110年 - 110 國立政治大學_碩士班暨碩士在職專班招生考試_應用數學系:微積分#102193

科目：研究所、轉學考(插大)-微積分 | 年份：110年 | 選擇題數：0 | 申論題數：15

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (15)

1. (14 points) Consider the following function:

Use the definition of the limit to show that

(a) (3 points) Find the partial derivatives at (0, 0) and the gradient vector ∇f(0,0).

(b)(6 points) Use the definition of the directional derivative to find D_u (0, 0) for all unit vectors u = (a, b).

(c) (6 points) Is f continuous at (0,0)? How about differentiability?

(a) (6 points) Let f(x) be a continuous function. Show that if f(x) has two local maxima, it also must have a local minima.

(b) (8 points) Show that part (a) becomes wrong for a function of two variables by finding all local maxima and minima of the function:

(a) (7 points) Evaluate

(b) (7 points) Evaluate

(a) (7 points) Use a suitable change of variable to compute
where R is the square with vertices (1,0), (0, 1), (-1, 0) and (0, -1).

(b) (7 points) Evaluate

(a) (3 points) Find radius and interval of convergence.

(b) (6 points) Show that for Ixl < 1:

(c) (6 points) The function on the right-hand side of (1) has a removable discontinuity at x = 1. Remove it and then show that (1) also holds for x = 1.

(a) (6 points) Consider
Compute f'(π).

(b) (8 points) Suppose that f(0) = 0 and

Find f(x).