Curve motion: find dy/dt

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.

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 \(x = 1\).

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.

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 \(x = 1\).

7 The equation of a curve is \(y = \frac { 12 } { x ^ { 2 } + 3 }\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the normal to the curve at the point \(P ( 1,3 )\).
3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

10 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_433_969_1475_587} The diagram shows an open rectangular tank of height \(h\) metres covered with a lid. The base of the tank has sides of length \(x\) metres and \(\frac { 1 } { 2 } x\) metres and the lid is a rectangle with sides of length \(\frac { 5 } { 4 } x\) metres and \(\frac { 4 } { 5 } x\) metres. When full the tank holds \(4 \mathrm {~m} ^ { 3 }\) of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is \(A \mathrm {~m} ^ { 2 }\).

1. Express \(h\) in terms of \(x\) and hence show that \(A = \frac { 3 } { 2 } x ^ { 2 } + \frac { 24 } { x }\).
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) is a minimum, showing clearly that \(A\) is a minimum and not a maximum.
   [0pt] [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 \(A\).

The equation of a curve is \(y = 4\sqrt{x} + \frac{2}{\sqrt{x}}\).

1. Obtain an expression for \(\frac{dy}{dx}\). [3]
2. A point is moving 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 change of the \(y\)-coordinate when \(x = 4\). [2]