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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
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題組內容
11. Let V be an inner product space and
(i) (10 分) Prove the Cauchy-Schwarz Inequality
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(i)(7分)
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(ii) (10 分) P.S. For (i) you can use the fact
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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.
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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.
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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.
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5.(8分)Calculatewhere C is the path enclosing the annular region R shown in Figure 1.
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(i) (10 分) Show that the Taylor series of f(x) at x = 0 is
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(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.
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(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.
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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＜x3
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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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