93年 - 93 國立暨南大學轉學生招生考試試題_微積分_(財金系)#125306

 (i) Find the inverse function of f(x). 

(ii) Write down the domain and the range of f(x) and separately, and construct their relationship. 

(iii) Check your results in (i) algebraically by evaluating f() and (f(x)) both equal to x. 

(iv) Sketch the graphs of f(x) and . Are the graphs of f(x) and reflections of each other? If they were, described the reflection line and prove it； if they were not, explained your results. 

(v) Do f(x) and define themselves as functions? Are they one-to-one functions? How do you get your results? 

  (i) If y = f(x)g(x), then y' = f'(x)g'(x).

(ii) If f(x) is an nth-degree polynomial, then = 0.

(iii)

 (i) h(x) = x²[2x+3]

(ii)

(iii) h(x) = ln|csc(x²) - cot(x²)|

(iv)

(v)

 (i)

(ii)

(iii)

(iv)

(v) ∫sec³(x) dx

(vi)

5. Sketch the graph of the function . Label the intercept, relative extremes, points of inflection, and asymptotes. State the domain of the function.

6. Apply Taylor's Theorem to find the power series (centered at 0) for the function , and find the radius of convergence.