\( \operatorname { Lim } _ { v \rightarrow + \infty } \frac { 6 e ^ { 2 v } - v ^ { 2 } + 1 } { e ^ { 2 v } - 4 \hat { u } + 1 } \)

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.

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

Find differenthal equation of the primitue \( x y+x^{2} y+y^{2}=13 \)

Given that \( \int_{0}^{1}\left(2+3 \cos \frac{1}{2} x\right) d x=\int_{-1}^{2} e^{2 x+1} d x \), find the value of \( k \), giving your answer correct to 3 decimal places.