2. Let C be the part of the curve z = y in the plane x = 1 from (1,2,4) to (1,0,0). Consider a cable in the shape of C with mass per unit length p(x, y, z) =√√4+18.xyz. (a) Write down a parametrisation for C in terms of an increasing parameter t. (b) Use MATLAB to sketch C in R³. (c) Determine the mass of the cable.

Suppose f(z) = u(x, y) +iv(x, y) is differentiable at zo = xo+iyo, with f'(zo) # 0. Show that the vectors Vu(xo, yo) and Vv(xo, yo) are perpendicular and have the same magnitude. Express this magnitude in terms of f'(zo).

3. Let i, j, k denote the unit vectors along the three coordinate axes. Let v(t) = ti+ sintj + costk and w(t) = 3ti + 2k. (a) Compute (v.w)' (t) directly and check your answer by using the product rule (for the dot product, stated in the class). Note that the dot above, is the dot product of vectors and denotes the derivative. (b) Compute (v x w)'(t) directly and check your answer by using the product rule (for the cross product, again stated in the class). Note that the x above, is the cross product of vectors.

Consider the curve C with parametrisation where t E R. r(t) = = cos(5t) cosh(t) i+ sin(5t) cosh(t) -j+tanh(t)k, (a) Show that C lies on the surface of the sphere described by the equation x² + y² + z² = 1. (b) Find the vector equation of the line L that is tangent to C at the point r(0).

Consider the curve C with parametrisation where t E R. r(t) = = cos(5t) cosh(t) i+ sin(5t) cosh(t) -j+tanh(t)k, (a) Show that C lies on the surface of the sphere described by the equation x² + y² + z² = 1. (b) Find the vector equation of the line L that is tangent to C at the point r(0).