This exam will be over material covered in class from Monday 14 February through Tuesday 8 March, corresponding to sections in the text.

Transcription

1 Math 275, section 002 (Ultman) Spring 2011 MIDTERM 2 REVIEW The second midterm will be held in class (1:40 2:30pm) on Friday 11 March. You will be allowed one half of one side of an sheet of paper (total surface area: ) for handwritten notes, the table of basic derivatives and integrals posted on the course website, and a scientific calculator. No other notes or technology will be allowed for this exam. This exam will be over material covered in class from Monday 14 February through Tuesday 8 March, corresponding to sections in the text. Suggested review problems: Your review for the material in section 12.8 is the current homework assignment (due Friday 11 March). For sections , look over your previous homework assignments and quizzes. Some additional problems that will give you a good workout are: sec # 13 18, 24, 39, 41 sec # 29, 30, 33, 34 sec # 39, 58, 59, 64 sec # 39, 40, 41, 45, 46, 48 sec # 5, 15, 21, 24, 27, 32 sec # 5, 23, 24, 29, 39, 48, 52, 55 sec # 11, 41 43, 45, 46, 54 Some of these have appeared in your homework. Try them again without looking at your previous work.

2 INTRODUCTION TO SCALAR-VALUED FUNCTIONS OF TWO OR THREE VARIABLES The domain of a function of two variables is the set points (ordered pairs of numbers) in R 2 for which the function is defined. Similarly, the domain of a function of three variables is the set of points (ordered triples) in R 3 for which the function is defined. The range of a scalar-valued function is the set of all values in R taken on by the function over its domain. Unless otherwise stated, the functions under discussion here are assumed to be scalar-valued functions of two or three variables. The graph of a function of two variables is a surface in R 3. The surface z = f(x, y) can be a useful tool in analyzing the behavior of the function f. On the other hand, the graph of a function of three variables requires four coordinate axes, so the graph is not a natural way to give a visual representation of a function of three variables. The level sets of a function are the set of points in its domain on which the function value remains constant. If f is a function of two variables, the level set f(x, y) = c is a curve (or collection of curves). If f is a function of three variables, the level set f(x, y, z) = c is a surface (or collection of surfaces). Level sets can be useful tools for visualizing the behavior of functions of two and three variables. They can also be useful ways to define curves and surfaces. A function of two (respectively three) variables has a limit at a point P 0 if, for any fixed number ε (no matter how small), one can find a disk (resp. sphere) of positive radius centered at P 0 so that the function values for all points within that disk (resp. sphere) are within a distance of ε of the function value at P 0. A function is continuous at a point if its limit exists and equals the function value at that point. There are various properties of limits and continuous functions that can be called upon for evaluating limits and determining continuity. 2

3 DERIVATIVES OF FUNCTIONS OF TWO OR THREE VARIABLES Geometrically, the derivative of a function of a single variable is the slope of the tangent line to the graph of the function. For a function of more than one variable, there are potentially infinitely many lines tangent to its graph. This leads to the question: what does it mean for a function of more than one variable to be differentiable? A partial derivative of a function is the instantaneous rate of change of the function with respect to one variable as the others are held constant. If f is a function of two variables, f x (resp. f y ) has the geometric interpretation as the slope of the line tangent to the curve of intersection of the surface z = f(x, y) and a plane parallel to the xz-plane (resp. yz-plane). Partial derivatives, where defined, are also functions; higher-order partial derivatives of a function are defined by taking successive partial derivatives. If second-order partial derivatives are continuous on an open region, then order of differentiation does not matter when computing mixed partial derivatives on this region: that is, f xy = f yx. A function of more than one variable is said to be differentiable at a point if there is a linear function that is a good approximation of the function. If the partial derivatives of a function are continuous on an open region, then the function is differentiable on this region and the linear function can be expressed in terms of the partial derivatives. For example, if f is a function of two variables and the partial derivatives are continuous on a region containing (x 0, y 0 ), then f is differentiable at (x 0, y 0 ) and the linearization of f at (x 0, y 0 ) is: L(x, y) = f(x 0, y 0 ) + f x (x x 0) + f y (y y 0) where the partial derivatives are evaluated at (x 0, y 0 ). The graph of this linearization L is the tangent plane to f at (x 0, y 0, f(x 0, y 0 )). If f is differentiable, the differential df reflects the change in f over an infinitesimal displacement in the domain. Differentials are sometimes used to approximate changes in f over small (but not infinitesimal) displacements in the domain. NB: the differential is not a derivative: derivative (instantaneous) rate of change of f differential (infinitesimal) change in f 3

4 Partial derivatives reflect the rate of change of a function along lines in the domain parallel to the coordinate axes. If a function is differentiable, there are also directional derivatives: rates of change of the function relative to any straight line in the domain traveled at unit speed. In addition, for differentiable functions one can consider the rate of change of the function relative to any parameterized curve r(t) in the domain: this gives rise to the chain rule. The gradient of a differentiable function is a vector field that has several useful properties. Its direction is that in which the function increases most rapidly, and its magnitude gives the maximum value of the directional derivatives of the function. Similarly, the function decreases most rapidly in the direction opposite that of the gradient, and the negative of the magnitude of the gradient gives the minimum value of the directional derivatives of the function. The gradient of a function is orthogonal to the function s level sets. This fact can be used to find equations of tangent and normal lines to level curves or tangent planes and normal lines to level surfaces. Furthermore, the differential, directional derivatives and the chain rule can all be expressed in terms of the dot product of the gradient with the appropriate vector in the domain. The gradient piece of the dot product comes from the function and contains information about how the function is changing, while the vector piece contains information about the path being travelled in the domain. This theme will resurface later in the course. differential: df = f d r d r = dx î + dy ĵ + dz ˆk directional derivative: Dûf = f û û a unit vector chain rule: df dt = f v v the velocity to the curve r(t) 4

5 LOCAL AND ABSOLUTE EXTREMA & OPTIMIZATION SUBJECT TO CONSTRAINT To find the local maxima and minima of a function of two variables, one first finds the critical points (that is, points where f = 0 or points where one or both of the partial derivatives fail to exist), then classifies these as either local maxima or minima, or saddle points. If the second partial derivatives of a function are continuous on an open region containing a critical point where f = 0, the second derivative test can expedite the classification. But be aware that there are instances where the second derivative test either cannot be used (for example, f(x, y) = x 2 + y 2 ) or is inconclusive. To find the absolute maxima and minima of a function of two variables over a closed and bounded region, first find all critical points in the interior of the region using the gradient, then find all critical points on the interior of the boundary curves. Evaluate these critical points together with the endpoints of the boundary curves. To maximize or minimize a function of two or three variables subject to some constraint in the domain, use the method of Lagrange Multipliers. 5