93年 - 93 中原大學_進修學士班招生考試_理工、化學、工學(進修部)群組二年級_應用數學系：微積分#124603

1.Let f(x) = be a continuous function on all real numbers, then ab=_________

2.=_________

3. =_________

4. Let f(x)=, then =_________

5. Let y=mx+b be the equation for the line tangent to the graph of the equation x²y²=16 at the point (2, 2). Then m+b=_________

6. Let F be a function such that F(1)=10,F(3)=40 and F'(x)=ax⁴+bx²+c, where a, b, and c are constants. Then =_________

7. If y=, then =_________

8. =_________

9. The interval of convergence of the power series is _________

10.=_________

11.If f(x,y)=x³-3xy+4y² and , then the directional derivative =_________

12.The global minimum value of f(x,y)=2x²+3y²-4x+6 on the region R= is _________

1. If is a vector-valued function and let c be a constant. Show that if is differentiable
on come interai I and if =0 for all t ∈ I be I then are orthognal for all t ∈ I .

2. Let f be differentiable on an open interval I. Use the mean value theorem to show that if
f'(x)>0 for all real x in I, then f increases on I.