7 → V by a(f)(x) = x² ƒ"(x), where ƒ" (x) is the double differential of a polynomial f(x). pjection? re exist a subspace WCV of V such that V = Im(a) + W?

If u and v are differentiable vector functions, c is a scalar, and f is a real-valued function, write the rules for differentiating the following vector functions. (a) u(t) + v(t) (b) cu(t) (c)f (t) u (t) (d) u(t) . v(t) (e) u(t) x v(t) (f ) u (f (t))

a. Sketch the graph of r(t) = ti+t2j. Show that r(t) is a smooth vector-valued function but the change of parameter t = 73 produces a vector-valued function that is not smooth, yet has the same graph as r(t). b. Examine how the two vector-valued functions are traced, and see if you can explain what causes the problem.

(a) Let vt) be a differentiable vector valued function of t. If v. dy = 0 for all t, can we say anything about |v|? Justify your answer and give it a meaningful interpretation.