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1 Announcements Topics: In the Functions of Several Variables module: - Section 1: Introduction to Functions of Several Variables (Basic Definitions and Notation) - Section 2: Graphs, Level Curves + Contour Maps - Section 3: Limits and Continuity - Section 4: Partial Derivatives To Do: - Read sections 1, 2, 3, and 4 in the Functions of Several Variables module - Work on Assignments and Suggested Practice Problems posted on the webpage under the SCHEDULE + HOMEWORK link

2 Contour Maps and Level Curves In general, sketching the graphs of functions of two variables (surfaces) is difficult so instead we sketch 2-dimensional representations of these surfaces in 2 R called contour maps.

3 Contour Maps and Level Curves Level Curves: The level curves of a function f of two variables are the curves with equations f (x, y) = k where k is a constant in the RANGE of the function. A level curve f (x, y) = k is a curve in the domain of f along which the graph of f has height k.

4 Contour Maps and Level Curves Contour Maps: A contour map is a collection of level curves. To visualize the graph of f from the contour map, imagine raising each level curve to the indicated height. The surface is steep where the level curves are close together and it is flatter where they are farther apart.

5 Contour Maps and Level Curves Examples: Draw a contour map for the following functions showing several level curves. Compare them to the surfaces we drew previously. (a) f (x, y) = 6 3x 2y (b) f (x, y) = 4 x 2 y 2

6 Limit of a Function in R 2 Definition: x a f (x) = L y means that the y-values can be made arbitrarily close (as close as we d like) to L by taking the x- values sufficiently close to a, from either side of a, but not equal to a. L a y=f(x) x

7 Existence of a Limit in R 2 The it exists if and only if the left and right its both exist (equal a real number) and are the same value.

8 Existence of a Limit in R 2 Example: Evaluate the following its or show that they do not exist. (a) (b) (c) x 1 f (x) where f (x) = x x 0 x x 0 1 x 2 $ x when x <1 & % 1 when x 1 2 ' & x

9 Existence of a Limit in R 2 It is relatively easy to show that this type of it exists since there are only two ways to approach the number a along the real number line: either from the left or from the right

10 Limit of a Function in R 3 Definition: (x,y) (a,b ) f (x, y) = L means that the z-values approach L as (x,y) approaches (a,b) along every path in the domain of f.

11 Existence of a Limit in R 3 In general, it is difficult to show that such a it exists because we have to consider the it along all possible paths to (a,b).

12 Existence of a Limit in R 3 However, to show that a it doesn t exist, all we have to do is to find two different paths leading to (a,b) such that the it of the function along each path is different (or does not exist).

13 Existence of a Limit in R 3 Example: Show that the following its do not exist. (a) (c) ( x, y) (0,0) (x,y) (0,0) y 2 x 2 2x 2 + 3y 2 x 2 + sin 2 y 2x 2 + y 2 (b) ( x, y) (0,0) 6x 3 y 2x 4 + y 4

14 Limit Laws Theorem: Assume that f (x, y) and (x,y) (a,b ) exist (i.e. are real numbers). Then g(x, y) (x,y) (a,b ) (a) (x,y) (a,b ) ( f (x, y) ± g(x, y) ) = f (x,y) ± (x,y ) (a,b) g(x,y) (x,y ) (a,b) (b) (x,y) (a,b ) ( c f (x, y) ) = c (x,y ) (a,b ) f (x,y), where c is any constant.

15 Limit Laws Theorem (continued): (c) (x,y) (a,b ) ( f (x, y) g(x, y) ) = (x,y ) (a,b) f (x, y) g(x,y) (x,y ) (a,b ) (d) (x,y) (a,b ) f (x, y) g(x, y) = (x,y) (a,b ) (x,y) (a,b ) f (x, y) g(x, y), provided (x,y ) (a,b) g(x,y) 0.

16 Some Basic Rules For the function f (x, y) = x, (x,y) (a,b ) f (x, y) = x = a (x,y) (a,b ) For the function f (x, y) = y, (x,y) (a,b ) f (x, y) = y = b (x,y) (a,b ) For the function. f (x, y) = c, (x,y) (a,b ) f (x, y) = c = c (x,y) (a,b )

17 Evaluating Limits Example #10: Using the properties of its, evaluate ( x, y) (2, 2) 1 xy 4. Solution: ( x, y) (2, 2) = = = 1 xy 4 1 ( x, y) (2, 2) xy 4 ( x, y) (2, 2) x ( x, y) (2, 2) 1 2 ( 2) 4 ( ) 1 ( x, y) (2, 2) ( x, y) (2, 2) y 4 ( x, y) (2, 2) = 1 8

18 Direct Substitution Theorem: If f (x, y) is a polynomial or rational function (in which case (a,b) must be in the domain of f ), then (x,y) (a,b ) f (x, y) = f (a,b).

19 Continuity of a Function in R 3 Intuitive idea: A function is continuous if its graph has no holes, gaps, jumps, or tears. A continuous function has the property that a small change in the input produces a small change in the output.

20 Continuity of a Function in R 3 Definition: A function at the point f (a,b) is continuous if (x,y) (a,b ) f (x, y) = f (a,b)

21 Continuity of a Function in R 3 Example: Determine whether or not the function # f (x, y) = x 2 + y if (x, y) (0,0) $ % 1 if (x, y) = (0,0) is continuous at (0,0).

22 Which Functions Are Continuous? A function is continuous if it is continuous at every point in its domain. Basic Continuous Functions: ü polynomials ü rational functions ü exponential functions ü logarithmic functions ü trigonometric functions ü root functions

23 Which Functions Are Continuous? Combining Continuous Functions: The sum, difference, product, quotient, and composition of continuous functions is continuous where defined. Example: Find the largest domain on which is continuous. f (x, y) = e x 2y + x + y 2

24 Limits of Continuous Functions By the definition of continuity, if a function is continuous at a point, then we can evaluate the it simply by direct substitution. Example: Evaluate ( x,y) (0, 1) ( e x2y + x + y 2 )

25 Derivative of y=f(x) Recall: Definition of the Derivative in Single Variable Calculus: df dx = f '(x) = h 0 f (x + h) f (x) h instantaneous rate of change of f with respect to x

26 Partial Derivatives of z=f(x,y) The partial derivative of a function of several variables is a way to measure the rate of change of the function as one of its variables changes.

27 Partial Derivatives of z=f(x,y) Example: Dynamics of Prey Consumption Consider the type-2 functional response model c(n,t h ) = an 1+ at h N where c(n,t h ) is the number of prey captured (in some fixed time interval), T h is the handling time, and N is the density of prey. How does the number of rabbits captured depend on the handling time and the density?

28 Partial Derivatives of z=f(x,y) The partial derivative of f with respect to x is the real-valued function defined by f / x f x (x, y) = h 0 f (x + h, y) f (x, y) h provided that the it exists. This function tells us the rate of change of f in the x-direction at all points (x,y) for which the it exists.

29 Partial Derivatives of z=f(x,y) The partial derivative of f with respect to y is the real-valued function defined by f / y f y (x, y) = h 0 f (x, y + h) f (x, y) h provided that the it exists. This function tells us the rate of change of f in the y-direction at all points (x,y) for which the it exists.

30 Partial Derivatives of z=f(x,y) Example: Using the definitions, compute for f (x) = x 2 y. f / x and f / y

31 Partial Derivatives of z=f(x,y) Rule for finding partial derivatives of z=f(x,y): 1. To find f x, treat y as a constant and differentiate f(x,y) with respect to x. 2. To find f y, treat x as a constant and differentiate f(x,y) with respect to y.

32 Partial Derivatives of z=f(x,y) Example: Find the first partial derivatives of the following functions. (a) f (x, y) = x 4 y 3 + 8x 2 y (b) z = x y (c) " z = arctan y % $ ' # x &

33 Geometric Interpretation of the Partial Derivatives of z=f(x,y) S Let z=f(x,y) be a function of two variables whose graph is the surface S. Fix y=b (constant) and let x vary. The curve c 1 on the surface S is defined by z=f(x,b). (Note: this is now only a function of the variable x)

34 Geometric Interpretation of the Partial Derivatives of z=f(x,y) T 1 S The partial derivative of f with respect to x at (a,b) is the slope of the tangent T 1 to the curve c 1 at the point P.

35 Geometric Interpretation of the Partial Derivatives of z=f(x,y) Now, fix x=a (constant) and let y vary. S The curve c 2 on the surface S is defined by z=f(a,y). (Note: this is now only a function of the variable y)

36 Geometric Interpretation of the Partial Derivatives of z=f(x,y) T 2 T 1 S S The partial derivative of f with respect to y at (a,b) is the slope of the tangent T 2 to the curve c 2 at the point P.

37 Geometric Interpretation of the Partial Derivatives of z=f(x,y) Example: Determine the signs of f x (1,2) and f y (1,2) on the graph below.

38 Partial Derivatives of z=f(x,y) Example: If f (x, y) = 4 x 2 y 2, find f x (1,0) and f y (1,0) and interpret geometrically.

39 Partial Derivatives of z=f(x,y) Example: Wind Chill The table below contains values of the wind chill index, or simply wind chill, W(T,v) based on measurements of air temperature T (in degrees Celsius) and wind speed v (in kilometres per hour). T=-25 T=-20 T=-15 T=-10 v= v= v= Estimate W T (-20, 30) and interpret the result.