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Unformatted text preview: 1. a) Diﬀerentiating implicitly yz = ln(x + z ) (assuming this equation deﬁnes
z as a function x and y ), ﬁnd the rate of change of z in the direction of xaxis,
when x = 0, y = 0, z = 1.
b) Show that if u(t, x) = f (x + ct) + g (x − ct),then utt = c2 uxx .
2. The temperature at a point (x, y ) is T (x, y ) and Tx (5, 1) = 2, Ty (5, 1) = 3.
A bug crawls so that its position after t seconds is given by x = 1 + t2 , y =
cos (πt). How fast is the temperature rising on the bug’s path in 2 seconds.
3. A right circular cylinder is measured to have a radius of 10 ± 0.02 inches
and a height of 6 ± 0.01 inches. Estimate the error in the volume calculation
(the formula V = πr2 h was used).
4. Write the equation of the tangent plane to the level surface S : f (x, y, z ) =
ln xy 2 z 3 = 8 at (e, e2 , e).
√
5. Consider f (x, y ) = x + e4y .
a) Find the maximal increase rate at (3, 0) and the direction in which it
occurs.
b) Find the rate of change of f (x, y ) at (3, 0) toward (4, 1).
c) Find the vector orthogonal to the level curve f (x, y ) = 2 at (3, 0).
d) Write linearization of f (x, y ) at (3, 0) and approximate f (2.9, 0.1).
6 a) Find local extremums and saddle points of f (x, y ) = x4 + y 3 − 3y − 4x.
b) Find absolute min. and max. of f (x, y ) = x2 y subject to g (x, y ) =
2
x + y 2 ≤ 3.
7. a) Evaluate by changing the order of integration
1
0 1
√ 1 + x3 dxdy ;
y b) For D = {(x, y ) − ∞ < x < ∞, y ≥ 0} evaluate
e−(x 2 +y 2 )/2 dydx. D 8. Evaluate
xdV , where E is enclosed by the planes z = 0, z = x + 5
E
and the cylinder x2 + y 2 = 4. 1 ...
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 '07
 KAMIENNY
 Math, Calculus, Rate Of Change

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