\( \lim _ { n \rightarrow \infty } [ \frac { 1 } { \sqrt { 4 n ^ { 2 } - 1 } } + \frac { 1 } { \sqrt { 4 n ^ { 2 } - 2 ^ { 2 } } } + \cdots + \frac { 1 } { \sqrt { 3 n ^ { 2 } } } ] \)

4. Let \( g \) be a function such that \( g(1)=3 \) and whose derivative is \( g^{\prime}(x)=6 \cos (2 \pi x)+3 \). Solve analytically a. Use a linear approximation to estimate the value of \( g(1.2) \). b. Find the solution to the differential equation \( d g=(6 \cos (2 \pi x)+3) d x \)

Given the differential equation \( \frac{dy}{dx} = 3y \) and the initial condition \( y(0) = 2 \), find the particular solution.

\( \lim _ { z - 1 } \frac { 4 z + 3 } { 2 z + 1 } \)

Find equations of the tangent line and normal line to \( x^{2}+4 x y+y^{2}=13,(2,1) \)

\( f ( x ) = 4 x ^ { 3 } - 3 x ^ { 2 } - x - 1 , x = - 1 \)