−0.178571428571429 as a faction
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Answer: Option c. D: all real numbers R: (y ≥ 8.75)
Solution:
y=x^2+x+9
Domain: all real numbers = ( - Infinite, Infinite)
y=ax^2+bx+c
a=1 > 0, the parabola opens upward: Range: y ≥k
b=1
c=9
Vertex: V=(h,k)
h=-b/(2a)
h=-1/(2(1))
h=-1/2
h=-0.5
k=y=h^2+h+9
k=(-0.5)^2+(-0.5)+9
k=0.25-0.5+9
k=8.75
Range: y≥k
Range: y≥8.75
The curve 
is parameterized by
so in the line integral, we have
You are mistaken in thinking that the gradient theorem applies here. Recall that for a scalar function
, we have gradient 
. The theorem itself then says that the line integral of 
along a curve 
parameterized by 
, where 
, is given by
Specifically, in order for this theorem to even be considered in the first place, we would need to be integrating with respect to a vector field.
But this isn't the case: we're integrating
, a scalar function.
so in the line integral, we have
You are mistaken in thinking that the gradient theorem applies here. Recall that for a scalar function
Specifically, in order for this theorem to even be considered in the first place, we would need to be integrating with respect to a vector field.
But this isn't the case: we're integrating