The link to the extra problems for Homework 10 was omitted above. This has been corrected.
After it was pointed out that the previous grading policy shortchanged students who had performed particularly well on the second exam, I decided on a last grading change. The grade on your second midterm will now be recorded as if the total were 80 points, not 100. Thus, in the previous example, if you scored 85% (i.e., 68/80) on the first midterm and 78/100 on the second midterm, then your grade on the second midterm would be recorded as 78/80, or 97.5%. Thus, a score of 91/100 on the final would replace the first midterm, giving recorded scores of 91%, 97.5%, and 91%. If, instead, your scores were 70/80, 40/80, and 87/100, then your recorded grades would be 87.5%, 87%, and 87%. I believe that this accurately reflects the achievement of people who did particularly well on the second (or first) exam.
I announced a change in the grading today. The grade for the final will now be allowed to replace one of your midterm grades—whichever one would be most advantageous. That is, if you scored 85 on the first midterm, 78 on the second midterm, and 91 on the final, then your recorded grades would be 85, 91, and 91, respectively.
In today's class, we defined derivatives and finished our proof of the limit law that states that the limit of a product is the product of the limits (if both limits exist). The proof involved both a clever algebraic manipulation (adding and subtracting a term to allow unexpected factorisation) and the realisation that the limits statements occurring in the hypotheses can actually be viewed as ‘machines’ for transforming εs into δs.
We did not prove the corresponding law for quotients, but the trick is similar (though slightly more involved). Though the quotient limit law is not immediately available to compute derivatives, we can sometimes perform some algebraic trickery that will bring derivatives into a form computable by the limit laws. We illustrated this by computing the derivatives of our familiar ‘proving ground’ functions, namely constant functions, the identity function, and the squaring function. The next class of functions of interest, the trigonometric ones, are covered on this week's homework.
In Thursday's class, we will finish up our treatment of calculus and, time permitting, move on to the number theory results discussed in the class notes and Chapter 5 of the text.
In today's class, we proved that our calculation (from the end of last Thursday's class) of the limit of sin(x)/x as x → 0 was correct by proving the squeeze theorem. We then put this calculation into a general context by interpreting sin(x)/x as a slope of a certain line (a secant line) joining two points on the graph of y = sin(x). The resulting limit is (by definition) the slope of the tangent line to the graph of y = sin(x) at x = 0. One can make a similar definition for any real function f at any point x0 in its domain. The resulting limit, if it exists, is called (by visual analogy with the case of ordinary slopes) (df/dx)(x0) or f′(x0). As an aid towards streamlining further limit calculations, we stated the limit laws, a set of tools for figuring out new limits given old ones. We laid the algebraic foundations in this class for the proof in the next class that the limit of a product is the product of the limits (if both exist).
A handout containing more detailed versions of some of the hand-drawn sketches made during lecture will be available tomorrow. For now, I have posted the Mathematica notebook I am using to generate those sketches.
In preparation for the exam, there will be extended office hours this Tuesday only, 4–5:20 PM. The grader has also agreed to return homeworks by Monday, so that they can be picked up during office hours or by appointment. I will answer e-mails every day this weekend, but might not be able to answer them very quickly.
If you believe that you need to take the final exam at an alternate time, please contact me as soon as possible about scheduling a time on Wednesday, April 16. Although I will do my best to accomodate people's needs, only certain conflicts justify an alternate time.
In Tuesday's class, we reviewed the definition of continuity, discussing, in particular, the fashion in which δ, ε, x, and x0 all can (or can not) depend on one another. We also showed, by returning to the example of the function x ↦ x^2 discussed in last Thursday's class, how one can recognise a complete proof of continuity—namely, by finding a value for δ, then showing directly (in our situation, by a chain of inequalities) that it behaves as desired.
Having discussed power functions at some length, we moved on to the trigonometric functions. These have power series expansions—in fact, they are defined by these expansions in an analysis course—but we view them as being defined by lengths associated to points on a circle. We drew a reference circle and, by superimposing many triangles on it, managed to compute enough lengths and areas to obtain some interesting results in today's class.
Today, we used the inequalities from Tuesday's class (and the homework) to show that the sine and cosine functions are continuous at x0 = 0. (This does not show continuity at other points. That is one of the problems on this week's homework.) We also showed that, though the function x ↦ sin(x)/x is not defined at x = 0, it does have a limit there, namely, cos(0). We will discuss why one might compute this limit, and what is the significance of this value, on Thursday's class (after the exam on Tuesday).
Please note an important clarification about the definition of continuity from Tuesday's class. This definition involves a point x0 (informally thought of as a stationary point), a positive number δ (informally thought of as measuring the error in x), and a positive number ε (informally thought of as measuring the error in y). It is vital for the definition of continuity that δ be allowed to depend on x0, as well as on ε. The wording of the definition given on Tuesday and at the beginning of class on Thursday wrongly suggested that δ could depend only on ε.
In Thursday's class, we generalised our notion of continuity to give the ideas of limits, which can sometimes be used to describe the notion of what the value of a discontinuous function at a point “should be”, and of continuity at a point. We then proved that x ↦ c, x ↦ x, and x ↦ x2 are continuous functions (with domain and target the set of real numbers), and discussed the fact that “the δs” in the first two examples, but not in the third, could be chosen independently of “the x0”. (This phenomenon is called uniform continuity.) We will begin the next class by discussing continuity of some trigonometric functions.
I will not have e-mail access in the evenings and early mornings from tomorrow until at least 11 March. I apologise for the inconvenience.
No new homework was assigned today.
In today's class, we formalised our observation (of 22 February) about the behaviour of squares of nearby numbers as the notion of continuity. To do so involved first defining the idea of a function (so that we have some objects whose continuity to discuss!); we gave an informal definition in class, which will be workable for our purposes, but the interested student can consult p. 70 of the text for a formal definition. We then worked from the informal idea of continuity (“nearby inputs produce nearby outputs”) to the formal definition (in terms of εs and δs).
In the next class, we will discuss some other ways to think about this definition, and how one might prove that a given function is (or is not) continuous.