REVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections

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1 REVIEW I MATH 254 Calculus IV Exam I (Friday, April 29 will cover sections Functions of multivariables The definition of multivariable functions is similar to that of functions of one variable. For example, a function f of two variables (x, y is a rule that assigns to each point (x, y in a region D in the plane a unique real number f(x, y. The region D is called the domain of f. If the domain of a function f is not specified, then it is understood to be the set of all points (x, y such that f(x, y is well-defined. In what follows below let f(x, y be a function of two variables with domain D. 2. Limits and Continuity. We say the limit of f(x, y exists as (x, y tends to a point (x 0, y 0 if there is a unique real number L such that f(x, y L can be as small as you want as long as (x, y is very close (but not equal to (x 0, y 0. The number L is called the limit of f(x, y as (x, y tends to a point (x 0, y 0. We use the following notation: lim f(x, y = L. (x,y (x 0,y 0 A function f is continuous at (x 0, y 0 if lim f(x, y = f(x 0, y 0. (x,y (x 0,y 0 If f is continuous at every point in its domain we say f is continuous. 3. Partial Derivatives. By definition, f x (x f(x, y 0 f(x 0, y 0 0, y 0 = lim. x x0 x x 0 1

2 2 So when computing the partial derivative of f(x, y with respect to x, the key is to view the other variable y as a constant. The notation f x (x, y is also used very often to denote the partial derivative f. x Just as in the case of functions of one variable, We can define second order partial derivatives: x = ( f = (f 2 x x = f xx, x x y x = ( f = (f x y = f xy, y x and higher order partial derivatives such as 3 f x = (, 3 x x 2 x y = ( f = (f y x = f yx, x y y = ( f = (f 2 y y = f yy, y y 4 f y x = ( 3 f. 3 y x 3 The important Clairaut Theorem says that f xy = f yx as long as both are continuous. This is a very convenient fact to use. The Laplacian of f is defined to be f xx + f yy. 4. Gradient, Differential and Directional Derivatives. The gradient of f is ( f gradf = f = x, f. y which is a vector field (vector valued function of (x, y, that is, at each point (x, y, f is a vector in R 2. The differential of f is df = f f dx + x y dy. This is a very important concept in mathematics. But for our purpose, it is in some sense a linear approximation to the change of function f. The directional derivative of f in direction u = (u 1, u 2 : f D u f(x = u f(x = u 1 x + u f 2 y.

3 Here u = (u 1, u 2 is a unit vector: u = (u u 2 2 1/2 = 1. This represents the rate of change of f in the direction u. We see that D u f = u f u f = f since u = 1. So if D u f 0 at a given point, D u f is maximized for u = f/ f. This says that the function f increases most rapidly in the direction of gradient. For a function f(x, y, z of three variables, the gradient of f is while the differential is f = (f x, f y, f z df = f x dx + f y dy + f z dz. Similarly, the directional derivative D u f for u = (u 1, u 2, u 3 is f D u f = u f(x = u 1 x + u f 2 y + u f Tangent Planes and Linear Approximation. The graph of f is the set of all points (x, y, z in space such that z = f(x, y and (x, y is in D. This is a surface in space. For example, the graph of the function f(x, y = x 2 + y 2 is the elliptic paraboloid z = x 2 + y 2. For function f(x, y of two variables, the tangent plane to the graph of f at a point (x 0, y 0, z 0 (so z 0 = f(x 0, y 0 is given by the equation The linear function z z 0 = f x (x 0, y 0 (x x 0 + f y (x 0, y 0 (y y 0. L(x, y = f(x 0, y 0 + f x (x 0, y 0 (x x 0 + f y (x 0, y 0 (y y 0 is called the linearization or linear approximation function of f at (x 0, y 0. We see that the graph of L is exactly the tangent plane of f at (x 0, y 0, z 0. For (x, y close to (x 0, y 0 we have the approximation f(x, y L(x, y. 3

4 4 We say f is differentiable at (x 0, y 0 if this gives a good approximation for (x, y close enough to (x 0, y 0. More precisely, f is differentiable at (x 0, y 0 if f(x, y = f(x 0, y 0 +f x (x 0, y 0 (x x 0 +f y (x 0, y 0 (y y 0 +R 1 (x x 0 +R 2 (y y 0 where R 1, R 2 0 as (x, y (0, 0. Theorem. If f x and f y are continuous then f is differentiable. 6. The Chain Rule. Suppose z = f(x, y and x, y are in turn functions of a variable t: x = g(t, y = h(t. The chain rule states that dz dt f(g(t, h(t = f x(g(t, h(tg (t + f y (g(t, h(th (t, or equivalently, dz dt = dx x dt + dy y dt. Similarly, if x = g(s, t and y = h(s, t, then and s = f g x s + f h y s t = f g x t + f h y t. 7. Level Curves and Implicit Differentiation. For a function f(x, y of two variables, the level curves of f are the curves defined by the equation f(x, y = c where c is constant. For example, the level curves of function f(x, y = x 2 +y 2 are the circles x 2 + y 2 = c of radius c for c > 0. Let (x 0, y 0 be a point on a level curve f(x, y = c (so f(x 0, y 0 = c. The gradient of f at (x 0, y 0 is perpendicular to the tangent line to the level

5 curve at (x 0, y 0. That is f(x 0, y 0 T = 0 where T is a tangent vector to the level curve. If f y (x 0, y 0 0. Then the level curve determines a function y = φ(x near (x 0, y 0. This means f(x, φ(x c. We can find the derivative φ (x of this function by implicit differentiation: d d f(x, φ(x = dx dx c = 0 since c is a constant. By the Chain Rule, So f x (x, φ(x + f y (x, φ(xφ (x = 0. φ (x = f x(x, φ(x f y (x, φ(x. This is also often written in the form dy dx = f x(x, y f y (x, y. For a function f(x, y, z of three variables, a surface defined by the equation f(x, y, z = c is called a level surface of f. At a point (x 0, y 0, z 0 on the surface, the gradient f = (f x, f y, f z is a normal vector to the level surface. So the equation of the tangent plane to the level surface at (x 0, y 0, z 0 is f x (x 0, y 0, z 0 (x x 0 + f y (x 0, y 0, z 0 (y y 0 + f z (x 0, y 0, z 0 (z z 0 = 0. Similarly, the implicit differentiation formula takes the form x = f x(x, y, z f z (x, y, z, Here are some example of level surfaces: y = f y(x, y, z f z (x, y, z. a Sphere: x 2 + y 2 + z 2 = a 2 ; f(x, y, z = x 2 + y 2 + z 2. b Cylinder: x 2 + y 2 = a 2 ; f(x, y, z = x 2 + y 2. c Cone: z 2 = (x 2 + y 2 ; f(x, y, z = x 2 + y 2 z 2. d Paraboloid: z = x 2 + y 2 ; f(x, y, z = x 2 + y 2 z. e Planes: ax + by + cz = d; f(x, y, z = ax + by + cz. 5

6 6 8. Maximum and Minimum Values A point (x 0, y 0 is called a critical point of f(x, y if f x (x 0, y 0 = 0 and f(x 0, y 0 = 0 or one of the partial derivatives does not exist. Theorem. If f has a local maximum or local minimum at a point (x 0, y 0 then (x 0, y 0 is a critical point of f. To determine whether f has a local maximum or local minimum or neither at a critical point, sometimes we can use the Second Derivative Test: Theorem. Suppose f has continuous second partial derivatives and suppose (x 0, y 0 is a critical point of f, that is, f(x 0, y 0 = 0. Let Then D = f xx (x 0, y 0 f yy (x 0, y 0 (f xy (x 0, y 0 2. (a If D > 0 and f xx (x 0, y 0 > 0, then f(x 0, y 0 is a local minimum. (b If D > 0 and f xx (x 0, y 0 < 0, then f(x 0, y 0 is a local maximum. (c If D < 0 then f(x 0, y 0 is not a local maximum or local minimum. 9. Lagrange Multipliers. Let f and g are functions of two variables (x, y. To find the maximum and minimum values of f(x, y subject to the constraint g(x, y = k where k is a constant, we can use the method of Lagrange multipliers: (a Find all values of x, y and λ such that f(x, y = λ g(x, y, and g(x, y = k. (b Evaluate f at all the point (x, y obtained in the previous step (a; the largest of these values is the maximum value of f (on the curve g(x, y = k; the smallest is the minimum value of f.