For each problem, approximate the area under the given function and above the \( x \)-axis on the give 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 c) Midpoint Rectangles d) Trapezoids

Derivada de \( 2 f(x)=\sqrt{x^{2}+4 x-5} \)

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

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

Use the quotient rule to find the derivative of the following. \( y=\frac{4 x+9}{\sqrt{x}} \)

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