)-axis at \(C\). Find the area of triangle \(A B C\).

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

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Explain why the curve has no stationary points. At the point \(P\) on the curve, \(x = 2\).
3. Show that the normal to the curve at \(P\) passes through the origin.
4. A point moves along the curve in such a way that its \(x\)-coordinate is decreasing at a constant rate of 0.06 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

10 A curve has equation \(y = \frac { 1 } { 2 } ( 4 x - 3 ) ^ { - 1 }\). The point \(A\) on the curve has coordinates \(\left( 1 , \frac { 1 } { 2 } \right)\).

1. (a) Find and simplify the equation of the normal through \(A\).
   (b) Find the \(x\)-coordinate of the point where this normal meets the curve again.
2. A point is moving along the curve in such a way that as it passes through \(A\) its \(x\)-coordinate is decreasing at the rate of 0.3 units per second. Find the rate of change of its \(y\)-coordinate at \(A\).

10 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-18_979_679_262_731} The diagram shows part of the curve \(y = 1 - \frac { 4 } { ( 2 x + 1 ) ^ { 2 } }\). The curve intersects the \(x\)-axis at \(A\). The normal to the curve at \(A\) intersects the \(y\)-axis at \(B\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(B\).
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-06_462_878_258_635} The dimensions of a cuboid are \(x \mathrm {~cm} , 2 x \mathrm {~cm}\) and \(4 x \mathrm {~cm}\), as shown in the diagram.

1. Show that the surface area \(S \mathrm {~cm} ^ { 2 }\) and the volume \(V \mathrm {~cm} ^ { 3 }\) are connected by the relation $$S = 7 V ^ { \frac { 2 } { 3 } }$$
2. When the volume of the cuboid is \(1000 \mathrm {~cm} ^ { 3 }\) the surface area is increasing at \(2 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the volume at this instant.

2 A curve has equation \(y = \frac { 2 + 3 \ln x } { 1 + 2 x }\).
Find the equation of the tangent to the curve at the point \(\left( 1 , \frac { 2 } { 3 } \right)\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5 \includegraphics[max width=\textwidth, alt={}, center]{3966e088-0a2f-434a-94fc-40765cd157a7-06_376_848_269_644} The diagram shows the curve with parametric equations $$x = 4 \mathrm { e } ^ { 2 t } , \quad y = 5 \mathrm { e } ^ { - t } \cos 2 t$$ for \(- \frac { 1 } { 4 } \pi \leqslant t \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-06_526_947_276_591} The diagram shows the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \left( x ^ { 2 } - 5 x + 4 \right)\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a maximum at the point \(C\).

1. Find the exact gradient of the curve at \(B\).
2. Find the exact coordinates of \(C\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9d93ad8c-0a22-4de7-8342-387606e4e510-3_584_675_945_735} The diagram shows the part of the curve \(y = \mathrm { e } ^ { x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The curve meets the \(y\)-axis at the point \(A\). The point \(M\) is a maximum point.

1. Write down the coordinates of \(A\).
2. Find the \(x\)-coordinate of \(M\).
3. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } e ^ { x } \cos x d x$$ giving your answer correct to 2 decimal places.
4. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (iii).

6 It is given that the curve \(y = ( x - 2 ) \mathrm { e } ^ { x }\) has one stationary point.

1. Find the exact coordinates of this point.
2. Determine whether this point is a maximum or a minimum point.