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106年 - 106 國立臺灣師範大學_轉學生招生考試試題_資訊工程學系二年級：微積分#120316

科目：研究所、轉學考(插大)-微積分 | 年份：106年 | 選擇題數：0 | 申論題數：11

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 (11)

1. (10 分) Use Newton's method to find the x-coordinate of the point where the curve y = x² - 2x crosses the horizontal line y = 1. Specifically apply Newton's method with the starting value x0 = 1 to get the first approximation x₁ = ? and then the second approximation x₂ = ? and omit all the other approximations xₙ, n ＞ 2.

2. (10 分) Find the length of the parameterized curve x = 666a6713c1365.jpg, 0 ≤ t ≤ 3.

(a) (10 分) Find666a673870a8f.jpgdx.

(b) (10 分) Solve the differential equation 666a675b67f3b.jpg.
(c) (10 分) Solve the differential equation 2y' = eˣ/² + y.

(a) (6 分) Find the limit of the sequence {aₙ}, where aₙ = 666a677d9fe2f.jpg.

(b) (8 分) Find the radius and interval of convergence of the series 666a67acb17d0.jpg.

(c) (6 分) Evaluate666a682559749.jpg.

(a) (8 分) Find the value of 666a685f00e9e.jpg at the point (1, 1, 1) if the equation

xy + z³x - 2yz = 0

defines z as a function of the two independent variables x and y and the partial derivative exists.
(b) (10 分) Find the derivative of f(x,y,z) = x³ - xy² - z at point P₀ (1,1,0) in the direction of the vector v = 2i - 3j + 6k.

7. (12 分) The usual way to evaluate the improper integral666a68d2a1f02.jpgdx (from 0 to ∞) is first to calculate its square: 666a68e630e36.jpgdxdy (double integral from 0 to ∞ for both x and y). Evaluate the integral and solve the resulting equation for I.