106年 - 106 台灣聯合大學系統(清華、交通、陽明、中央四校聯招)學士班轉學生考試_A3、A4、A7：微積分#112636

1. How many critical points does the function f(x) =|x² - 1|have? Answer :________

2.If = 1, then a-b=? Answer :_________

3. Suppose f" is continuous on [0,1], f(1) = 2, f'(1) = 2 and the average value of f on [0, 1] is 2. Evaluate dx. Answer :__________

4. Find g', where g(x) is the inverse of f(x) = . Answer :___________

5. Find the directional derivative of f(z,y) = at P₀(1,1) in the direction of v= .Answer :__________

6. Evaluate dydx. Answer:_______

7. Let Rbe the solid inside x² + y²+ z² = 1, outside z = and above the xy-plane. Find the volume of R. Answer :___________

8. Evaluate the line integral dy, where C : boundary of the region lying between the graphs of the ellipse = 1 and the circle x² + y² = 1.
 Answer:_______

1Find any extrema of the function subject to the constraint x² + y²≤ 1.

2. For the function
   show that fx(0,0) and fy(0,0) both exist, but that f is not differentiable at (0, 0).

3. Determine whether the seriesconverges absolutely or conditionally, or di-verges.