3. Given the curve \( \mathrm{r}(t)=\left\langle 4 t,-\frac{2 t^{3}}{3}, 2 t^{2}\right\rangle \), find the followings. (a) (5 pts) Find \( \mathrm{a}(t) \) and \( \mathrm{T}(1) \). (b) ( 8 pts) Find the decomposition of \( \mathrm{a}(t) \) into tangential component, \( a_{T} \), and normal com- ponents, \( a_{N} \), at \( t=1 \). (c) ( 5 pts) Find the curvature \( \kappa(1) \).

Attention, les réponses aux QCM de rattrapage ne rapportent que \( 25 \% \) des points. \[ f: x \mapsto\left(-4 x^{4}-5 x\right)^{4} \] Quelle est la dérivée de la fonction \( f \) ? On admettra qu'un ensemble de dérivabilité existe pour cette fonctio \[ 1 \]

Solt la fonction \( f \) définie ci-dessous: \[ f: x \mapsto(x+1) e^{-8 x+8} \] Déterminer la dérivée de \( f \).

Given that \( \int_{1}^{4} f(x) \mathrm{d} x=5 \), evaluate \( \begin{array}{ll}\text { (a) } \int_{4}^{1} 2 f(x) \mathrm{d} x, & \text { (b) } \int_{1}^{3} \mathrm{f}(x) \mathrm{d} x+\int_{3}^{4}[f(x)+3] \mathrm{d} x\end{array} \)

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 c) Midpoint Rectangles d) Trapezoids (a)

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