100年 - 100 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#122243

科目：研究所、轉學考(插大)-微積分 | 年份：100年 | 選擇題數：0 | 申論題數：17

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所屬科目：研究所、轉學考(插大)-微積分

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申論題 (17)

1. (8 分)In some instances we can find end behavior models for rational functions. If the degree of the numerator is one greater than the degree of the denominator, the graph of the rational function f(x) has an oblique (slanted) asymptote. Find the oblique asymptote for the graph of

(i)Find an equation for the line tangent to the curve at the point defined by t = -1;

(ii) Find the value of d²y/dx² at this point.

3. (8 分) Use implicit differentiation to find dy/dx and d²y/dx² for the curve 2 = x - y.

(i) (6 分)Find the curve y = f(x) in the xy-plane that passes through the point (9, 4) and whose slope at each point is ;

(ii) (6 分)Find dy/dx if y =dt；

(iii) (6 分) Find ∫[sin²x + x²sin(x³)]dx;

(iv)Find dx;

(v) (6 分)Solve the differential equation 2y' = e^x/2 + y.

(i) (8 分)A curved wedge is cut from a cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder. The second plane crosses the first plane at a 45° angle at the center of the cylinder. Find the volume of the wedge.

(ii) (6 分) Find the lengths of the curve x = t³, y = (3t²) / 2, 0 ≤ t ≤ .

(i) (6 分)Evaluate ∫dx.

(ii) (6 分) Find the area under the curve y =from x = 1 to x = ∞.

(i) (4 分) Find.

(ii) (4 分) For what value of a does the series converge?

(iii) (4 分) For what value of x does the power series converge?

(iv) (6 分)Find the length of the curve given by the polar coordinate equation r = -1 + cos θ.