105年 - 105 國立清華大學碩士班考試入學試題_科技管理研究所/乙組:微積分#110849

(a)

(b)

(c)

3. (a) Evaluate

(b) Evaluate

4. Solve the initial value problem

5. Suppose f is differentiable in a neighborhood of c and f'(c) ＞ 0. Show that there is a δ ＞ 0 such that f(a) ＜ f(c) ＜ f(b) for all a in (c-δ,c) and all b in (c,c +δ).

6. Let T(x,y) be the temperature at the point (x, y) on the ellipse
  
 and suppose that ∂T/∂x = y, ∂T/∂y = x. Locate the maximum and minimum temperatures on the ellipse by examining dT/dt and d²T/dt².

7. (a) Expand f(x) = as a power series.

(b) Use part (a) to find the sum of the series