In this question you may use the values in the table below.

Variables \(x\) and \(y\) are related by the equation \(y=\sin5x\), where \(0\leqslant x\leqslant\frac{\pi}{10}\).

Use calculus to find the approximate change in \(x\) when \(y\) increases from \(\frac{\sqrt3}{2}\) by the small amount \(0.01\).

Given that \(y=x^{2} \tan \frac{x}{2}\), use calculus to find the approximate change in \(y\) as \(x\) increases from \(\frac{\pi}{3}\) to \(\frac{\pi}{3}+h\), where \(h\) is small.

(a) It is given that \(y=\mathrm{e}^{3 x+2} \tan x\). Use calculus to find the approximate change in \(y\) as \(x\) increases from 0.1 to \(0.1+h\), where \(h\) is small.

(b) A curve is such that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\sin (3 x+\pi)\).

The curve passes through the point \(\left(\frac{\pi}{9}, \frac{4}{3}\right)\). Find the exact \(y\)-coordinate of the point on the curve where \(x=\frac{5 \pi}{12}\).

Variables \(x\) and \(y\) are related by the equation \(y=x \sqrt{1+2 x}\). (a) Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). (b) It is given that when \(y=12, x=4\). Find the approximate change in \(x\) when \(y\) increases from 12 by the small amount 0.06 . (c) Find the \(x\)-coordinate of the stationary point on the curve \(y=x \sqrt{1+2 x}\).

Variables \(x\) and \(y\) are related by the equation \(y=\frac{x}{\ln 3 x}\). Use differentiation to find the approximate change in \(y\) when \(x\) increases from 1 to \(1+h\), where \(h\) is small.

It is given that \(y=\frac{\ln \left(2 x^{2}+1\right)}{x+2}, x \neq-2\). (a) Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). (b) Given that \(x\) increases from 1 to \(1+h\), where \(h\) is small, find the approximate corresponding change in \(y\). (c) When \(x=1\), the rate of change in \(y\) is 3 units per second. Find the corresponding rate of change in \(x\).

Variables \(x\) and \(y\) are such that \(y=\cos x \sin ^{2} x\). Use differentiation to find the approximate change in \(y\) as \(x\) increases from 3 to \(3+h\), where \(h\) is small.

\(y=\frac{4x^3+2\sin 8x}{1-x}.\)

Use differentiation to find the approximate change in \(y\) as \(x\) increases from \(0.1\) to \(0.1+h\), where \(h\) is small.

The variables \(x\) and \(y\) are related by

\(y=\frac{\sqrt{3x^2-2}}{x-4}\).

(a) Show that \(\frac{dy}{dx}\) can be written in the form

\(\frac{Ax+B}{(x-4)^2\sqrt{3x^2-2}},\)

where \(A\) and \(B\) are integers to be found.

(b) When \(x=3\), \(x\) is increased by a small amount \(h\). Find the approximate change in \(y\), in terms of \(h\).

(a) Given that

\(y=\frac{1+\cos^2x}{\tan x},\)

use differentiation to find the approximate change in \(y\) as \(x\) increases from \(\frac{\pi}{4}\) to \(\frac{\pi}{4}+h\), where \(h\) is small.

(b) Given that

\(y=\frac{1}{(x-3)^3},\)

show that

\(y-\frac{dy}{dx}-\frac13\frac{d^2y}{dx^2}\)

can be written as

\(\frac{(x+1)(x-4)}{(x-3)^5}.\)

A curve has equation

\(y=\frac{(2x+1)^{3/2}}{x+5},\qquad x\ge0.\)

(a) Show that

\(\frac{dy}{dx}=\frac{(2x+1)^{1/2}}{(x+5)^2}(Ax+B),\)

where \(A\) and \(B\) are integers to be found.

(b) Show that there are no stationary points on this curve.

(c) Find the approximate change in \(y\) when \(x\) increases from \(1\) to \(1+p\), where \(p\) is small.

(d) Given that when \(x=1\) the rate of change in \(x\) is \(2.5\) units per second, find the corresponding rate of change in \(y\).

The variable \(x\) is measured in radians. Given that

\(\displaystyle y=\frac{(1+\sin 3x)^4}{\sqrt{x}}\),

use differentiation to find the approximate change in \(y\) as \(x\) increases from 1.9 to \(1.9+h\), where \(h\) is small.

Variables \(x\) and \(y\) are related by the equation \(\displaystyle y=1+\frac{2}{x}+\frac{1}{x^2}\), where \(x\gt 0\). Use differentiation to find the approximate change in \(x\) when \(y\) increases from 4 by the small amount 0.01.

It is given that

\(y=\frac{(3x^2-2)^{2/3}}{x-1},\)

for \(x\gt 1\).

(a) Write \(\frac{dy}{dx}\) in the form

\(\frac{(3x^2-2)^{-1/3}}{(x-1)^2}\left(x^2+Ax+B\right),\)

where \(A\) and \(B\) are integers.

(b) Find the approximate increase in \(y\) as \(x\) increases from \(2\) to \(2+p\), where \(p\) is small.

Variables \(x\) and \(y\) are such that

\(y=e^{\frac{x}{2}}+x\cos2x,\)

where \(x\) is in radians. Use differentiation to find the approximate change in \(y\) as \(x\) increases from \(1\) to \(1+h\), where \(h\) is small.

Variables \(x\) and \(y\) are such that

\(y=\frac{\sin x}{\cos x}.\)

Use differentiation to find the approximate change in \(y\) as \(x\) increases from \(-\frac{\pi}{4}\) to \(h-\frac{\pi}{4}\), where \(h\) is small.

A curve has equation

\(y=\frac{e^{3x}\sin x}{x^2}.\)

Use differentiation to find the approximate change in \(y\) as \(x\) increases from \(0.5\) to \(0.5+h\), where \(h\) is small.

Variables \(x\) and \(y\) are such that

\(y=\sin x+e^{-x}.\)

Use differentiation to find the approximate change in \(y\) as \(x\) increases from \(\frac{\pi}{4}\) to \(\frac{\pi}{4}+h\), where \(h\) is small.

(a) Differentiate

\(y=\tan(x+4)-3\sin x\)

with respect to \(x\).

(b) Variables \(x\) and \(y\) are such that

\(y=\frac{\ln(2x+5)}{2e^{3x}}.\)

Use differentiation to find the approximate change in \(y\) as \(x\) increases from \(1\) to \(1+h\), where \(h\) is small.

Variables \(x\) and \(y\) are related by the equation \(y=\dfrac{\ln x}{e^x}\).

(i) Show that \(\dfrac{dy}{dx}=\dfrac{1-\ln x}{xe^x}\).

(ii) Hence find the approximate change in \(y\) as \(x\) increases from \(2\) to \(2+h\), where \(h\) is small.

It is given that

\(y=\frac{\ln(4x^2+1)}{2x-3}.\)

(i) Find \(\frac{dy}{dx}\).

(ii) Find the approximate change in \(y\) as \(x\) increases from \(2\) to \(2+h\), where \(h\) is small.

It is given that

\(y=(1+e^{x^2})(x+5).\)

(i) Find \(\frac{dy}{dx}\).

(ii) Find the approximate change in \(y\) as \(x\) increases from \(0.5\) to \(0.5+p\), where \(p\) is small.

(iii) Given that \(y\) is increasing at a rate of \(2\) units per second when \(x=0.5\), find the corresponding rate of change in \(x\).

(i) Differentiate

\(y=(3x^2-1)^{-\frac13}\)

with respect to \(x\).

(ii) Find the approximate change in \(y\) as \(x\) increases from \(\sqrt3\) to \(\sqrt3+p\), where \(p\) is small.

(iii) Find the equation of the normal to the curve \(y=(3x^2-1)^{-\frac13}\) at the point where \(x=\sqrt3\).

The variables \(x\) and \(y\) are such that

\(y=\ln(3x-1),\qquad x\gt\frac13.\)

(i) Find \(\dfrac{dy}{dx}\).

(ii) Hence find the approximate change in \(x\) when \(y\) increases from \(\ln(1.2)\) to \(\ln(1.2)+0.125\).

The variables \(x\) and \(y\) are such that

\(y=\ln(3x-1),\qquad x\gt\frac13.\)

(i) Find \(\dfrac{dy}{dx}\).

(ii) Hence find the approximate change in \(x\) when \(y\) increases from \(\ln(1.2)\) to \(\ln(1.2)+0.125\).

Variables \(s\) and \(t\) are such that \(s=4t+3e^{-t}\).

(i) Find the value of \(s\) when \(t=0\).

(ii) Find the exact value of \(t\) when \(\dfrac{ds}{dt}=2\).

(iii) Find the approximate increase in \(s\) when \(t\) increases from \(\ln5\) to \(\ln5+h\), where \(h\) is small.

A curve has equation

\(y=6x-x\sqrt{x}.\)

(i) Find the coordinates of the stationary point of the curve.

(ii) Determine the nature of this stationary point.

(iii) Find the approximate change in \(y\) when \(x\) increases from \(4\) to \(4+h\), where \(h\) is small.

The variables \(x\) and \(y\) are such that \(y=\ln(x^2+1)\).

(i) Find an expression for \(\dfrac{dy}{dx}\).

(ii) Hence, find the approximate change in \(y\) when \(x\) increases from \(3\) to \(3+h\), where \(h\) is small.