MTH 277 Test 2 – show your work
Name
1. Limit and Continuity
a. lim(x,y)→(1,2)
x2 +y
x2 −y 3
?
b. Does lim(x,y,z)→(0,0,0)
xy+yz+xz
x2 +y 2 +z 2
exist? If yes, what is the limit value? If not, why?
2. Level curve, gradient and directional derivative :
a. Let z = f (x, y ) = x2 − xy + y 2 + y , consider the level curve f (x, y ) = 2, what is the
gradient ∇f at (1, 1)?
b. What is the equation of the tangent line at (1, 1) for f (x, y ) = 2?
c. What is the equation of the normal line at (1, 1) for f (x, y ) = 2?
d. Given a vector uˆ = ( √12 , - √12 ), what is the directional derivative Dû at (1, 1)?
e. Let w = g (x, y , z) = x2 − xyz + y 2 + y + z 2 − 4 , consider the level curve
g (x, y , z) = 0, what is the gradient ∇g at (1,− 1, − 2)?
f. What is the equation of the tangent plane at (1,− 1, − 2) for g (x, y , z ) = 0?
g. Consider the (x, y) points of g (x, y , z) = 0 particles moving along a unit circle C,
find a parametric form, i.e. x = u(t), y = v(t)
h. Find the velocity on the z direction, i.e. derivative dd zt use the chain rule as a
function of t
i. Find the acceleration on z direction as a function of t
3.
M ixed partial and higher derivatives, chain rule
a. Given f (x, y, z) = x3 − x 3y + ln ( yz ) − sin(x) , find the gradient ∇f
b. z = f (x, y ) = x3 + y 3 − 6 x2 + 9y 2 + 12x + 27y + 19 ,, find its relative min and max if
j.
any, and the saddle points
Given g(x, y, z) = x2 + y 2 + z 2 = 9, consider (x, z) as particles moving along an
elliptical curve 4x 2 + 9z 2 = 1 , find a parametric form as , i.e. x = u(t), z = v(t)
k. Find the velocity on the y direction, i.e. derivative dd yt use the chain rule as a
function of t
l. Find the acceleration on y direction as a function of t
4. Tangent plane and Tangent line
a. Given a surface A as x2 − 2x + y 2 + (z − 1)2 = 4, f ind its tangent plane at ( 1,1,3)
b. Given a surface B as x2 + y 2 + z = 4, f ind its tangent plane at ( 1,1,3)
c. A intersects B on a curve at (1, 1, 3) find the tangent line along this curve (for both A
and B) at this point
5. Double integral (over a region), polar coordinate, surface area
a. Let f (x, y ) = x2 − xy + y 2 + y , and a region D = [-1, 1] by [-1, 1] for (x , y) find
∬D f (x, y ) da = ?
b. Find the volume under the surface z = f (x, y ) = √1 − x2 and above the
triangular region formed by y = x, x = 1, and the x-axis
c. Find the volume defined by z = f (x, y ) = √4 − x2 − y 2 above the circular region
bounded by the two axes and the circle x2 + y 2 = 4 in the first quadrant
d. Find the surface area defined by the part c.
6. Triple integral: the tetrahedron is defined by corners at (0, 0, 0), (0, 3, 0), (2, 3, 0), and (2, 3, 5)
bdf
a. If the volume is given by ∫ ∫ ∫ dz dy dx , what are a, b, c, d, e and f?
ace
bdf
b. If the volume is given by ∫ ∫ ∫ dx dy dz , what are a, b, c, d, e and f?
ace
bdf
c. If the volume is given by ∫ ∫ ∫ dy dx dz , what are a, b, c, d, e and f?
ace
d. Compute the volume use the integral a.
e. Compute the volume use the integral b.
f. Compute the volume use the integral c.
7. Surface area and Integral
Given a right circular cone S with both height and base radius are a
a. Write the surface equation f(x, y, z) = 0
b. Find its volume V
c. Find its surface area
d. Let g(x,y,z) = x+y+z is defined on S, find
∫ ∫ g ds
S