Math 241 - Exam #2 Solutions

1a. limit = <2,4,8> or 2i + 4j + 8k
1b. T = <1,2t,3t^2>/sqrt(1+4t^2+9t^4)
1c. r'(0) = <1,0,0>, r''(0) = <0,2,0>
k = |r' x r''|/|r'|^3 = |<0,0,2>|/1^3 = 2

2. r' =
|r'| = sqrt(3) e^t
L = int_0^2pi |r'| dt = sqrt(3) (e^(2pi)-1)

3. v(t) =
r(t) =

4. False, for example f(x,y) = (x+y)^2/(x^2+y^2), with f(0,0) = 0 the limit doesn't exist.

5. z_x = (1 + x/sqrt(x^2+y^2))/(x + sqrt(x^2+y^2))
z_y = y/sqrt(x^2+y^2)/(x + sqrt(x^2+y^2))

6. f_xx = 5(x^2+y^2)^(1/2)(4x^2+y^2)
f_xy = f_yx = 15 xy (x^2+y^2)^(1/2)
f_yy = 5(x^2+y^2)^(1/2)(x^2+4y^2)

7. at (1,-1,0), z_x = 1, z_y = 1, so the plane is
z -0 = 1(x-1) + 1(y+1), or
z = x + y

8. dz/ds = 2s cos(x+y) - 2t cos(x-y)
dz/dt = 2t cos(x+y) - 2s cos(x-y) or

dz/ds = 2(s-t)cos x cos y - 2(s+t)sin x sin y
dz/dt = 2(t-s)cos x cos y - 2(s+t)sin x sin y

ccollins@math.utk.edu
Last Modified: Feb. 24, 1999