MATH 250, Study Guide for Midterm Test 2

11/15/2014

Test coverage

1. Derivative for functions of several variables (Chapter 14)
   1. Functions of several variables, domains, ranges, graphs (14.1)
   2. Partial Derivatives (14.3)
   3. Tangent Planes and Linear Approximations. Differentials (14.4)
   4. Differentiability, Continuity and Limits (14.4, 14.2)
   5. The Chain Rule (14.5)
   6. Directional Derivatives and the Gradient Vector (14.6)
   7. Maximum and Minimum Values (14.7)
   8. Method of Lagrange Multipliers (14.8)
2. Multiple Integrals (Chapter 15)
   1. Double Integrals over Rectangles (15.1)
   2. Iterated Integrals (15.2)

Key concepts (review from your lecture notes or the textbook)

• Domains for functions of two and three variables
• Graphs and level sets (level curves in 2D, level surfaces in 3D). Contour maps
• Partial derivatives, slopes of tangent lines to the graph
• Equation of the tangent plane. Linear approximation L(x,y), differential df
• Definition of continuity in the case of two and three variables; examples of continuous and discontinuous functions
• Limits: techniques of polar coordinates, and path limits
• Chain rule in the case of several variables
• Directional derivatives. Computing directional derivatives using the chain rule (the dot product formula)
• Gradient vector. Geometric significance of the gradient vector
• Tangent plane and normal line to a surface
• Local extrema (maxima and minima). Critical points
• The Second Derivative Test for local extrema
• Absolute maxima and minima in domains with boundaries
• Lagrange multiplier method for solving problems with constraints: cases of one and two constraints
• Double integrals over rectangles: Geometric and Algebraic (Riemann sum) definitions
• Computing double integrals in simple examples (prisms, cylinders, spheres) using the geometric definition
• Computing double integrals using iterated integrals

Basic types of exam questions (the list is not all-inclusive, but it covers most of the typical midterm questions; "R" refers to problems in Chapter Review)

• Find the domain of a function and sketch a few level curves/surfaces (14.1: homework; 14.R: 1, 2, 5, 6)
• Determine the region of continuity of a function (14.2: 29-38)
• Find a limit using (a) continuity or (b) polar coordinates or (c) show that limit does not exist using polar coordinates/path limits (14.2: 7, 9, 12, 13, 39-41; 14.R: 9, 10)
• Compute partial derivatives, verify that partial derivatives satisfy a certain equation (14.3 76; 14.5: 47; 14.R: 23)
• Find the linear approximation or the differential of a function at a given point (14.4: 25; 14.R: 32)
• Use differentials to estimate increments of functions (14.4: 31, 33, 38, 39, 41; 14.R: 34)
• Applied problems on the chain rule: examples from geometry, physics and other sciences (14.5: Example 2, problems 35, 36, 38-41, 43)
• Find the rate of steepest increase/decrease at a point; find a direction with the specified rate of increase/decrease (14.6: 23-25, 27, 28, 29; 14.R: 47, 48)
• Find a tangent plane and/or normal line to a surface (14.4: 5, 42; 14.6: 41, 43, 44, 51, 52; 14.R: 25, 31)
• Find all critical points of a function and determine their type (apply the second derivative test) (14.7: 7, 11, 15, 17, 19, 20, 39, 43)
• Find the global maximum and minimum of a function in a closed bounded region (14.7: 29, 31, 35)
• Solve a constraint optimization problem using Lagrange's method (one or two constraints) (14.8: 2 (b), 3, 5, 15, 17, 43; 14.R: 53, 55, 61-63
• Set up and compute a Riemann sum for a given choice of sample points (upper right, lower left, midpoint...) (15.1: homework problems)
• Evaluate a double integral by using volumes of elementary figures (cylinders, prisms or spheres) (15.1: homework problems)
• Compute a double integral by reducing it to an iterated integral (15.2: homework problems)

Test preparation

Good luck on the exam!