1.Differentiate $y = \sin(3x^2 + 5x)$ with respect to $x$.
Sol: $\frac{dy}{dx} = \cos(3x^2 + 5x) \cdot (6x + 5) = (6x + 5)\cos(3x^2 + 5x)$.
2.Evaluate $\int_0^\pi \sin^2(x) dx$.
Sol: $\int_0^\pi \frac{1 - \cos(2x)}{2} dx = \left[ \frac{x}{2} - \frac{\sin(2x)}{4} \right]_0^\pi = \frac{\pi}{2} - 0 = \frac{\pi}{2}$.
3.Find the maximum and minimum values of $f(x) = x^3 - 3x + 2$.
Sol: $f'(x) = 3x^2 - 3 = 0 \implies x = \pm 1$. $f''(x) = 6x$. At $x=-1$, $f''(-1)=-6 < 0$ (Max $y = 4$). At $x=1$, $f''(1)=6 > 0$ (Min $y = 0$).
4.If $v(t) = 4t^2 - 2t$, find displacement $s(t)$ from $t=0$ to $t=3\text{ s}$ assuming $s(0)=0$.
Sol: $s(t) = \int_0^3 (4t^2 - 2t) dt = \left[ \frac{4t^3}{3} - t^2 \right]_0^3 = \frac{4(27)}{3} - 9 = 36 - 9 = 27\text{ m}$.
5.Use binomial approximation to calculate $(1.002)^{10}$ correct to 4 decimal places.
Sol: $(1 + 0.002)^{10} \approx 1 + 10(0.002) = 1 + 0.02 = 1.0200$.
6.Find the angle $\theta$ in radians if $\sin \theta = \frac{1}{2}$ in the second quadrant.
Sol: $\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\text{ radians} = 150^\circ$.
