大学の数学の問題です。
数学の問題です。
よく分からないのでa,b,cすべての解答を教えてください。
cはどういうグラフか説明をお願いします。
あと、英訳もしていただけると助かります。
A map α : I → R^3 is called a curve of class C^k if each of the coordinate functions in the expression α(t) = (x(t),y(t),z(t)) has continuous derivatives up to order k. If α is merely contiuous, we say that α is of class C^k. A curve α is called simple if the map α is one-to-one.
Let α : I → R^3 be a simple curve of class C^2. We say that α has a weak tangent at t = t。∈ I if the line determined by α(t。+ h) and α(t。) has a limit position when h → 0. We say that α has a strong tangent at t = t。 if the line determined by α(t。 + h) and α(t。+ k) has a limit position when h,k → 0. Show that
a. α(t) = (t^3,t^2), t ∈ R, has a weak tangent but not a strong tangent at t = 0.
b. If α : I → R^3 is of class C^1 and regular at t = t。, then it has a strong tangent at t = t。.
c. The curve given by α(t) = (t^2,t^2) (t≧0) α(t) = (t^2,-t^2) (t≦0) is of class C^1 but not of class C^2. Draw a sketch of the curve and its tangent vectors.
お礼
回答ありがとうございます