課程名稱︰微積分乙
課程性質︰醫學系必修
課程教師︰薛克民
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2018/01/08
考試時限（分鐘）：110
試題 :
Autumn Semester, 2017
MATH 1209 Calculus B National Taiwan University
Final Examination
Date: 13:20-15:10, 01/08, 2018
Total scores: 80 points
1. (10 points) Find the linear approximation to the function
2x
f(x,y) = e cos(3y)
at the point (0,0). Use your approximation to get an estimate of the value
of f(0.1,0.1).
2. (15 points) Let u(x,y) = f(r,s), r = x + 2y, s = x - 2y.
2
∂ u
(a) (10 points) Use the chain rule to calculate ───── in terms of the
∂x ∂y
partial derivatives of f.
2
3 2 ∂ u
(b) (5 points) Suppose that f(r,s) = r + s . What is ───── ?
∂x ∂y
2
(y-x )
3. (15 points) Let f(x,y,z) = xz + e .
(a) (5 points) Compute the gradient ▽f.
(b) (5 points) Find the directional derivative of f at (x ,y ,z ) = (0,0,1)
0 0 0
along the direction to (x ,y ,z ) = (1,1,2).
1 1 1
(c) (5 points) Find the direction along which f increases most rapidly at the
point (0,0,1).
2 2 2
4. (15 points) Consider the function f(x,y) = 2x y - x - y . Find and classify
the critical points of f as local maxima, minima, or saddles.
↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓本 題 送 分↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓
5. (15 points) Find the potential extrema of the function
2 2
f(x,y) = x + 3xy + y - x + 3y
subject to the constraint that
2 2
g(x,y) = x - y + 1 = 0.
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑本 題 送 分↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑
6. (10 points) Evaluate
16 4 3
∫ ∫ sin y dy dx.
0 √x