所屬科目:研究所、轉學考(插大)-微積分
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 byx(t) = 3 cos ), y(t) = 3 sin) and z(t) = 4for 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.