Multivariable Calculus Exam 2 Preparation Math 28 (Spring 2) Exam 2: Thursday, May 2. Friday May, is a day off! Instructions: () There are points on the exam and an extra credit problem worth an additional points. (2) Unless indicated otherwise, show your work. No credit for correct answers alone. () No technology or notes of any type are allowed on the exam. () Remember that the vector r is shown in boldface as in the text, and not as r as I wrote on the board in class. Also on the course website isthesecondexamfromthisclassinspringquarter2and its solutions. Exam Covers: Here are the sections of the text we have covered since the first midterm. Also listed is additional information you are responsible for. Chapter 2 sections 5-9. Chapter sections -5. Group Project 2 All homework All lectures Quizzes -, worksheets, handouts and of course, all the material from the first part of the course that led to these sections and this material. Please recall that my exams tend to be computationally intensive and the only way to do well is to have done lots of problems in advance and to have learned actively. Notes: Last year, I did not cover Lagrange multipliers, explaining its omission from Exam 2. Last year s group project 2 was completely different. The problem related to Group 2 on that exam was problem 7, the extra credit problem. Topics functions of two and variables, graphs, level curves and surfaces partial derivatives tangent plane the chain rule directional derivatives and the gradient
optimization of functions of two and three variables. Lagrange Multipliers double integrals and the problems they solve (area, volume, mass) evaluating a double integral as an iterated integral polar coordinates evaluating a double integral as an iterated integral in polar coordinates triple integrals and the problems they solve (volume, mass) evaluating a triple integral as an iterated integral cylindrical coordinates evaluating a triple integral as an iterated integral in cylindrical coordinates spherical coordinates evaluating a triple integral as an iterated integral in spherical coordinates Sample problems Do not assume that these problems are representative of all problems on the exam or cover all topics on the exam!. Pages87-:56568798887 2. Pages 957-96: 5 8 5. Let f (x, y) =e (x2 +y 2 ) (a) Compute f x (x, y) (b) Compute f y (x, y) (c) Compute f xx (x, y) (d) Compute f xy (x, y). Let f (x, y) = p y x 2. (a) What is the domain of f. (b) Make a rough sketch and label each of the level curves f (x, y) =, f (x, y) = and f (x, y) =2. (c) Find f (, ). Add this vector to the sketch. (d) Find D u f (, ) where u is a unit vector in the direction of v =i +j. (e) Find the equation of the tangent plane to the graph of f at the point (, ). 2
5. The temperature at the point (x, y) in the plane is given T (x, y) =x 2 +9y 2. At time t =, a cold seeking particle is at the point (, ). Find the path of the particle, assuming that, at any point (x, y) visited by the particle, its velocity v = T (x, y). 6. (This is an exercise in using the chain rule.) Let z = f (x, y). Suppose that x = r cos θ and y = r sin θ. Bycomposition,define F (r, θ) =f (r cos θ, r sin θ) Assuming that all needed partial derivatives exist, compute the following: (a) F r (b) F θ (c) F rr (d) F rθ (e) F θθ 7. Find dw dt where w =ln(x +2y z 2 ), x =2t, y = t, and z = t 8. Let h (x, y) =xy and assume x = f (t) and y = g (t) are differentiable. Let F (t) = h (f (t),g(t)). Use the chain rule to find F (t). What famous rule from first year calculus have you rediscovered? 9. Find the critical point or points of the following functions. Identify each as a local maximum, local minimum or saddle point. (a) f (x, y) =x 2 y 2 +x +6y (b) f (x, y) =(x 2 +y 2 ) e (x 2 +y 2 ). Compute Z 2 Z x (x 2 +2y) dy dx. Writethedoubleintegral Z Z x 2 Z? Z? f (x, y) dy dx in the form f (x, y) dx dy.?? That is, supply the limits of the integrals when the order of integration is reversed. (There is no integral to evaluate here.)
2. Shown is that part of the surface x 2 +6y + z =6in the first octant. A solid S occupies the region in the firstoctantbetweenthissurfaceandthecoordinateplanes. z 2 2 y 2 x The density at the point (x, y, z) in the solid is given by δ (x, y, z) =xyz. Write the 6 iterated triple integrals whose value is the mass of the solid. Do not evaluate these expressions.. A thin plate occupies the region R in the x-y plane above the x-axis and enclosed by the cardioid r =+cosθ as shown. The density at the point (x, y) of this plate is given by δ (x, y) =y. Findthemassofthesolid. 6 8. Consider the cylindrical solid S described by x 2 + y 2 z 2 Compute the following integrals. (a) zdv (b) (x 2 + y 2 ) dv (c) (x 2 + y 2 ) zdv Hint for this problem. Use cylindrical coordinates.
5. A thin plate occupies a region R in the x-y plane. The density at the point (x, y) of the plate is given by δ (x, y) =x 2. Compute the mass of the the plate if (a) R is the region in the FIRST QUADRANT of the x-y plane enclosed by the parabola y = x 2 and the line y =, as shown on the left. (b) R is the region in the FIRST QUADRANT of the x-y plane enclosed by the circle r =,asshownontheright. diagram for part (a) diagram for part (b) 6. Compute the mass of the plate given in problem 5, part (a), if instead the density of the solid at the point (x, y) is given by δ (x, y) =y 2. 7. The solid S occupies the region between the hemisphere x 2 + y 2 + z 2 =, z, and the plane z =. (a) Find the volume of S. (b) Find the mass of the solid if its density at the point (x, y, z) is given by i. δ (x, y, z) =z ii. δ (x, y, z) =x 2 + y 2 8. Recall that r = 2cosθ is the polar equation in the x-y plane of the circle whose rectangular equation is (x ) 2 + y 2 =.Seethefigure on the left. (a) The top half of a sphere of radius 2, centered at (,, ) sits above the x-y plane. Write an iterated integral in polar coordinates whose value is the volume of the solid region consisting of all points that lie above the x-y plane, below the sphere and inside the cylinder (x ) 2 + y 2 =. The figures on the right show some views. (b) Evaluate the integral in part (a). 2 - the base of the cylinder view from above view from below 5