Midterm 2

Summary of concepts

• Function of two or more variables
• Graph of a function of two variables
• Level curves and surfaces, traces
• Definition of partial derivatives, various notations
• Geometric interpretation as slopes of tangents to curves
• Interpretation as rates of change
• Clairault's Theorem on equality of mixed partials
• Computation of tangent planes to surfaces z = f(x,y) via traces (x-curves and y-curves)
• Higher-order partial derivatives
• Linear approximation in one variable
• Linear approximation in several variables using the gradient
• Gradient of a function of several variables
• Directional derivatives via limits
• Directional derivatives in terms of the gradient (dot product)
• Geometric interpretation of the directional derivative
• Partial derivatives as directional derivatives
• Geometric interpretation of the gradient (direction of maximal increase)
• The gradient as a normal vector, application to computation of tangent planes
• Chain rule for functions of several variables, why it is immediate from linear approximation
• Implicit partial differentiation
• Definitions of local and absolute maxima and minima, pictures
• Necessary conditions: vanishing gradient OR an interior point where one of the partial derivatives doesn't exist OR a boundary point
• Definition of "critical point" (one of the first two kinds above!)
• How to find critical points
• Model pictures for the second derivative test in two variables
• Matrix of second partial derivatives and how it looks in the model cases
• Discriminant (capital Delta)
• The Second Derivative Test
• Local maxima, local minima, and saddle points
• When does the second derivative test give NO information?
• The Second Derivative Test is checking for one of the model geometries (up to a change of variables)
• Why solve constrained optimization problems?
• The "points on the boundary" case of local max/min leads us to such problems
• Method of Lagrange multipliers
• Geometry behind the method: "march up/downhill until you can't anymore"
• Using the method to narrow down the search for maxima and minima

(A Partial List of) Typical computational tasks

• *** new! contour plots, graphing level curves
• Computing partial derivatives of first and second order
• Computing derivatives (partial and ordinary) by implicit differentiation
• Computing tangent plane to surfaces of the form z = f(x,y) using x-curves and y-curves (traces)
• Computing tangent plane to surfaces of the form g(x,y,z) = 0 using the gradient of g
• Using the gradient to compute approximate values of a given function f(x,y) near a given point (a,b)
• Computing derivatives via the multi-variable chain rule
• Computing gradients and directional derivatives
• Finding critical points of a function of several variables using the first derivative test
• Classifying critical points using the second derivative test
• Finding maxima/minima with constraints by Lagrange multiplier method
• Application to optimization problems

Things to remember:

• Any formulas you need for the kinds of calculations mentioned above. It's a good idea to prepare a list of all the formulas you will need, and then memorize those. Of course, you shouldn't bring that list to the exam.