5. (10 points) From the equation

 
Solvein terms of x and y

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: 

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 integralwhereand 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.

一、下面的短文是一位著名國關理論學者2019年在一個重要期刊發表的文章摘要,請先閱讀再回答以下問題。 A century ago, Woodrow Wilson changed America's place in the world when he sent two million men to fight in Europe, but America withdrew into isolationism in the 1930s. After the Second World War, Harry Truman and others created a framework of permanent alliances and multilateral institutions that became known as the 'liberal international order' or 'Pax Americana'. Those terms have become obsolete as descriptions of the US place in the world, but the need for the largest countries to provide public goods remains. An open international order covers political-military affairs; economic relations; ecological relations; and human rights. It remains to be seen to what degree these depend on each other and what will temain as the 1945 package is unpacked. Wilson's legacy of developing international institutions continues to make sense. Leadership is not the same as domination, and it will need to be shared. There have always been degrees of leadership and degrecs of influence during the seven decades of American pre- eminence after 1945. Now with less preponderance and a more complex world, American exceptionalism in terms of its economic and military power should focus on sharing the provision of global public goods, particularly those that require 'power with' others. Wilson's century old insights about international institutions and a rules-based order will remain crucial, but America's place in that world may be threatened more by the rise of populist politics at home than the rise of other powers abroad. 請簡要說明何謂 “Pax Americana”·何謂“American exceptionalism",兩者有甚麼關係?並請評估作者的立論是基於哪個(或哪些)國際關係理論的 假定或視角,然後說明你是否贊同其說?理由為何?(25%)