a) is conservative if it is the gradient field for some scalar function . This would require
Integrating both sides of the first equation with respect to yields
Differentiate with respect to :
Differentiate with respect to :
We want to be independent of and ; we can make them both disappear by picking .
b) This is the so-called triple product, which has the property
Computing the determinant is easy with a cofactor expansion along the first column:
c) Let
Compute the partial derivatives and evaluate them at :
Then the tangent plane to at (1, 1, 1) has equation
d) In polar coordinates, is the set
Then the integral evaluates to
e) By the chain rule,
Eliminating the parameter, we find
so that when .
Compute derivatives:
Then at the point (1, 1), the derivative we want is
Answer:
15 oz/ 1 lb
Step-by-step explanation:
Answer:
36 feet
Step-by-step explanation:
you multiply 12 by three and you have your answer
Before the work was done, the cost was ...
($4/gal)*(8 gal)/(340 mi) = $32.00/(340 mi)
After the work was done, the cost was ...
($4/gal)*(7 gal)/(350 mi) = $28.00/(350 mi)
The savings is the difference of these costs:
(32.00/340 -28.00/350) $/mi ≈ $0.01/mi
After the mechanic worked on the car, it cost about $0.01 per mile less to run.