Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b) Evaluate F. dr, where r(t) = (e', e²" , r²) for 1 < 1 < 3. . (c) Evaluate F. dr for any path C from P = (}, 4, 2) to Q = (2, 2, 3) contained in the region x > 0, y > 0. (d) Why is it necessary to specify that the path lies in the region where x and y are positive?

If X and y are continuous functions of t on an interval /, then the equations X = x(t) and y = y(t) are called and t is called the %3D Name the curves represented by a) x =t - 3 and y = t b) x = 4sin(t) and y = 3cos(t)|

o'pg. 156-159 J #1.) Determine f' and F", if (x)=x²-14 X4 #2.) For y = x9-7x³ +2, find d²y dx2 #6) Determine the maximum and a f(x) = 2x³-9x², -2≤ x ≤ 4 minimum of each function on the given interval b) f(x) = 12x - x³, x= [-3,5] (²) f(x) = 2x + 18,1 fcx)=x²-2x + 6₁-1 ≤ x ≤7 b) f(x)= x³ + x², -3 ≤ x ≤ 3 Cf(x)= x³-12x + 2,-5 ≤ x ≤ 5 d.) f(x) = 3x5-5x³, -2≤x≤4

(2) Consider the function f: Nx N→Q given by f(x, y) = = your claim. X y + 1 Find the image of f and prove