(a) If there is a scalar function f(x, y) such that ∇ f(x, y) = x² i + y² j, then
∂f/∂x = x²
∂f/∂y = y²
Integrating both of these equations with respect to x and y (respectively) gives
f(x, y) = 1/3 x³ + g(y)
f(x, y) = 1/3 y³ + h(x)
Differentiating with respect to the other variable gives
∂f/∂y = g'(y) = y²
∂f/∂x = h'(x) = x²
so it follows that
f(x, y) = 1/3 x³ + 1/3 y³ + C
for some constant C.
(b) By the gradient theorem,
You write it as 46 divide by 1058. 46 x 2 is the closest you will get to 100. 46 x 2 is 92. put it in and you have 138 left. multiply 46 x 3 and you will get 138. amswer is 23. remember to bring down the next number whwn dividing
Decreased as getting more for less/same amount makes it so the worth of euro has went down
D. 32/47
You would subtract 1580 from 640 to get 940. You would then simplify 640/940 to get 32/47.
The factor of 11 is going to be 77 because you have to stay in between 70 to 80
and 7 times 11 is 77. It also has to be a number in the 11 timetables. It can't be 88 because we can't cross 80 so the factor is going to be 77.
Hope it helps
