1. Announcements Topics: To Do:
Systems of DEs (Predator-Prey Model) (8.5)
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
To Do:
Read section 8.5 in the hardcover textbook and sections 1, 2, and 3 in the “Functions of Several Variables” module
Work on Suggested Practice Problems in 8.5 in the hardcover textbook and sections 1, 2, and 3 in the “Functions of Several Variables” module (posted on webpage)
Finish Assignment 3. Work on Assignments 4, 11, and 12.
Holiday this weds, test next weds…info posted soon

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 R called contour maps. 2
Like in geography.
So back here. For this outer ring… the k value is 20… so we are looking for all points in the domain such that the mountain has a height of 20 over these points. They all lie on the circle labelled there.

3. Contour Maps and Level Curves
Level Curves: The level curves of a function f of two variables are the curves with equations where k is a constant in the RANGE of the function. A level curve is a curve in the domain of f along which the graph of f has height k.
In general, sketching the graphs of functions of two variables (surfaces) is difficult so instead we sketch 2-D representations of these surfaces in R2 called CONTOUR MAPS.
So, basically k is some height that your function can reach. You set your function equal to this height and figure out all of the points (x,y) that produce this height. Hopefully the set of these points forms some curve you easily know how to draw.

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.
You pick some equally spaced heights in the range of the function and find the equations of the level curves for this height. Then you draw them all in the domain and now you have a contour map. This should tell someone what the surface SHOULD look like even if you can’t draw it for them.

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) (b)
In general, sketching the graphs of functions of two variables (surfaces) is difficult so instead we sketch 2-D representations of these surfaces in R2 called CONTOUR MAPS.
This is all done in an attempt to represent the 3d object in 2d because most of us aren’t great at drawing in 2d…. With a little bit of training, your brain will be able to look at contour maps, raise the lines correctly, and create a 3d image even if your hand can’t draw it…

6. Contour Maps and Level Curves
Matching Example: Match the equation of each function to its graph and to its contour map. (a) (b) (c) (d) (e) (f)
Take a lot of time to do this…
Things you can look at/take into consideration:
cross-sections of the surface… by doing this, you can easily sketch in 2-d and try to visualize the whole surface based on properties of its cross sections…
Zeros of the function… along which lines/curves in the domain is the height of the surface 0. These help to tell the difference between various trigonometric functions because they have very distinctive level curves corresponding to the zeros of the function.
Relating it to something in single variable calculus… this should be a starting out point actually.

7. Contour Maps and Level Curves
Where to start:
Consider the type of function. What are its properties? What would the graph of a single-variable function of this type look like?
Sketch some cross-sections of the surface (start with x=0, then y=0). By doing this, you can easily sketch some curves in 2D and try to visualize the whole surface being pieced together
Look at the zeros of the function… (level curves in the domain along which the height of the surface 0). These are really helpful in differentiating between various trigonometric functions.
Where to start:
Things you can look at/take into consideration:
cross-sections of the surface… by doing this, you can easily sketch in 2-d and try to visualize the whole surface based on properties of its cross sections…
Zeros of the function… along which lines/curves in the domain is the height of the surface 0. These help to tell the difference between various trigonometric functions because they have very distinctive level curves corresponding to the zeros of the function.
Relating it to something in single variable calculus… this should be a starting out point actually.

8. Contour Maps and Level Curves
Graphs for Matching Example:
In general, sketching the graphs of functions of two variables (surfaces) is difficult so instead we sketch 2-D representations of these surfaces in R2 called CONTOUR MAPS

9. Contour Maps and Level Curves
Contour Maps for Matching Example:
In general, sketching the graphs of functions of two variables (surfaces) is difficult so instead we sketch 2-D representations of these surfaces in R2 called CONTOUR MAPS

10. Limit of a Function in R 2
Definition: 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.
This limit EXISTS!! Explain why even though the hole is not filled in. or ask… how do we check that a limit exists?
Okey, so recall the meaning of this expression in 2D… when we write this expression, we mean that the y-values approach L more and more closely as x approaches a (from either side of a) more and more closely but without touching a. the limit above exists because the y-values are heading towards some finite value L (not infinity) whether we approach a from the left or from the right.

11. Existence of a Limit in R2
The limit exists if and only if the left and right limits both exist (equal a real number) and are the same value.
So the limit in the previous picture exists because as we approach a, starting on the left of a OR on the right of a, the y-values approach the single value “L”.
to say a limit DNE, we mean that either the y-values approach different values from either side or that the y-values approach +/- infinity from one (or both) sides. (draw a picture of the infinite case… y-values do NOT approach a single real number L)

12. Existence of a Limit in R2
Example: Evaluate the following limits or show that they do not exist. (a) (b) (c)
First one exists… second one does not exist… notice how the second limit DNE because the left and right aren’t equal… the one on the previous page didn’t exist because

13. Existence of a Limit in R2
It is relatively easy to show that this type of limit 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
So its easy to check if a limit exists… just two cases.

14. Limit of a Function in R 3
Definition: means that the z-values approach L as (x,y) approaches (a,b) along every path in the domain of f.
In 3D, we consider what is happening to the z-values of a function f(x,y) as the point (x,y) moves closer to some point (a,b).
If the z-values (heights) are approaching a single value as (x,y) approaches (a,b) from every angle, then we say that the limit exists and equals the value (height) L. But how many ways can the point (x,y) approach the point (a,b)?
The formal definition states that we can make the z-values arbitrarily close (as close as we’d like) to L by taking the point (x,y) sufficiently close to the point (a,b) without actually touching it. And along every path to (a,b) within the domain.

15. Existence of a Limit in R3
In general, it is difficult to show that such a limit exists because we have to consider the limit along all possible paths to (a,b).
Cleary, we will not be doing this. Instead, to show a limit exists, we we first have to learn some rules for limits which will allow us to simplify first and then validate plugging into the expression, the same as we did before. To show a limit DOESN”T exist is actually an easier place to start (it is easier to FAIL something that PASS, right?)

16. Existence of a Limit in R3
However, to show that a limit doesn’t exist, all we have to do is to find two different paths leading to (a,b) such that the limit of the function along each path is different (or does not exist).
Show the details of this example. First, what is the domain of this function. With rational functions, you can usually expect to see a huge problem with the graph… a rip, tear, hole, asymptote, etc…
The orientation of this is weird… origin is in the middle. x is next to x-values so along y-axis means along the line x=0.
Note: undefined at (0,0)

17. Existence of a Limit in R3
Example: Show that the following limits do not exist. (a) (b) (c)
Check these examples.
All we have to do is to find two different paths leading to (a,b) such that the limit of the function along each path is different.
Maybe add in graphs of these?

18. Limit Laws
Theorem: Assume that and exist (i.e. are real numbers). Then (a) (b)
OKEY, now to calculate some limits!! First we need some rules that will allow us to evaluate limits. Then we determine the limits of really simple functions (so we can “prove” the value of this limits from their graphs). Then, we have rules which tell us that combining these simple pieces for which limits all exisit and we can calculate will produce a function who’s limit also exisits and we can just break it down into pieces to evaluate it.

19. Limit Laws Theorem (continued): (c) (d)
In general, sketching the graphs of functions of two variables (surfaces) is difficult so instead we sketch 2-D representations of these surfaces in R2 called CONTOUR MAPS

20. Some Basic Rules For the function .
Show why the first one is true. Graph is a plane. Lift the floor up at a 45 degree angle to the xy-plane. The z-value depends purely on the x-value. So we can think of this as a function of just one variable and we have established how to calculate this limit. Others follow similarily.

21. Evaluating Limits
Example #10: Using the properties of limits, evaluate
Break these guys down into blocks so that all we need are the 3 basic results to evaluate the limit of the combination of these parts.
MAYBE JUST ONE EXAMPLE.

22. Direct Substitution
Theorem: If is a polynomial or rational function (in which case must be in the domain of ), then .
Instead of using these bloody limit laws all the time we can prove the following theorem which allows us to directly substitute in the point for rational and polynomial functions when the point is in the domain.
So the previous two fall under this category and instead of breaking them down into pieces, we can use direct substitution to evaluate their limits.

23. Continuity of a Function in R3
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.
Continuity: Formally, if the limit of the function as (x,y) -> (a,b) exisits and equals the value of the function AT (a,b) then the function is continuous at (a,b). This means that its graph has no holes, asymptotes, breaks, “cliffs”,etc. and that a small change in the input produces a small change in the output. We like continuous functions because they allow us to use direct substitution to evaluate limits and also because we are free from problems with these guys.
Examples: Review a couple of discontinuous functions (problems with the limit) maybe a piecewise? Work with continous functions.

24. Continuity of a Function in R3
Definition: A function is continuous at the point if
We like continuous functions because they allow us to use direct substitution to evaluate limits and also because we are free from problems with these guys.
Put in 3 implicit conditions.
Examples: Review a couple of discontinuous functions (problems with the limit) maybe a piecewise? Work with continous functions.

25. Continuity of a Function in R3
Example: Determine whether or not the function is continuous at (0,0).
We like continuous functions because they allow us to use direct substitution to evaluate limits and also because we are free from problems with these guys.
Put in 3 implicit conditions.
Examples: Review a couple of discontinuous functions (problems with the limit) maybe a piecewise? Work with continous functions.

26. Which Functions Are Continuous?
A function is continuous if it is continuous at every point in its domain.
Basic Continuous Functions:
polynomials
rational functions
exponential functions
logarithmic functions
trigonometric functions
root functions
Good idea to be able to easily identify which functions are continuous without going through all that work for each new function.
Okey, so this basically looks like every single function we’ve dealt with is continuous on its domain… any exceptions??

27. 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 each function is continuous. (a) (b)
Each PIECE is continuous so the combinations are continuous where they are defined.
Determine where they are defined (ie. Their domains) then you will know where they are continuous.
Really, for our simple, ‘normal’ functions, asking where a function is continuous amounts to just finding the domain of the function. For some weird functions, the domain could be everything yet the function is no where continuous…. For example, piecewise functions…

28. Limits of Continuous Functions
By the definition of continuity, if a function is continuous at a point, then we can evaluate the limit simply by direct substitution. Example: Evaluate each limit. (a) (b)
By the definition of continuity, blah blah blah, if a function is continuous at a, then we can evaluate the limit simply by direct substitution.
Since each piece is continuous, the combination is continuous so we can evaluate each by direct substitution.