3. Given the curve \( r(t)=\left\langle 4 t,-\frac{2 t^{3}}{3}, 2 t^{2}\right\rangle \), 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 \( \kappa(1) \).

Given the following velocity function of an object moving along a line, find the position function with the given initial position.

Resuelve la ecuación diferencial \( \frac{dy}{dx} = 3y^{2} \) dado que \( y(0) = 1 \).

Given the following velocity function of an object moving aiong a line, find the position function with the given initial position. \[ v(t)=2 \sqrt[3]{t}, s(0)=3 \] The position function is \( s(t)=\square \)

Evaluate the following definite integral using the Fundamental Theorem of Calculus. \( \frac{1}{7} \int_{0}^{\ln 4} e^{x} d x \)

Find the point(s) at which the function \( f(x)=10-|x| \) equals its average value on the interval \( [-10,10] \) The function equals its average value at \( x=\square \) (Type an integer or a fraction. Use a comma to separate answers as needed.)