114年 - 114 台灣聯合大學系統_學士班轉學生考試試題:微積分#137762

(1) Evaluate  .

(2) Define the function

F(x) = f(ln(2 sinx)), 0 ＜x ＜π,

where f is differentiable on (-00, 00) and f'(0) = 2. Find F(π/6).

(3) Find the following integral

(4) Find all critical points of the function

f(x,y) =x³- бху - у², -∞ ＜x ＜∞ and - o∞ ＜y＜ 0∞,

and classify each critical point as a location where a local maximum, local minimum, saddle point occurs.

(5) Find the rational number that is represented by the repeating decimal

(6) Calculate the area of the region that is completely enclosed by the graphs of the functions f(x) = x² and g(x) = 4-x².

(7) Evaluate the double integral

 where R = {(x, y): 0 ≤ x ≤ 3,0 ≤y ≤2}.

(8) Find the domain of the following function:

二、計算、證明題:
(1) Find the Taylor series of the function

 and determine its radius of convergence.

(2) Use the separation of variables to solve the following differential equation:

= 2(100 - Q),

with the initial condition Q(0) = 3.

(3) To create an open box from a square piece of cardboard, we can cut out identical squares from each corner and then fold up the resulting flaps. If the cardboard measures 10 inches on each side, determine the dimensions of the box that will yield the maximum volume. Note: Points will be awarded only if calculus is used in the solution.