3- (a) Let \( \mathbf{F}=f(r) \mathbf{e}_{\theta} \) be a vector field written in cylindrical coordinates. Draw \( \mathbf{F} \) on the \( z=0 \) plane for \( f(r)=r^{2} \). Find the curl of \( \mathbf{F} \) for generic \( f(r) \). When does the curl vanish? Does it match your intuition about curl? (b) Repeat part (a) for \( \mathbf{F}=f(r) \mathbf{e}_{r} \). 4- The vector field \( \mathbf{F}=\hat{\mathbf{r}}=\mathbf{e}_{1} \) (in polar coordinates) has constant magnitude, but not constant direction. So its derivative is non-zero. What are \( \boldsymbol{\nabla} \cdot \mathbf{F} \) and \( \boldsymbol{\nabla} \times \mathbf{F} \) ? Repeat for \( \mathbf{F}=\hat{\boldsymbol{\theta}}=\mathbf{e}_{2} \).

(a) Determine whether \( f(x)=\frac{|x-4|-2}{x} \) is continuous at \( x=4 \). [6 marks] (b) Given \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \). By finding the limit at appropriate values, obtain the horizontal and vertical asymptotes, if they exist. Hence, sketch the graphs of \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \).

3. \( \lim _{x \rightarrow \infty} \frac{x-1}{2 x+5} \)

nction définie par \( f(x)=\frac{L}{x^{2}+1} \)

If the slope of the tangent to the curve: \( y=f(x) \) at any point on it equals \( 6 x+a \), where \( a \) is constant if the equation of the tangent to the curve at the point \( (1,-1) \) is \( y=4- \) \( 5 x \), then the equation of the curve is ... \( \begin{array}{llll}\text { a) } y=3 x^{2}-4 & \text { b) } y=3 x^{2}-11 x+7 & \text { c) } y=3 x^{2}-4 x & \text { d) } y=3 x^{2}-x-3\end{array} \)

Determine the indefinite integral \( \int (6x^3 + 4x^2) \, dx \)