105年 - 105 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：微積分#145937

1. Find the limit (if possible): .

2. Let fbe the function given by . Find all values of a and k so that f is differentiable at x=1 .

3. Let , and f'(2)=-5 Find G'(2)

4. Decide whether the series converges or diverges: .

5. Find the open intervals on which the function is increasing.

6. Find the equation of the tangent line to the graph of the function at x=2.

7. Find a function f(x) that satisfies the differential equation and the initial condition f(0)=3

8. Find the indefinite integral .

(a) Compute the gradient of f at the point (1,1,3).

(b) Find the equation of the tangent plane to the surface f(x,y,z)=0 at the point

10. Find the area of the region R that lies below the parabola y=4x-x² above the x-axis, and above the line y=-3x+6 .