Answer all Question
[Total marks: 140 Marks]
Question 1
[Total Marks: 32 Marks]
1.1 Find the derivatives of the following functions. (CO2, PO1, C2)
a) $y = \frac{2x^2+1}{e^{2x}}$
[3 Marks]
b) $f(x) = \ln\left(\frac{x^2-1}{\sqrt{x-1}}\right)$
[3 Marks]
1.2 Find the $\frac{dy}{dx}$ in terms of x and y. (CO2, PO1, C2)
$y = \sqrt{\frac{6x}{x+2}}$
[3 Marks]
1.3 Find $\frac{dy}{dx}$ in terms of t. (CO2, PO1, C2)
$x = \frac{1-t^2}{1+t^2}$ and $y = \frac{t}{1+t^2}$
[3 Marks]
1.4 A curve has parametric equations $x = 2t - \ln(2t)$, $y = t^2 - \ln(t^2)$ where $t > 0$. Find the value of t at the point on the curve where $\frac{dy}{dx} = 2$. (CO2, PO1, C2)
[5 Marks]
1.5 Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ of $y = (3+4x)e^{-2x}$. (CO2, PO1, C2)
[5 Marks]
1.6 A manufacturer produces cylindrical metal cans and wants to minimise the amount of material used while maintaining a fixed volume of 1000 cm$^3$. (CO3, PO1, C3)
i. Express the surface area, S of the can in terms of its radius, r.
ii. Find the radius that minimises material usage.
iii. Determine the corresponding height and verify that the result is a minimum.
[10 Marks]
