106年 - 106 台灣綜合大學系統學士班轉學生聯合招生試題：微積分A#98983

1. (10 points) Given a curve C in defined by

In(1-x³+y³)-4=0.

Find the point on C at which the tangent line is vertical.

2. (10 points) If the function

is continuous at x= 0, then (a,b) =?

3. (10 points) Write down the first three terms (three lowest order terms) of the

Taylor series of

at O. (Hint:

)

4. (10 points) Evaluate the following integral:

5. (10 points) From the equation

Solvein terms of x and y

6. (10 points) Find all values of a so that the series
is divergent.

7. (10 points) Compute the following improper integral

8. (10 points) Compute the line integral

where

and C is the closed loop formed by traveling from (-2,0) to (4,0) to (3,3) to (-1,3) and back to (-2,0) by straight lines.

9. (10 points) Given the function F :

At (0,0,0), find the direction along which the function decreases most rapidly and find the corresponding rate of change.

10. (10 points) A rectangular box is formed by cutting four equal corners from a square of side 3 and then folding up (see the figure below). Find the maximum possible volume of the box.

106年 - 106 台灣綜合大學系統學士班轉學生聯合招生試題：微積分A#98983

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