1.State the derivative of $\sin(x)$ and $\cos(x)$ with respect to $x$.
Sol: $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$.
2.Evaluate the definite integral $\int_1^3 x^2 dx$.
Sol: $\int_1^3 x^2 dx = \left[ \frac{x^3}{3} \right]_1^3 = \frac{27 - 1}{3} = \frac{26}{3}$.
3.What is the formula for the roots of quadratic equation $ax^2 + bx + c = 0$?
Sol: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
4.State the small angle approximation for $\sin \theta$ and $\cos \theta$ when $\theta \ll 1$ in radians.
Sol: $\sin \theta \approx \theta$ and $\cos \theta \approx 1 - \frac{\theta^2}{2} \approx 1$.
5.Find $\frac{dy}{dx}$ if $y = x^3 \sin(x)$.
Sol: Using product rule: $\frac{dy}{dx} = x^3 \cos(x) + 3x^2 \sin(x)$.
6.Evaluate $\int_0^{\pi/2} \cos(2x) dx$.
Sol: $\int_0^{\pi/2} \cos(2x) dx = \left[ \frac{\sin(2x)}{2} \right]_0^{\pi/2} = \frac{\sin(\pi) - \sin(0)}{2} = 0$.
7.Find the maximum value of $y = 5 \sin x + 12 \cos x$.
Sol: Max value = $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = 13$.
8.Differentiate $y = e^{3x^2 + 2}$ with respect to $x$.
Sol: Using chain rule: $\frac{dy}{dx} = e^{3x^2 + 2} \cdot \frac{d}{dx}(3x^2 + 2) = 6x e^{3x^2 + 2}$.
