\begin{tabular}{l} For each problem, approximate the area under the given function and above the \( x \)-axis on the given \\ interval using the specified number of rectangles/trapezoids. You may use your calculator. \\ 1.) \( f(x)=x^{2}-3 x+4,[1,4], n=6 \) \\ a) Left Rectangles \\ b) Right Rectangles \\ d) Trapezoids \\ c) Midpoint Rectangles \\ (n) \\ \\ \hline\end{tabular}

The area bounded by the curve \( y=f(x) \) where \( y=n-\frac{n^{2}}{4}, x=\frac{n}{2} \) and \( X \)-axis equals....... square unit. \( \begin{array}{llll}\text { (a) } \frac{7}{3} & \text { (b) } \frac{3}{4} & \text { (c) } \frac{4}{3} & \text { (d) } \frac{3}{7}\end{array} \)

3. Find the value of \( \lim _{x \rightarrow 0} \frac{x^{2}+3 x}{x} \) if it exists.

Find the first and second derivatives of \( f(x)=x^{3}(2 x-5) \)

Osrucuims fuzrarenui inmerpan: \( \int_{1}^{4} \sqrt{7 x-3} d x \)

Solve the second order ordinary differential equation \[ \begin{array}{l}\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+l(l+1) y=0 \\ \text { and with the knowledge that Legendre's polynomials are defined such that } \mathrm{P}_{1}(\mathrm{x}=1)=1 \text {, show that } \\ \qquad P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)\end{array} \]