Q1 (9 points) Let z(x, y) and w(x, y) be smooth functions that satisfy the equations w+w³ + x²yz + 4 = 0, z³+w+xy/w+1=0 and also satisfy 1 z = 1, w = -1 when x = -29 y 2 Using differentials, estimate the value of z(-1.95, -.35) and calculate the percent relative error (the actual value is z(-1.95, —.35) = 0.910558).

Find the directional derivative of the function at the point P in the direction of the point Q. f(x, y, z) = xy²z³, P(2, 1, 1), Q(0, -3, 5) D (2, 1, 1) = Need Help?

//welro) Find a local linear approximation for the function 5х + 6 f(x) valid for x near 5 5x + 6 Answer: Hint: The local linear approximation at x = 5 is the tangent line at x = 5. The equation for this tangent line, in slope-intercept form, is у 3 т(х — 5) +b where m f' (5) and b = f(5). So compute m and b, put these into the equation for the tangent line, and then enter the right-hand side of that equation as your answer.

Find the gradient of the curve y = a²xe-b*x at x = 1.

2. Examine the function f(x, y) = x' - 15xy +y³ +7 for relative extrema and saddle points. a. saddle point: (0, 0, 7): relative minimum: (5, 5, –118) b. relative minimum: 0. 0, ); relative maximum: (5, 5, -118) c. saddle points: (0, 0, 7) (5, 5, –118) d. saddle point: (5, 5, –118). relative minimum: (0, 0, 7) (5, 5,-118). relative maximum: (0, 0, 7) e. relative minimum:

Calculate fz(3, 6, 9), where f(x, y, z) = xyz. fz(3, 6, 9) =

6. (a) Let f(x, y) = x³y*. Find the directional derivative of f in the direction of T = (3, 4) at the point (2, 2) (b) Evaluate |Vf(2,2)|. Which number is larger; your answer from part (a) or your answer from part (b)? Why?