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7. (8 分) Show that the area of a triangle with the vertices (x₁,y₁), (x₂,y₂),and (X₃,y₃) is
where0＜x₁＜x₃2and 0＜y₁＜y₂＜y₃.

相關申論題

(ii) (10 分) P.S. For (i) you can use the fact

2. (8 分) Calculate the volume of the solid formed by revolving the region bounded by the graphs of y= x3 +x+ 1,y=1 and x = 1 about the line x = 2.

3.(8 分) Let T(x, y, z) = 20 + 2x + 2y +Z2 represent the temperature at each point on the sphere x2 + y2 +Z2 = 11. Calculate the mini- mum and maximum temperatures on the curve formed by the inter- section of the plane x + y + z = 3 and this sphere.

4. (8 分) Let D be the region bounded by the lines x-2y=0,x-2y+4=0x+j-4=0,x+y-1=0. Calculate the integral 3xydxdy.

5.(8分)Calculatewhere C is the path enclosing the annular region R shown in Figure 1.

(i) (10 分) Show that the Taylor series of f(x) at x = 0 is

(ii) (8 分) By integrating f(-x2) over a suitable region and using the formula of verify the value of Here you need only calculate it without the rigorous analysis.

(iii) (8 分) Let Find a matrix B such that = I + A. Hint: (i) is a key for obtaining a B, where you shall notice16 = 24.

(i) (8 分) If S =[v1, v2,...,vn} is an orthogonal set of nonzero vectors in V, then S is linearly independent.

相關試卷

110年 - 110 國立清華大學碩士班考試入學試題_計算與建模科學研究所:數學分析#105592

110年 · #105592

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