114年 - 114 臺灣綜合大學系統_學士班轉學生考試試題：微積分B#137819

1. Let g(x) be the function
　　　　　　　　　　　　g(x) = ex^-2.
Compute g'(1) and g"(1).

2. Find the extreme values of the function
　　　　　　　　　　f(x) =.
on the interval [1, 5].

3. Evaluate the following definite integral:
∫₀¹ xe²x dx.

4. Evaluate the following definite integral:
dx.

5. Let f(x) = 2x^-1, and T = aₙ(x – 2)ⁿ the Taylor series of f(x). What is a₁₀₀?

6. Suppose that x and y satisfy the equation
                                                 y³ - x² = 4.
Find dy/dx and d²y/dx² when (x, y) = (2, 2).

7. Find the length L of the space curve given by
x(t) = 3 cos ), y(t) = 3 sin) and z(t) = 4
for t = 0 to 1.

8. Let 
                       f(x, y) = x + sin(x + 2y).
Find the unit vector in the direction in which f increases most rapidly at the point (0, 0) and give the rate of change of f in that direction.

10. Evaluate the double integral
              ∬_R (x + y)dA,
where R is the region bounded by y = x and y = x².