104年 - 104 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#118783

1. A rectangle is bounded by the x-axis and the semicircle. What width and length should the rectangle have so that its area is a maximum? ( 10 分)

2. Consider a function described by f(x)=(I+x)². Find the points on the graph of y=(1+x)²that are closest to the point, (5,3).(10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 plane.

3. Find an equation of the tangent line to the graph of at x=2. (10分)

4. Evaluate the integral, ∫(x² -2)exp(x³ -6x)dx, in which exp(●) denotes the natural exponential function.(10 分)

5. Find the volume under the surface z=exp(-x-y) and above the triangle with vertices (0,0),(1,0),and (0,1).(10分)

6. Find the Taylor series of the function f(x)=In(x) about x=1, in which In(●) denotes the natural logarithmic function. Moreover, find the radius of convergence and the interval of convergence for this series.(10分)

7. Evaluate the integral, exp(-x)sin(x)dx.(10分)

8. In which directions does the directional derivative of the function f at point (1,1)vanish if f(x,y)=x²+2y²?(10分)

9. Find. (10分)

10. Evaluate the integral, cos(2nx) sin(x)dx, where n is an integer. (10 分)

104年 - 104 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#118783

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