114年 - 114 國立臺灣大學_轉學生招生考試_物理學系/土木工程學系/生物環境系統工程學系/生物機電工程學系:微積分(B)#137745

1. =      (1)     .

(a) Find the Taylor series for centered at a = 0.      (2)     .

(b) Find the Taylor series for f(x) centered at a =2.      (3)      

(c)=     (4)     

(a) The linearization of y(x) at x =1 is      (5)      .

(b) y"(1)=      (6)     

4.=     (7)     

 (a) The tangent plane to f(r, y, z) = 10 at (x, y, z) = (0, 1, 2) is      (8)     

(b)

and .Then Dag(0,1,2) =      (9)     

(c) Assume that f(x,y, z) attains the maximum value at (x, y, z) = (0,1,2) when restricted to a level surface h(x, y, z) = 1, where h(x,y, z) is differentiable and(0, 1,2) = 6. Using linear approximation, estimate the maximum value of f(x, y, z) subject to the nearby level surface h(x,y, 2) = 0.9. The estimated value is approximately      (10)     

 (a) Find and by the Mean Value Theorem. Is f(x) differentiable at =2? Justify your answers.

(b)  List all critical numbers of f(x) in the interval (0, 6). Where does f(x) attain a local maximum? Where does f(x) attain a local minimum?

(c)  Find intervals on which y = f(x) is concave upward. Find intervals on which y = f(x) is concave downward. Find the inflection points of f(x).

(d) Sketch the graph of f(x) for x ∈ [0, 6]. 

(e) Suppose that , for ＞ 6. -f(6).

(a) Parametrize the surface S and evaluate curlF .dS directly.

(b) Let C be the boundary curve of S. Decompose C as the union of differentiable parametric curves and evaluate F.dr directly. Show that the line integral equals JC the surface integral in part (a) which is consistent with Stokes' Theorem