Problem 1.Find the critical points of the functionf(x, y) = 2x3− 3x2y − 12x2− 3y2and determine their type i.e. local min/local max/saddle point. Are there any global min/max?Problem 2.Determine the global max and min of the functionf(x, y) = x2− 2x + 2y2− 2y + 2xyover the compact region−1 ≤ x ≤ 1, 0 ≤ y ≤ 2.Problem 3.Using Lagrange multipliers, optimize the functionf(x, y) = x2+ (y + 1)2subject to the constraint2x2+ (y − 1)2≤ 18.Problem 4.Consider the functionw = ex2ywherex = u√v, y =1uv2.Using the chain rule, compute the derivatives∂w∂u,∂w∂v.Problem 5.(i) For what value of the parameter a, will the planesax + 3y − 4z = 2, x − ay + 2z = 5be perpendicular?(ii) Find a vector parallel to the line of intersection of the planesx − y + 2z = 2, 3x − y + 2z = 1.(iii) Find the plane through the origin parallel toz = 4x − 3y + 8.(iv) Find the angle between the vectorsv = (1, −1, 2), w = (1, 3, 0).1(v) A plane has equationz = 5x − 2y + 7.For what values of a is the vector(a, 1,12)normal to the plane?Problem 6.(i) Compute the second degree Taylor polynomial of the functionf(x, y) = ex2−yaround (1, 1).(ii) Compute the second degree Taylor polynomial of the functionf(x) = sin(x2)around x =√π.(iii) The second degree Taylor polynomial of a certain function f(x, y) around (0, 1) equals1 − 4x2− 2(y − 1)2+ 3x(y − 1).Can the point (0, 1) be a local minimum for f? How about a local maximum?Problem 7.(i) The temperature T (x, y) in a long thin plane at the point (x, y) satisfies Laplace’s equationTxx+ Tyy= 0.Does the functionT (x, y) = ln(x2+ y2)satisfy Laplace’s equation?(ii) For the functionf(x, y) = sin(x2+ y2) ln(x4y4+ 1) tan(xy)is it true thatfxyxyy= fyyxyx?Problem 8.Consider the function f(x, y) =x2y4.(i) Carefully draw the level curve passing through (1, −1). On this graph, draw the gradientof the function at (1, −1).(ii) Compute the directional derivative of f at (1, −1) in the direction u =45,35. Use thiscalculation to estimatef((1, −1) + .01u).(iii) Find the unit direction v of steepest descent for the function f at (1, −1).(iv) Find the two unit directions w for which the derivative fw= 0.Problem 9.Consider the functionf(x, y) =pln(e2xy3).(i) Write down the tangent plane to the graph of f at (2, 1).(ii) Find the approximate value of the numberpln(e4.1(1.02)3).Problem 10.Suppose thatz = e3x+2y, y = ln(3u − w), x = u + 2v.Calculate∂z∂v,∂z∂w.Problem 11.(i) Find z such that1 +1z+1z2+1z3+ . . . = 3.(ii) Calculate the series13+232+2233+ . . . +2993100.Problem 12.The probability density function for the outcome x of a certain experiment isp(x) = Ce−x, for x ≥ 0.(i) What is the value of the constant C?(ii) What is the cumulative distribution function?(iii) What is the median of the experiment?(iv) What is the mean of the experiment?(v) What is the probability that the outcome of the experiment is bigger than 1?Problem 13.Consider the function f(x, y) = 5 − (x + 1)2− y2.(i) Draw the cross section corresponding to x = 1.(ii) Draw the contour diagram of f showing at least three levels.(iii) Draw the graph of f.(iv) What is the equation of the tangent plane to the graph of f at (1, 0, 1)?Problem 14.Find the point on the plane2x + 3y + 4z = 29that is closest to the origin. You may want to minimize the square of the distance to the origin.Problem 15.Find the critical points of the function f(x, y) = 2x3+ 6xy + 3y2and describe their