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103年 - 103 國立臺灣師範大學_學士班轉學生招生考試試題_機電工程學系二年級：微積分#120325
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8. The function f is defined by
. Evaluate f(1). (10 分)
其他申論題
5. A wire of given length L may be cut into pieces to form a square and an equilateral triangle. How should it be cut if the sum of the areas of the square and the equilateral triangle is to be a minimum? A maximum? (10 分)
#512569
6. What is the maximum directional derivative of the function f(x, y) = 75-2x² - y² at the point (5,5)? (5 分) Find a line tangent to the curve f(x, y) = 0 at this point. (5 分) Note:Point (a, b) means that the projections of the point on the x and y axes have coordinates a and b, respectively, in the rectangular coordinate system in two-dimensional space.
#512570
a. Sketch the region of integration for the given iterated integral. (5 分)
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b. Change or reverse the order of integration. In other words, rewrite the integral in the form of . (5 分)
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9. Find dr dθ. (10 分)
#512574
10. Evaluatecos(2nx)sinx dx, where n is an integer. (10 分)
#512575
11. Evaluate dy dx. (10 分)
#512576
12. Find the least squares line for the given data. (10 分)(3,2), (4,3), (5,2). Method of Least Squares: The straight liney = a + bxshould be fitted through the given points (x₁,y₁), ..., (xₙ,yₙ) so that the sum of the squares of the distances of those points from the straight line is minimum, where the distance is measured in the vertical direction (the y-direction).
#512577
(a) (5 分)
#512578
(b) (5 分)
#512579
. Evaluate f(1). (10 分)
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. Evaluate f(1). (10 分)