What is the slope of (-2,5) (3, -5) A. -1/2 B. -2 C. 2 D. 1/2
1 answer:
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The integral is path-independent if we can find a scalar function <em>f</em> such that grad(<em>f</em> ) = <em>A</em>. This requires
Take the first PDE and integrate both sides with respect to <em>x</em> to get
where <em>g</em> is assumed to be a function of <em>y</em> alone. Differentiating this with respect to <em>x</em> gives
which would mean <em>g</em> is *not* a function of only <em>y</em>, but also <em>x</em>, contradicting our assumption. So the integral is path-dependent.
Parameterize the unit circle (call it <em>C</em>) by the vector function,
with <em>t</em> between 0 and 2π.
Note that this parameterization takes <em>C</em> to have counter-clockwise orientation; if we compute the line integral of <em>A</em> over <em>C</em>, we can multiply the result by -1 to get the value of the integral in the opposite, clockwise direction.
Then
and the (counter-clockwise) integral over <em>C</em> is
and so the integral in the direction we want is -2π.
By the way, that the integral doesn't have a value of 0 is more evidence of the fact that the integral is path-dependent.
23.
Area formula of a triangle:
given: base = 9 cm height = 4 cm
A = 1/2 bh
=1/2(9)(4)
=18 cm²
24.
given: probability of rain - 0.03
of no rain - ?
1 - 0.03 = 0.97
25.
Volume formula of a cuboid:
V = length x base x height
= 6 x 4 x 2.5
= 60 cm³
