MATH 250, Study Guide for Midterm Test 2

04/16/2014

Test coverage

1. Derivative for functions of several variables (Chapter 14)
   1. Functions of several variables, domains, ranges, graphs (14.1)
   2. Limits and Continuity (14.2)
   3. Partial Derivatives (14.3)
   4. Tangent Planes and Linear Approximations. Differentials (14.4)
   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, interior, exterior and boundary points
• Graphs and level sets (level curves in 2D, level surfaces in 3D). Contour maps
• Limits: ε-δ-definition, squeezing principle, limits along different paths
• Definition of continuity in the case of two and three variables; examples of continuous and discontinuous functions
• Partial derivatives, slopes of tangent lines to the graph
• Equation of the tangent plane. Linear approximation L(x,y), differential df(x,y)
• 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
• Second Derivative Test for local extrema and saddle points
• 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 geometric definition
• Computing double integrals using iterated integrals

Basic types of questions (the list is not all-inclusive, but it covers most of typical midterm questions)

• Find the domain of a function and sketch a few level curves/surfaces (14.1)
• Compute a limit using the squeezing principle or show that limit does not exist(14.2)
• Check the continuity of a function or find out how to define the function at a point not in the domain so that the result is a continuous function (14.2)
• Compute partial derivatives, verify that partial derivatives satisfy a given relation (Laplace's equation is a typical example) (14.3)
• Find the equation of a tangent plane to a graph (14.4)
• Find the linear approximation or a differential of a function at a given point (14.4)
• Use differentials to estimate increments of functions (14.4)
• Compute derivatives of composite functions using the chain rule (14.5)
• Applied problems on chain rule: examples from geometry and physics/other sciences (14.5: Example 2, problems 35-44)
• Find the rate of steepest increase/decrease at a point; find a direction with given rate of increase/decrease (14.6)
• Find a tangent plane and/or normal line to a surface (14.6)
• Find all critical points of a function and determine their type (apply second derivative test) (14.7)
• Find the global maximum and minimum of a function in a closed bounded region (14.7)
• Solve a constraint optimization problem using Lagrange's method (one or two constraints) (14.8)
• Set up and compute a Riemann sum for a given choice of sample points (upper right, lower left, midpoint...) (15.1)
• Evaluate a double integral by using volumes of elementary figures (cylinders, prisms or spheres) (15.1)
• Compute a double integral by reducing it to an iterated integral (15.2)

Test preparation

Practice tests (for ~75 minutes):

Test 1: 14.3: 77, 14.2: 13, 14.6: 25, 14.7: 29, 14.8: 3, 14.R*: 23, 15.1: 13, 15.2: 27.

Test 2: 14.3: 76(d), 14.2: 38, 14.4: 14, 14.5: 38, 14.R*: 54, 14.8: 16, 15.1: 14, 15.2: 38.

(* 14.R is Chapter 14, Review Problems.)

Good luck on the exam!