85年 - 85 中原大學轉學生考試：微積分#124658

(a) (sec x - tan x)

(b)

2. (5 point) Let g(x) = x(x - 1)(x - 2)(x - 3). How many real roots has the equation
g'(x) = 0? For each, specify an interval containing that root and no others.

(a) (7 point) ∫ tan⁻¹ x dx

(b) (8 point)dy dx

4. (5) if z = f⁻¹(y) = y⁵ + y³, what is (f⁻¹)'(2)?

5. (15) Find all the local maximum values (局部極大值), local minimum values(局部極小值), and saddle points (鞍點), of the following function
f(x, y) = x³ + y³ + 3x² - 3y² - 8.

6. (10 point) Find
where C is the arc of the circle x² + y² = 1 from (1, 0) counterclockwise to ().

7. (10 point) Find dy/dx if y is a differentiable function of x satisfying the equation
2x³ + y³ - 5xy = 0.

(a) If f' is continuous, and if the graph of f passes through the points (1, 2) and (3, 5),
what is f'(x) dx?

(b) Let F(x) = f(t) dt. What is F'(x)?