ME5331 MIDTERM EXAMINATION

4:00-5:20pm, October 18, 2021

Closed book/note/No calculator

2

1. If a = (2, −1, 1) and b = (1, 1, −1), compute the area of the parallelogram spanned by the two vectors.

2. Expand

ln (2 + x),

by the Taylor series about x = 1 through the second degree.

3. If u = x  y and v = [( − x² + y²)/2], compute¹

⎢ ⎢ ∂(x, y) ∂(u, v) ⎢ ⎢ .

4. Compute

⌠ (⎜) ⌡ ∂u ∂n  v  d l,

where

u = x y,    v = x2 − y2 2 ,

and the integral path is along the unit circle ccw.

5. Given y³ − 3 x y − x = 1, compute y′(x).

6. Does

∞ ∑ n=1  ⎛ ⎝ ln (n3 + 1 n ) − 3 ln n ⎞ ⎠

converge or diverge ? Why ?

7. Find the shortest distance from (1,2) to the line, x+2y=2.

8. Exchange the order of integrations for

⌠ ⌡ 1 0  ⌠ ⌡ e ex  f(x, y) dy d x = ⌠ ⌡ ? ?  ⌠ ⌡ ? ?  f(x, y) dx dy.

9. Evaluate

I ≡ ⌠ ⌡ C  (3 x +2 x y +z2) dx + (x2 −ez y) d y + (y ez − 1) dz

where C is the integral path from (0, 0, 0) to (1, 0, 0) shown below. (Hint: This problem is easier than it looks.)

10. Compute

⌠ (⎜) ⌡ dl √ 4 x2 + 9 y2 ,

where the integral range is along the ellipse, 2 x² + 3 y² = 1.

Solution

1.

a ×b = (0,3,3),

so

|a ×b | = 3 √2.

2.

ln (2 + x) = ln (3 + (x − 1)) = ln 3 + ln  ⎛ ⎝ 1+ x−1 3 ⎞ ⎠ = ln 3 + ⎛ ⎝ x−1 3 ⎞ ⎠ − 1 2 ⎛ ⎝ x−1 3 ⎞ ⎠ 2   + …

3.

⎢ ⎢ ∂(u, v) ∂(x, y) ⎢ ⎢ = 1 ⎢ ⎢ ∂(x, y) ∂(u, v) ⎢ ⎢ = 1 ⎢ ⎢ ⎢ ⎢ y, x −x, y ⎢ ⎢ ⎢ ⎢ = 1 x2 + y2 .

4.

I = ⌠ (⎜) ⌡ ∂u ∂n  v  d l = ⌠ (⎜) ⌡ n·∇u  v d l = ⌠ ⌡ ⌠ ⌡ ∇·( ∇u  v) dS = ⌠ ⌡ ⌠ ⌡ ( ∆u v + ∇u ·∇v )d S.

Note

∆u = ∂2 u ∂x2 + ∂2 u ∂y2 =0,     ∇u = ⎛ ⎜ ⎜ ⎝ y x ⎞ ⎟ ⎟ ⎠ ,    ∇v = ⎛ ⎜ ⎜ ⎝ x −y ⎞ ⎟ ⎟ ⎠ ,     ∇u ·∇v = 0,

Hence,

I = 0.

5.

3 y2 y′−3 y − 3x y′− 1 = 0 → y′ = 1+3 y 3 (y2 − x)

6.

ln  ⎛ ⎝ n4 + 1 n ⎞ ⎠ − ln n3 = ln (1 + 1 n4 )  ∼  1 n4 .

As p=4 > 1, the sum of the series converges.

7. Define f^* as

f* = (x−1)2+(y−2)2−λ(x+2y−2).

Solving

∂f ∂x = 0, ∂f ∂y = 0, ∂f ∂λ = 0,

yields

x = 2 5 , y= 4 5 , λ = − 6 5 .

Therefore,

d = √   (x−1)2+(y−2)2   =   ⎛ √ ( 2 5 −1)2+( 4 5 −2)2   = 3 √5 .

8.

⌠ ⌡ 1 0  ⌠ ⌡ e ex  f(x, y) dy d x = ⌠ ⌡ e 1  ⌠ ⌡ ln y 0  f(x, y) d x dy.

9.

Note that z=0 and d z = 0. Also, (P, Q) = ( 3x+2 xy, x²−e^z y) is irrotational. Choose the shortest cut from (0,0,0) to (1, 0, 0), i.e. y=0,  x :0→ 1. Then,

I = ⌠ ⌡ 1 0  3 x d x = 3 2 .

10. Choose

n = ⎛ ⎜ ⎜ ⎜ ⎝ 2 x/ √   4 x2 + 9 y2   3 y/ √   4 x2 + 9 y2   ⎞ ⎟ ⎟ ⎟ ⎠ ,   v = ⎛ ⎜ ⎜ ⎝ x y ⎞ ⎟ ⎟ ⎠

I = ⌠ (⎜) ⌡ n·v dl = ⌠ ⌡ ⌠ ⌡ ∇·v dS = 2 ⌠ ⌡ ⌠ ⌡ dS = 2 ×π 1 √2 1 √3 = 2 π √6 .

Footnotes: