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The solutions to exam 2 of math 232, held on august 3, 2007. The exam covers various topics including limits, tangent planes, linear approximations, derivatives, extrema, integration, volumes, and mass center. Students are required to solve problems related to finding limits, calculating derivatives, integrating functions, finding extrema, and determining volumes and mass centers.
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Math 232 August 3, 2007
Name
x^2 yz x^4 + y^4 + z^4
does not exist. (6 pts)
x^2 + y^2 at the point (3, 0) then use this approx- imation to estimate the value of the function at (3, 1). (6 pts)
(a) fx if f (x, y) = 3x^4 y + 8x^0.^5 y^2 − 3 x^2
(b)
∂h
2 a
(2a + b)h
(c)
∂^2 z ∂y∂x
if z = ex+2y^ sin y.
(d) fx(π/ 3 , 1) if f (x, y) = x ln (y cos (x))
0
∫ √ 4 −x 2
−
√ 4 −x^2
e−x
(^2) −y 2 dy dx. (Hint: polar coordinates) (6 pts)
∂z ∂x
and
∂z ∂y
for the formula z^2 = x^2 − y^2. Now use these formulas to write the equation of the plane tangent to the surface at 5, − 3 , −4). (6 pts)
