If T x , y is the temperature at a point x , y on a thin metal plate in the xy -plane, then the level curves of T are called isothermal curves. All points on such a curve are at the same temperature. Suppose that a plate occupies the first quadrant and T x , y = x y . (a) Sketch the isothermal curves on which T = 1 , T = 2 , and T=3 . (b) An ant, initially at 1 , 4 wants to walk on the plate so generating the graph Off from various view so that the temperature along its path remains constant. What path should the ant take and what is die temperature along that path?
If T x , y is the temperature at a point x , y on a thin metal plate in the xy -plane, then the level curves of T are called isothermal curves. All points on such a curve are at the same temperature. Suppose that a plate occupies the first quadrant and T x , y = x y . (a) Sketch the isothermal curves on which T = 1 , T = 2 , and T=3 . (b) An ant, initially at 1 , 4 wants to walk on the plate so generating the graph Off from various view so that the temperature along its path remains constant. What path should the ant take and what is die temperature along that path?
If
T
x
,
y
is the temperature at a point
x
,
y
on a thin metal plate in the xy-plane, then the level curves of T are called isothermal curves. All points on such a curve are at the same temperature. Suppose that a plate occupies the first quadrant and
T
x
,
y
=
x
y
.
(a) Sketch the isothermal curves on which
T
=
1
,
T
=
2
,
and T=3
.
(b) An ant, initially at
1
,
4
wants to walk on the plate so generating the graph Off from various view so that the temperature along its path remains constant. What path should the ant take and what is die temperature along that path?
3. The cost C of manufacturing transformers in a factory varies with the output N per day
according to the relation: C = a + b/N + eN where a, b, and e are constants. For what
output will the cost be least?
4. A given cylindrical tank whose radius 'r' and altitude 'h' are observed to be varying. At a
certain instant, r = 10 inches and is increasing at 2 inch per second, while at this instant
the altitude h is 20 inches and is increasing at 0.5 inch per second. Determine the rate at
which the volume is changing?
1
(3) Consider the function f(x, Y) = 72+ y? +1'
1
(a) Sketch the level curves off for z =
1
and z =
10
(b) Find k such that the level curve of f is a single point.
2. An object moves along the graph y = x° – 2x – 17, starting from the point (0, –17) at time t = 0, where
time is in seconds and the units along the x- and y-axes are in meters. It moves so that its x-coordinate
%D
changes at a constant rate of 3 m/sec. At time t = 1 second, what is the rate at which its distance from
the origin is changing?
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