SECTION B Answer any two questions from this section 1. (a) Find the derivative of \( r(t)=a t \cos (3 t) i+b \sin ^{3} t j+c \cos ^{3} t k \) [8 marks]

To find the derivative of the vector function \( r(t) = a t \cos(3t) \, i + b \sin^3(t) \, j + c \cos^3(t) \, k \), we will differentiate each component with respect to \( t \). 1. For the \( i \) component: \[ \frac{d}{dt}(a t \cos(3t)) = a \cos(3t) + a t(-3\sin(3t)) = a \cos(3t) - 3a t \sin(3t) \] 2. For the \( j \) component: \[ \frac{d}{dt}(b \sin^3(t)) = b \cdot 3\sin^2(t) \cdot \cos(t) = 3b \sin^2(t) \cos(t) \] 3. For the \( k \) component: \[ \frac{d}{dt}(c \cos^3(t)) = c \cdot 3\cos^2(t)(-\sin(t)) = -3c \cos^2(t) \sin(t) \] Putting it all together, the derivative \( r'(t) \) is: \[ r'(t) = \left(a \cos(3t) - 3a t \sin(3t)\right) i + \left(3b \sin^2(t) \cos(t)\right) j + \left(-3c \cos^2(t) \sin(t)\right) k \] This result will give you the rate of change of the vector function \( r(t) \) with respect to \( t \).