110年 - 110 國立中央大學_碩士班招生考試_數學系/應用數學組(一般生、在職生):微積分#105396

Problem 1. (10%)Find

Problem 2. (10%) Show that

Problem 3. (10%) Evaluate

Problem 4. (10%) Find the indefinite integral

Problem 5. (10%) Evaluate

Problem 6. (10%) Find a positive integer n such that Explain your answer.

Problem 7. (10%) Find the relative extrema of the function Use the second derivative test when applicable.

Problem 8. (10%) Find the maximum and minimum value of the n-variable function x₁ + x₂+...+x_n  subject to the constraint  by the method of Lagrange multipliers.

Problem 9. (10%) Evaluate the double integral
  
where R is the trapezoidal region with vertices (1,0), (2,0), (0, -2) and (0, -1).

Problem 10. (10%) Find the area of the region bounded by the graphs of the polar equations r =