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

科目：研究所、轉學考(插大)-微積分 | 年份：103年 | 選擇題數：0 | 申論題數：13

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 (13)

1. Find the radius of convergence and interval of convergence for the given power series. (5 分)666a9028ddc60.jpg

2. Consider the function 666a904d72bc5.jpg. Find f'(x). (5 分)
3. Differentiate x² log₁₀(2x²+3). (5 分)
4. Approximate cosx by a fourth-degree polynomial for values of x close to 0. (5 分)
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)
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.
a. Sketch the region of integration for the given iterated integral. (5 分)

b. Change or reverse the order of integration. In other words, rewrite the integral in the form of 666a90d02eb39.jpg. (5 分)

8. The function f is defined by 666a90f887b2a.jpg. Evaluate f(1). (10 分)

9. Find 666a911aeef4e.jpg dr dθ. (10 分)

10. Evaluate666a913db99de.jpgcos(2nx)sinx dx, where n is an integer. (10 分)

11. Evaluate 666a91666148c.jpgdy dx. (10 分)
12. Find the least squares line for the given data. (10 分)
(3,2), (4,3), (5,2). 
Method of Least Squares: The straight line
y = a + bx
should 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).