Consider f and c below. f(x, y) = (3 + 4xy2)i + 4x2yj, c is the arc of the hyperbola y = 1/x from (1, 1) to 3, 1 3 (a) find a function f such that f = ∇f. f(x, y) = (b) use part (a) to evaluate c f · dr along the given curve c.

We want to find a scalar function such that .
So we need to have

Integrating both sides with respect to gives

Differentiating with respect to gives

So we find that

By the fundamental theorem of calculus, we then know the line integral depends only on the values of at the endpoints of the path. Therefore