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

6 A point \(P\) is moving along a curve in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per minute. The equation of the curve is \(y = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } }\).

1. Find the rate at which the \(y\)-coordinate is increasing when \(x = 1\).
2. Find the value of \(x\) when the \(y\)-coordinate is increasing at \(\frac { 5 } { 8 }\) units per minute.

10 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x ^ { 2 } - \frac { 4 } { x ^ { 3 } }\). The curve has a stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).

1. Determine the nature of the stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).
2. Find the equation of the curve.
3. Show that the curve has no other stationary points.
4. A point \(A\) is moving along the curve and the \(y\)-coordinate of \(A\) is increasing at a rate of 5 units per second. Find the rate of increase of the \(x\)-coordinate of \(A\) at the point where \(x = 1\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 The function f is defined by \(\mathrm { f } ( x ) = ( 4 x + 2 ) ^ { - 2 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
   A point is moving along the curve \(y = \mathrm { f } ( x )\) in such a way that, as it passes through the point \(A\), its \(y\)-coordinate is decreasing at the rate of \(k\) units per second and its \(x\)-coordinate is increasing at the rate of \(k\) units per second.
2. Find the coordinates of \(A\).

9 Water is poured into a tank at a constant rate of \(500 \mathrm {~cm} ^ { 3 }\) per second. The depth of water in the tank, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\). 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 the formula \(V = \frac { 4 } { 3 } ( 25 + h ) ^ { 3 } - \frac { 62500 } { 3 }\).

1. Find the rate at which \(h\) is increasing at the instant when \(h = 10 \mathrm {~cm}\).
2. At another instant, the rate at which \(h\) is increasing is 0.075 cm per second. Find the value of \(V\) at this instant.

4 A curve has equation \(y = x ^ { 2 } - 2 x - 3\). A point is moving along the curve in such a way that at \(P\) the \(y\)-coordinate is increasing at 4 units per second and the \(x\)-coordinate is increasing at 6 units per second. Find the \(x\)-coordinate of \(P\).

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { ( 3 x - 2 ) ^ { 3 } }\) and \(A ( 1 , - 3 )\) lies on the curve. A point is moving along the curve and at \(A\) the \(y\)-coordinate of the point is increasing at 3 units per second.

1. Find the rate of increase at \(A\) of the \(x\)-coordinate of the point.
2. Find the equation of the curve.

3 A curve has equation \(y = \frac { 1 } { 60 } ( 3 x + 1 ) ^ { 2 }\) and a point is moving along the curve.
Find the \(x\)-coordinate of the point on the curve at which the \(x\) - and \(y\)-coordinates are increasing at the same rate.

3 Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the radius of the balloon is increasing when the radius is 10 cm .

7 The point \(( 4,7 )\) lies on the curve \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } }\).

1. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 4\).
2. Find the equation of the curve.

10

1. Find \(\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\). \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726} The diagram shows the curve with equation \(y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the volume of revolution.
   The normal to the curve at the point \(( 1,1 )\) crosses the \(y\)-axis at the point \(A\).
3. Find the \(y\)-coordinate of \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The volume \(V \mathrm {~m} ^ { 3 }\) of a large circular mound of iron ore of radius \(r \mathrm {~m}\) is modelled by the equation \(V = \frac { 3 } { 2 } \left( r - \frac { 1 } { 2 } \right) ^ { 3 } - 1\) for \(r \geqslant 2\). Iron ore is added to the mound at a constant rate of \(1.5 \mathrm {~m} ^ { 3 }\) per second.
[0pt]

1. Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m . [3]
2. Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
   It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
2. Find \(\mathrm { f } ( x )\).
3. Find the coordinates of the other stationary point and determine its nature.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11

1. Find the coordinates of the minimum point of the curve \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\). \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-18_675_901_1270_612} The diagram shows the curves with equations \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\) and \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\). The curves intersect at the points \(( 0,18 )\) and \(( 4,6 )\).
2. Find the area of the shaded region.
3. A point \(P\) is moving along the curve \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\) in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing when \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A large industrial water tank is such that, when the depth of the water in the tank is \(x\) metres, the volume \(V \mathrm {~m} ^ { 3 }\) of water in the tank is given by \(V = 243 - \frac { 1 } { 3 } ( 9 - x ) ^ { 3 }\). Water is being pumped into the tank at a constant rate of \(3.6 \mathrm {~m} ^ { 3 }\) per hour. Find the rate of increase of the depth of the water when the depth is 4 m , giving your answer in cm per minute.

3 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-05_424_529_248_815} The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, \(x \mathrm {~cm}\), of each edge of the container increases at the constant rate of 0.01 cm per minute. The volume of the container at time \(t\) minutes is \(V \mathrm {~cm} ^ { 3 }\). Find the rate of increase of \(V\) when \(x = 20\).

9 A curve has equation \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\).

1. Find the equation of the normal to the curve at the point \(A ( 4,3 )\), giving your answer in the form \(y = m x + c\).
   A point is moving along the curve \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\) in such a way that at \(A\) the rate of increase of the \(x\)-coordinate is \(3 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of increase of the \(y\)-coordinate at \(A\).
   At \(A\) the moving point suddenly changes direction and speed, and moves down the normal in such a way that the rate of decrease of the \(y\)-coordinate is constant at \(5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
3. As the point moves down the normal, find the rate of change of its \(x\)-coordinate.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
2. Find the possible \(x\)-coordinates of \(A\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\) and \(P ( 1,5 )\) is a point on the curve.

1. The normal to the curve at \(P\) crosses the \(x\)-axis at \(Q\). Find the coordinates of \(Q\).
2. Find the equation of the curve.
3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 1\).

10 The equation of a curve is \(y = \sqrt { } ( 5 x + 4 )\).

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 \(x\) has the constant value 0.03 units per second. Find the rate of increase of \(y\) at the instant when \(x = 1\).
3. Find the area enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

2 Find the gradient of the curve \(y = \frac { 12 } { x ^ { 2 } - 4 x }\) at the point where \(x = 3\).

6 The function f is such that \(\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5\) for \(x \geqslant 0\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why f is an increasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]

4 A watermelon is assumed to be spherical in shape while it is growing. Its mass, \(M \mathrm {~kg}\), and radius, \(r \mathrm {~cm}\), are related by the formula \(M = k r ^ { 3 }\), where \(k\) is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and the mass is 3.2 kg . Find the value of \(k\) and the rate at which the mass is increasing on this day.

6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.

4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 8 ( 3 x + 4 ) ^ { - \frac { 1 } { 2 } }\).

1. A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the \(y\)-coordinate as \(P\) crosses the \(y\)-axis. The curve intersects the \(y\)-axis where \(y = \frac { 4 } { 3 }\).
2. Find the equation of the curve.