Evaluate ∫c∇f. dr using fundamental theorem of line integral, where f(x, y,z)=cos (πx)+sin(πy)-xyz and c is any path that starts at (1,1/2,2) and ends at (2,1,-1)

Two streets intersect at a 30- degree angle. at the intersection, the are four crosswalks formed that are the same length. what type of quadrilateral is formed by the crosswalks?

Complete the table for different values of x in the polynomial expression -7x2 + 32x + 240. then, determine the optimal price that the taco truck should sell its tacos for. assume whole dollar amounts for the tacos.

Find a parametrization for the curve「and determine the work done on a particle moving along γ in r3 through the force field f: r^3--r^3'where f(x, y,z) = (1,-x, z) and (a) im (γ) is the line segment from (0,0,0) to (1,2,1) (b) im (γ) is the polygonal curve with successive vertices (1,0,0), (0,1,1), and (2,2,2) (c) im (γ) is the unit circle in the plane z = 1 with center (0,0,1) beginning and ending at (1,0,1), and starting towards (0,1,1)