Saturday, October 24, 2009 / Primary Topic / Secondary Topic / Additional Topics
Limits / Continuity / Pre-Calculus Review and Derivatives
Comments / Lecture Notes
Pre-Calculus Review
Domain and Range
Domain – The values which the independent variable, usually , may assume in a function/equation.
Range – The values which the dependent variable, usually , may assume according to the -values in a function/equation
Periods
Not all of the trigonometric functions have the same period. There are two you must remember:
Period of and are all
Period of and are
Graphs
Basic graphs that we all need to remember in some way or another:
/ /
/ /
/ ; is a constant / ; is a constant
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/ /
/ /
Piecewise Functions
Know how to graph functions that are defined by multiple equations as well as how to evaluate points or values.
In the equation of if asked to find , you better not substitute the 2 into .
Graph Transformations
All functions on the AP exam do not come in their crystal clear easy forms. They are almost always functions that undergo some transformation. Understand how functions of the form may be transformed by the variables and in .
End Behavior
You don’t necessarily need to know how a graph will behave EVERYWHERE (it would be cool though), but you should have an idea how a graph will behave in the long run. Will the graph go towards positive infinity? Negative infinity? Rise fast? Slowly move up? Mellow out with an asymptote? Review this topic briefly.
Why do we need to know all this from that class we took last year?
Why not? In reality, you need Pre-Calculus to make sure that the problems you are working on and the graphs you analyze on the AP exam make sense. There will be times where you CAN eliminate answers because you understand that the domain or range of a function does not include some numbers listed in the multiple choice. When you need to graph a function on the free response section, you will be able to set your window much faster and with less difficulty. It just makes life easier if you have a strong mastery of the basics.
Limits
The Path vs. The Point
A point is not the same as a limit‼ In talking about limits, we are not interested in the point, or the -value that corresponds to an -value. We are asking the question about the path traveled along the curve as we approach an -value.
Intent vs. Existence
Another way to consider limits is to altogether abandon the point. Imagine that the point at does not exist. Now imagine that you are a dot on a calculator going towards that value, but you don’t know that there is no point at . As you travel towards your destination, you have some idea in your head of where you are going to end up. Along the way on this journey, wherever you think you are ending up at, that is the limit. The point does not have to exist for you to have this journey.
True Story – I wanted to go to eat at Norms Restaurant. As I take the 10 Freeway Eastbound, I expect to be eating at Norms at Valley and Del Mar. That is the limit, my destination. However, Norms was closed the week before and therefore does not exist. My limit exists, as I certainly intended to eat there, but the point was nonexistent. End of story. *sigh*
Nonexistent Limits
Now, after that weird story about Norms, we get to the concept of nonexistent limits. Nonexistent limits usually happen whenever we have an asymptote or we are talking about a function out at the infinities. The best case we can refer to here is the standard hyperbola . If we trace the right arm of the hyperbola in the first quadrant out towards the infinities, we get closer and closer to the -value of zero, which is our limit. We’ll never get to our destination here, but at least we know that zero is our destination. Now, on the same arm of the hyperbola, trace backwards. What we see this time is that the hyperbola goes up and up and up as we get to the -value of zero. THIS time, we do not have a limit. The limit does not exist.
WHY??? Why does the limit exist as we move along the graph to the right, but not the left here? The problem has to deal with the fact that no matter how far to the right we move, we see that we are getting closer and closer to zero, so zero is our limit. However, as we move more and more to the left, we closer and closer to the -axis, but our -value keeps on going up. We have no limit as to what -value we will ultimately reach. Because of this reason, our limit does not exist.
Limits approaching from the Left and Limits approaching from the Right
The really cool thing about that sad Norms story from above is that the idea can be transferred over to when we discuss limits from one direction. As we have studied before, the following 3 criteria need to be satisfied in order for a limit to exist:
1.) must exist
2.) must exist
3.)
Yan’s sad story to help you remember the three criteria:
We have two people who are going to meet each other at location . The reason: at , there is a famous lover’s cliff, where they plan to confess to each other. The first person, Chris, goes to from the left and ends up at the bottom of the cliff and waits. The second person, Alex, goes to from the right and ends up at the top of the cliff. Each of them has an intent of where to go and therefore, their own individual limit. The problem is, well, unless Alex goes off the cliff and lands near Chris, they’ll never meet.
In order for our star-crossed lovers to meet, their limits must be the same. Unfortunately, they have different limits. Let us answer this question: Do they have the same intended meeting spot? The answer is no. They have different intentions and because of that, they do not have a limit.
Limits at Infinity
Limits! More and more limits! At the infinities, some funny things happen to our functions. Every time we examine a limit going towards the infinities, we are really looking the end behaviors. Is a function growing or not growing? Are they approaching a set value (asymptote)? Remember that the strategy here is to divide by the highest powers to simplify the function if the function is rational. Additionally, remember that may be interpreted as zero.
Sandwich Theorem a.k.a. Squeeze Theorem
The sandwich theorem is a very general idea that is very no-brainer, but the theorem still needs to be stated. Anytime you have three functions; and , and , never has any choice but to be between and . In this scenario, our function really has no choice as to where it wants to be. Now, in talking about the limits of the functions, we get the following generalization. If , poor has no alternative and submit to the notion that must also be .
From the sandwich theorem, there are two properties that come out as “vital, you must remember these.”
Nonzero/Zero, Zero/Nonzero, Zero/Zero, and Infinity/Infinity
These four topics usually drive people batty and nuts because they are so similar and yet so drastically different from each other. A quick review:
Nonzero/Zero à Bad. Nothing may be divided by zero. If you get this as a result, the limit does not exist. Just imagine being told that you have to give out ten dollars to people before you get home and you are dropped off on an abandoned island. You’re pretty much not going anywhere. is often also called “undefined” because this cannot happen.
Zero/Nonzero à Good. This result is nice and conclusive; you get a zero.
Zero/Zero and Infinity/Infinity à We don’t know. This form is called an “indeterminate” form because they provide us very little information about the limit. We generally need to do more work to simplify the limit before anything meaningful can be determined.
Limit Strategies
The following are all valid strategies that you will use when determining limits. I have them listed in order from “easiest” to “the last resort.”
1.) Substitution – Plug in the number you are finding the limit at, pray you get a number or a nonzero divided by zero as a result, celebrate.
2.) Factor and Simplify – If substitution failed, then try factoring the problem out. Usually when you factor, you will find factors in the numerators and denominators that simplify each other out of existence. Upon this being successful, substitute the number you are finding the limit at and pray again that you get a number or a nonzero divided by zero as a result. If true, celebrate. Else, go to 3.)
3.) Conjugates – Conjugates provide the most involved method of determining limits. Quite differently from Algebra II and Pre-Calculus where you had to use the conjugate of the denominator only, you have the option of using the conjugate of the numerator as well. The ultimate goal of multiplying by the conjugate is to create the scenario where you can simplify factors in the numerator and denominator again. Afterwards, substitute again and you should get your hard earned value.
4.) L’Hospital’s Rule – or also known as L’Hopital’s Rule is a shortcut for the indeterminate forms you find in limit problems. Anytime you have an indeterminate form, take the derivatives of the numerator and denominator without using the quotient rule. With this result, substitute the -value in and you will have your limit. If you get another indeterminate form, re-differentiate.
Continuity
Continuity implies no breaks in the graph. Another way to consider and think of continuity is how possible it is to draw a graph without lifting up your pencil. If you can, it’s continuous. If you can’t, then it obviously isn’t.
Three Conditions of Continuity
There are three conditions of that must be upheld in order for a graph to be continuous at a point.
1.) The limit must exist at the point.
2.) The point must exist.
3.) The limit must equal the point.
Continuity story time: Let us say that a family is going to the circus.
1.) The family must want to go to the circus. (limit exists)
2.) The circus must be in town. (point exists)
3.) The desire of the family to go to the circus must coincide with the circus being in town (limit = point)
Removable versus Non-Removable Discontinuities
Not all discontinuities are the same or built the same way. Some of them are what we call removable while others are non-removable. The difference is dependent on whether you have an asymptote or a hole at the point.
Removable Discontinuity – A removable discontinuity exists whenever you have a limit type problem where you could factor parts of the numerator and denominator and then simplify them. These are the types of problems where you have a hole in the function. The reason they are called removable is because you can plug in a temporary point into that hole to turn it into a function and remove it later.
Non-Removable Discontinuity – These usually occur when you have either asymptotes or piecewise functions. Quite literally, Non-Removable discontinuities exist when a function has a left and right hand side of a value that do not match.
Derivatives
Definition of the Derivative
Main Definition:
Alternate Definition
Derivatives and Continuity
Differentiability implies Continuity
Continuity does not imply Differentiability
UCLA AP Readiness 2009