#Zero limits free pdf how to#

More precisely: Slide 8 Definition 4 Consider a function f : D IR 2 R IR. Partial derivatives The partial derivatives of f computed at any point (x, y) D define the functions f x (x, y) and f y (x, y), the partial derivatives of f. Discuss the geometrical meaning of the partial derivative using the graph of the function. Example: Find the partial derivative of f(x, y) = x 2 + y 2 /4 with respect to y at (, 3). To compute the partial derivative of f(x, y) with respect to y at (a, b), one follows the same idea: First, evaluate the function at x = a, that is compute f(a, y) second, compute the usual derivative of single variable functions evaluate the result at y = b, and the result is f y (a, b). Example: Find the partial derivative of f(x, y) = x 2 + y 2 /4 with respect to x at (, 3). h 0 hД¦ Math 20C Multivariable Calculus Lecture 2 6 So, to compute the partial derivative of f(x, y) with respect to x at (a, b), one can do the following: First, evaluate the function at y = b, that is compute f(x, b) second, compute the usual derivative of single variable functions evaluate the result at x = a, and the result is f x (a, b). h 0 h The partial derivative of f(x, y) with respect to y at (a, b) D is denoted as f y (a, b) and is given by f y (a, b) = lim. Slide 7 The partial derivative of f(x, y) with respect to x at (a, b) D is denoted as f x (a, b) and is given by f x (a, b) = lim. Partial derivatives Definition 3 (Partial derivative) Consider a function f : D IR 2 R IR. Composition of continuous functions are continuous, example f(x, y) = cos(x 2 + y 2 ).ДҐ Math 20C Multivariable Calculus Lecture 2 5 Partial derivatives Slide 6 Definition of Partial derivatives. Slide 5 Rational functions are continuous on their domain, for example, f(x, y) = P n(x, y) Q m (x, y), f(x, y) = x2 + 3y x 2 y 2 + y 4 x 2 y 2, x В±y. (x,y) (x 0,y 0) Continuity Examples of continuous functions: Polynomial functions are continuous in IR 2, for example P 2 (x, y) = a 0 + b x + b 2 y + c x 2 + c 2 xy + c 3 y 2. x x 0 x x 0 x x 0 ), Definition 2 (Continuity) A function f(x, y) is continuous at (x 0, y 0 ) if lim f(x, y) = f(x 0, y 0 ). Slide 4 If the limits lim x x0 f and lim x x0 g exist, then ( ) ( В± lim (f В± g) = x x 0 lim f x x 0 lim g x x 0 ( ) ( ) lim (fg) = lim f lim g. Then, one knows that lim y 0 y = 0, therefore the squeeze theorem says that lim (x,y) (0,0) x 2 y x 2 + y 2 = 0.Д¤ Math 20C Multivariable Calculus Lecture 2 4 Limits and Continuity Limit laws for functions of a single variable also holds for functions of two variables. (proof: 0 y 2, then x 2 (x 2 + y 2 ).) Therefore, one has the inequality y x2 y x 2 + y 2 y, (x, y) IR2, (x, y) (0, 0). However, this argument does not prove that the limit actually exists.this can be done with the squeeze theorem. Therefore, one cannot conclude that the limit does not exist. Then so one has that f(x, mx) = x2 mx x 2 + m 2 x 2 = mx + m 2, lim f(x, mx) = 0, (x,mx) (0,0) m IR. Consider the curves C m given by y = mx, with m a constant. (2) Let us first try the side limit theorem, to try to prove that the limit does not exist. Does the following limit exist? lim (x,y) (0,0) x 2 y x 2 + y 2. Then f(x, mx) = 3x 2 x 2 + 2m 2 x 2 = 3 + 2m 2,ДЈ Math 20C Multivariable Calculus Lecture 3 so one has that lim f(x, mx) = 3 (x,mx) (0,0) + 2m 2 is different for each value of m. (x,0) (0,0) Notice that in the above example one could compute the limit for arbitrary lines, that is, C m given by y = mx, with m a constant. Then, then Therefore, the limit in () does not exist. (x,0) (0,0) Let us now pick up the curve C 2 as the y-axis, that is, x = 0. Let pick the curve C as the x-axis, that is, y = 0. Does the following limit exist? lim (x,y) (0,0) 3x 2 x 2 + 2y 2.

#Zero limits free pdf how to#

(x,y) (x 0,y 0) Example: How to use the side limit theorem. Slide 3 A tool to prove that a limit exists is the following: Theorem 2 (Squeeze) Assume f(x, y) g(x, y) h(x, y) for all (x, y) near (x 0, y 0 ) Assume Then lim f(x, y) = L = lim h(x, y), (x,y) (x 0,y 0) (x,y) (x 0,y 0) lim g(x, y) = L. Review: Limits Slide 2 Definition Given a function f(x, y) : D IR 2 IR and a point (x 0, y 0 ) IR 2, we write lim f(x, y) = L, (x,y) (x 0,y 0) if and only if for all (x, y) D close enough in distance to (x 0, y 0 ) the values of f(x, y) approaches L.Дў Math 20C Multivariable Calculus Lecture 2 A tool to show that a limit does not exist is the following: Theorem (Side limits) If f(x, y) L along a path C as (x, y) (x 0, y 0 ), and f(x, y) L 2 along a path C 2 as (x, y) (x 0, y 0 ), with L L 2, then lim f(x, y) (x,y) (x 0,y 0) does not exist. 1 Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit.