Chapter 33

If \(f(x)=4 x^{2}-3 x+2\) find: \(f(0), f(3), f(-1)\) and \(f(3)-f(-1)\)

Given that \(f(x)=5 x^{2}+x-7\) determine: (i) \(f(2) \div f(1)\) (ii) \(f(3+a)\) (iii) \(f(3+a)-f(3)\) (iv) \(\frac{f(3+a)-f(3)}{a}\)

Differentiate from first principles \(f(x)=x^{2}\) and determine the value of the gradient of the curve at \(x=2\)

Find the differential coefficient of \(y=5 x\)

Find the derivative of \(y=8\)

Differentiate from first principles \(f(x)=2 x^{3}\)

Find the differential coefficient of \(y=\) \(4 x^{2}+5 x-3\) and determine the gradient of the curve at \(x=-3\)

Using the general rule, differentiate the following with respect to \(x\) : (a) \(y=5 x^{7}\) (b) \(y=3 \sqrt{x}\) (c) \(y=\frac{4}{x^{2}}\)

Find the differential coefficient of \(y=\frac{2}{5} x^{3}-\frac{4}{x^{3}}+4 \sqrt{x^{5}}+7\)

If \(f(t)=5 t+\frac{1}{\sqrt{t^{3}}}\) find \(f^{\prime}(t)\)

Differentiate \(y=\frac{(x+2)^{2}}{x}\) with respect to \(x\)

Differentiate the following with respect to the variable: (a) \(y=2 \sin 5 \theta\) (b) \(f(t)=3 \cos 2 t\)

Find the differential coefficient of \(y=7 \sin 2 x-3 \cos 4 x\)

Differentiate the following with respect to the variable: (a) \(f(\theta)=5 \sin (100 \pi \theta-0.40)\) (b) \(f(t)=2 \cos (5 t+0.20)\)

An alternating voltage is given by: \(v=100 \sin 200 t\) volts, where \(t\) is the time in seconds. Calculate the rate of change of voltage when (a) \(t=0.005 \mathrm{~s}\) and (b) \(t=0.01 \mathrm{~s}\)

Differentiate the following with respect to the variable: (a) \(y=3 \mathrm{e}^{2 x}\) (b) \(f(t)=\frac{4}{3 \mathrm{e}^{5 t}}\)

Differentiate \(y=5 \ln 3 x\)

Find the gradient of the curve \(y=3 x^{2}-7 x+2\) at the point \((1,-2)\)

If \(y=\frac{3}{x^{2}}-2 \sin 4 x+\frac{2}{\mathrm{e}^{x}}+\ln 5 x\), determine \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)

If \(f(x)=4 x^{5}-2 x^{3}+x-3\), find \(f^{\prime \prime}(x)\)

Given \(y=\frac{2}{3} x^{3}-\frac{4}{x^{2}}+\frac{1}{2 x}-\sqrt{x}\), determine \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\)

The length \(L\) metres of a certain metal rod at temperature \(t^{\circ} \mathrm{C}\) is given by: \(L=1+\) \(0.00003 t+0.0000004 t^{2}\). Determine the rate of change of length, in \(\mathrm{mm} /{ }^{\circ} \mathrm{C}\), when the temperature is (a) \(100^{\circ} \mathrm{C}\) and (b) \(250^{\circ} \mathrm{C}\)

The luminous intensity \(I\) candelas of a lamp at varying voltage \(V\) is given by: \(I=5 \times 10^{-4} \mathrm{~V}^{2}\). Determine the voltage at which the light is increasing at a rate of \(0.4\) candelas per volt.

Newton's law of cooling is given by: \(\theta=\theta_{0} \mathrm{e}^{-k t}\), where the excess of temperature at zero time is \(\theta_{0}^{\circ} \mathrm{C}\) and at time \(t\) seconds is \(\theta^{\circ} \mathrm{C}\). Determine the rate of change of temperature after \(50 \mathrm{~s}\), given that \(\theta_{0}=15^{\circ} \mathrm{C}\) and \(k=-0.02\)

The pressure \(p\) of the atmosphere at height \(h\) above ground level is given by \(p=p_{0} \mathrm{e}^{-h / c}\), where \(p_{0}\) is the pressure at ground level and \(c\) is a constant. Determine the rate of change of pressure with height when \(p_{0}=10^{5}\) Pascals and \(c=6.2 \times 10^{4}\) at 1550 metres.