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

The equation of a curve is \(y = (5-2x)^{\frac{1}{2}} + 5\) for \(x < \frac{5}{2}\).

1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\). [4]
2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is \(-3\). Find the equation of the perpendicular bisector of \(AB\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]

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]

A curve is such that \(\frac{d^2y}{dx^2} = -4x\). The curve has a maximum point at \((2, 12)\).

1. Find the equation of the curve. [6]

A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at 0.05 units per second.

1. Find the rate at which the \(y\)-coordinate is changing when \(x = 3\), stating whether the \(y\)-coordinate is increasing or decreasing. [2]

\includegraphics{figure_2} The diagram shows the curve \(y = 2x^2\) and the points \(X(-2, 0)\) and \(P(p, 0)\). The point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.

1. Express the area, \(A\), of triangle \(XPQ\) in terms of \(p\). [2]

The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(PQ\) remains parallel to the \(y\)-axis.

1. Find the rate at which \(A\) is increasing when \(p = 2\). [3]

\includegraphics{figure_10} The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{3}}\) and the tangent to the curve at the point \(A\). The \(x\)-coordinate of \(A\) is 4.

1. Find the equation of the tangent to the curve at \(A\). [5]
2. Find, showing all necessary working, the area of the shaded region. [5]
3. A point is moving along the curve. At the point \(P\) the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of \(P\). [3]

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

1. Find \(\frac{dy}{dx}\). [2]

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.

1. Find the possible \(x\)-coordinates of \(A\). [4]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{\ln(2x + 1)}{x + 3}\). The curve has a maximum point M.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. Show that the x-coordinate of M satisfies the equation \(x = \frac{x + 3}{\ln(2x + 1)} - 0.5\). [2]
3. Show by calculation that the x-coordinate of M lies between 2.5 and 3.0. [2]
4. Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]

\includegraphics{figure_5} The diagram shows part of the curve with equation \(y = \frac{x^3}{x + 2}\). At the point \(P\), the gradient of the curve is 6.

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt[3]{12x + 12}\). [4]
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 3.8 and 4.0. [2]
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Show the result of each iteration to 5 significant figures. [3]

The function \(\text{f}\) is defined by \(\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)\) for \(0 \leqslant x < \pi\).

1. Find the exact value of \(\text{f}'\left(\frac{\pi}{3}\right)\). [3]

A curve has parametric equations $$x = \ln(t + 1), \quad y = t^2 \ln t.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Find the exact value of \(t\) at the stationary point. [2]
3. Find the gradient of the curve at the point where it crosses the \(x\)-axis. [2]

A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$

1. Find the equation of the tangent to the curve at the origin. [5]
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

\includegraphics{figure_9} A container in the shape of a cuboid has a square base of side \(x\) and a height of \((10 - x)\). It is given that \(x\) varies with time, \(t\), where \(t > 0\). The container decreases in volume at a rate which is inversely proportional to \(t\). When \(t = \frac{1}{10}\), \(x = \frac{1}{2}\) and the rate of decrease of \(x\) is \(\frac{20}{37}\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac{dx}{dt} = \frac{-1}{2t(20x - 3x^2)}$$ [5]
2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(x\). [6]

Let \(\text{f}(x) = \frac{e^{2x} + 1}{e^{2x} - 1}\), for \(x > 0\).

1. The equation \(x = \text{f}(x)\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5. [2]
2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
3. Find f\('(x)\). Hence find the exact value of \(x\) for which f\('(x) = -8\). [6]

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

1. Find the equation of the curve. [4]
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 \((2, 5)\). [2]
3. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \frac{\mathrm{d}y}{\mathrm{d}x}\) is constant. [2]

A particle \(P\) of mass \(2 \text{ kg}\) moving on a horizontal straight line has displacement \(x \text{ m}\) from a fixed point \(O\) on the line and velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\). The only horizontal force acting on \(P\) is a variable force \(F \text{ N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]

\includegraphics{figure_1} Figure 1 shows a sketch of a segment \(PQRP\) of a circle with centre \(O\) and radius \(5\) cm. Given that • angle \(PQR\) is \(\theta\) radians • \(\theta\) is increasing, from \(0\) to \(\pi\), at a constant rate of \(0.1\) radians per second • the area of the segment \(PQRP\) is \(A\) cm²

1. show that $$\frac{dA}{d\theta} = K(1 - \cos \theta)$$ where \(K\) is a constant to be found. [2]
2. Find, in cm²s⁻¹, the rate of increase of the area of the segment when \(\theta = \frac{\pi}{3}\) [4]

The function f is given by $$f(x) = \frac{3(x + 1)}{(x + 2)(x - 1)}, \quad x \in \mathbb{R}, x \neq -2, x \neq 1.$$

1. Express \(f(x)\) in partial fractions. [3]
2. Hence, or otherwise, prove that \(f'(x) < 0\) for all values of \(x\) in the domain. [3]

Given that \(y = \arctan \frac{x}{\sqrt{1 + x^2}}\) show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2}}\) [4]

Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A\) cm\(^2\), after \(t\) seconds is given by $$A = (p + qt)^2,$$ where \(p\) and \(q\) are positive constants. Given that when \(t = 0\), \(A = 4\) and that when \(t = 5\), \(A = 9\),

1. find the value of \(p\) and show that \(q = \frac{1}{5}\), [5]
2. find \(\frac{\mathrm{d}A}{\mathrm{d}t}\) in terms of \(t\), [3]
3. find the rate at which the area of the stain is increasing when \(t = 15\). [2]