Taylor Polynomial's Tango

為微積分學生設計的動畫：(Followed by english description) 可將動畫暫停(用鍵盤上的空格鍵)然後思考 (1) 在一次泰勒多項式，可否看出在展開點一次微分的正負？ (2) 在二次泰勒多項式，可否看出在展開點二次微分的正負？ 此時兩曲線的凹凸性質有何關係？ 何種情況可從此二次泰勒多項式看到展開點的一次微分？ (3) 可否從n次泰勒多項式的圖，看出展開點n次微分的正負？ __________________ For calculus students: You can pause the animation (use space bar on keyboard) and try. (1) At the 1st degree Taylor polynomial, What's the sign of the 1st derivative at contact point? (sign means positive or negative) (2) At the 2nd degree Taylor polynomial, What about the sign of the 2nd derivative at the contact point? How about the concavity of this two curves? In what case, we can tell the sign of the 1st derivative at the contact point from this 2nd degree Taylor Polynomial? (3) At nth degree Taylor polynomial, think about the sign of the nth derivative at the contact point. ___________________________ The fixed curve is y=1-sqrt(1.1+sin(x)). (sqrt means the square root)