Not far to go now! Only 1-1/2 weeks left in the term, and we have one project, one test, and one final exam.

The most recent quiz was returned. I hoped it would go better than it did ... maybe it was just a product of lots of things due in other classes near the end of that week.

From now until the end of the semester, we'll discuss applications of integrals. Today's wasn't too hard. Our very first application was to find the area under a curve for a positive function. We can now expand to find the area of a region between two curves. By way of Riemann sums, we found that if f is the function for the top curve and g the function for the bottom curve, then the area of the region between them from a to b is given by



Sometimes, we're given all the information we need to set up this integral. Most times, we will have to discover the boundaries ourselves, as in the parabolas example we did in class. Usually, we just set the two functions equal to each other and solve for x to get our a and b values if they're not already given. One way to think of what we're doing is as if a laser scanner is being run across our region from left to right, measuring the length of its beam as we go... sometimes, as we saw, it's easier to run the scanner from bottom to top instead of from left to right.

Wednesday, we move on to computing volumes of particular types of solids using integration.

I hope everyone had a great Thanksgiving break and I hope you don't have to run as far as I do to counterbalance the amount you ate! In case you've forgotten, I am Professor Poodiack and this is the blog for your Calculus class.

Before the break in Calculus I, we finally finished up a long example started the previous Friday. (Actually, the class finished it while I enjoyed a hot steaming cup of coffee.) We then established some properties of definite integrals, some of which were just like properties of indefinite integrals.

Finally, a lot of hemming, hawing, and teasing, came to an end as we started talking about the Fundamental Theorem of Calculus, Part I. (Most textbooks refer to this as part I, even though MathWorld refers to it as the "second theorem." Maybe Wolfram ranks in terms of amount of use.) After slogging our way through antiderivatives -- indefinite integrals -- and Riemann sums for definite integrals, we're convinced that the similar names are not a coincidence.

The Fundamental Theorem of Calculus explicitly states the links between derivatives, antiderivatives, and definite integrals. In fact, Part I shows that essentially definite integrals and derivatives are inverse operations:

Fundamental Theorem of Calculus (Part I): Let f be a continuous function on [a,b]. Then the function



is continuous on [a,b] and differentiable on (a,b). Moreover,

Another way to say this is by way of Liebniz notation:


In other words, (kinda sorta) the derivative of the integral of f leaves us with f. In reality here, what we have is another derivative formula.

Before we broke, we discovered the ultimate link between derivatives, antiderivatives, and definite integrals by way of the fundamental theorem of calculus, part deux. After a couple of days of working to find limits of Riemann sums, we were ready for (and deserved) a shortcut.

Fundamental Theorem of Calculus (Part II)

Let f be a continuous function on the interval [a,b]. If F is any antiderivative of f, then



This is an incredibly deep result because it ties together a purely algebraic object (the antiderivative, or indefinite integral) with a geometric object (the definite integral, visualized as a sum or difference of rectangle areas).

We decided first off that since the function F can be any antiderivative, we'll always choose the one with constant of integration 0 (so we don't have to worry about the "+ C" part). Also we have some new notation:



So, for example, we can compute the area under the curve from x = 0 to x = 1 as



Our quest to reverse differentiation will now turn into finding ways of reversing well-known rules. The first one we're trying to reverse is the Chain Rule.

One way of handling the Chain Rule was by setting a new variable u to be the "inside" of our function -- a substitution, if you will. We will try the same trick. If we have a difficult integration to perform, we will try setting a new variable u to be the "inside" of the most difficult part. This substitution will hopefully transform our difficult integral into one of the ones that from our basic table of antiderivatives.

For example, to calculate , we set . This means that and



If we were, say, missing the factor of 3 in the differential, we'd have to compensate by inserting a factor of 1/3 when we switch the integral over.

For definite integrals, we have two ways to go. We can find the antiderivative as a side calculation. For example, we have



We can also change the limits of integration as we go and not have to worry about changing back to x's at the end. For the above example with , we know that if x = 0, then u = 1 and if x = 2, then u = 9. Therefore,



We'll start working on applications of integrals in Chapter 6. See you Monday.

The solutions to Quiz #8 are available from the Test and Quiz Solutions page.

Thursday in Calculus, we introduced an object called a definite integral of f from a to b as the limit of Riemann sums for f, where unlike the area formulation we used yesterday, f does not have to be nonnegative. That is,



where f is a continuous function on [a,b], the 's are sample points for the i th subinterval, and is the width of that subinterval. If f is nonnegative on [a,b], then the definite integral equals the area under the curve from a to b. If f becomes negative somewhere in that interval, the integral equals the area above the x-axis minus the area below the x-axis. If f is a velocity function, we saw today that the definite integral will give the exact distance traveled between time a and time b.

It is quite difficult to compute definite integrals as limits of Riemann sums. For example, suppose with a = 0, b = 3, and n = 6 right rectangles. Visually, here is what we have:

Pasted Graphic 5

We can see that there is more area below the x-axis than above, so we expect a negative answer. (We get - 3.9375 in fact.)

We figured out that for known areas (circles, rectangles, and triangles), we can find the value of certain definite integrals really easily, without having to deal with rectangles, sums, and limits. For the unknown areas, we have to go back to that technique.

On Monday, we'll find an easier way to compute the definite integral.

This week in MA121 we seemed to veer off the path we were on. Instead of continuing with antiderivatives, we talked about how to compute the area under a curve.

The idea is this: Suppose we have a curve between x = a and x = b, and that in that interval. We're going to try to compute the area of the region under our curve but above the x-axis. In most cases, we won't have a region whose area fits one of our well-known formulas (rectangle, triangle, circle). So in the typical calculus manner, we're going to sneak up on the answer.

We do this by overlaying the region with rectangles, the region whose area formula is the easiest we know. Formally what we do is split the interval from a to b into say, n, subintervals by picking partition points so that



(In practice we usually pick the points to be equal distances apart, but you don't have to. Sometimes in numerical integration it's advantageous not to do so.)

So we have these subintervals . Let's call the width of the first interval , the width of the second interval , etc.

In each of these subintervals, we pick one point called a sample point, where we're actually going to compute the value of the function. Let's name these sample points .

Now we draw rectangles with width and height . This means the area of each rectangle is . We can approximate the area under our curve by adding up the areas of all the rectangles we've drawn. That is,



This sum is what is called a Riemann sum. It will come up quite often in defining quantities for us down the line.

We also noticed that when we use more rectangles, our approximation is better. So we surmised that


as long as the limit exists. (Here and above, we have used sigma notation, which represents long sums of numbers.)

We will examine this sum in more detail on Friday. See you then.

Believe it or not, we know most of what there is to know about differentiation. It is useful in calculus to be able to reverse the differentation process. Given a function f, can we find a function F such that ? (For instance, knowing the velocity of an automobile may help us find the overall position function of the car if we know where the car was at one particular moment.)

Such a function F is called an antiderivative (or indefinite integral) for f. Because of the rule we learned at the end of the previous section, we can write the most general antiderivative of f as



The number C is any real number and is given the name the constant of integration. The above is called an integral equation.

For example, we have that since .

Any derivative equation we can write can be rewritten in integral form as above. There is a small integral table on the hard card in the back of your book. We learned about how to reverse the Power Rule (when our power is not -1), and about finding antiderivatives for the natural logarithm, exponential, and basic trigonometric functions.

One type of reversal we learned was how to reverse the General Power Rule and the Chain Rule. The idea is to take a difficult integral and write it as one of the ones on our basic integral table. For example, if we let u = sin x, then du = cos x dx and we could transform the following integral.



One thing to remember here is not to proceed with the integration until the entire integrand is expressed in terms of u. (No x's can remain.)

We will see the usefulness of this in Chapter 5 and a connection to computing area of a region in the xy-plane. We'll work on it after Friday's exam.

I'm behind in my lecture blogging, so here's a quick attempt to catch up before the exam on Wednesday.

The section on the Mean Value Theorem is transitional for us. On the one hand, it continues in the vein of figuring out what the derivative says about the graph of a function f. On the other hand, we are getting towards the point of wondering what we can say about an unknown function if we know some things about its derivative.

The Mean Value Theorem (MVT) is simply Rolle's Theorem with the restriction on the function values at a and b removed. In that case, then there is a number such that



(Informally, the Mean Value Theorem is a "rotated" version of Rolle's Theorem. See Section 4.6.) That is, there is a point c where the tangent line is parallel to the secant line connecting the endpoints. An application comes with the fact that a car that has traveled 120 miles in 3 hours must have been going precisely 40 mph at some point in its journey.

Two applications that pave the way to the future for us are:

• If the derivative of a function on an interval (a,b) is zero everywhere on (a,b), then the function is constant on (a,b).
• Two functions with the same derivative must differ by only an additive constant (the "+C").

We will use this last fact to introduce antidifferentiation.

Thursday in Calc I, we spent some time using everything we've learned in Chapter 4 to develop a technique for graphing a function from beginning to end.

The main things we examined were:

• Domain -- the set of "good x"s for our function. This helps us set up a general locale for our graph.
• Intercepts -- where the graph hits the x- and y-axis.
• Symmetry -- whether the function is even (; graph is symmetric about the y-axis), odd (; graph is symmetric about the origin), or none of the above. If there are trig functions involved, we should check whether our function is periodic.
• Asymptotes -- vertical asymptotes generally occur where the denominator is 0 (but not the numerator); horizontal asymptotes are lines the graph follows arbitrarily closely on the far left or far right. Depending on the function, we may need to check for slant or oblique asymptotes.
• Intervals of increase and decrease -- take the first derivative and check where it's positive or negative.
• Local maxima or minima -- Use the first derivative test to examine critical points found in the previous step.
• Concavity / inflection points -- Take the second derivative and see where it's positive or negative. The inflection points will be the points in the domain where the second derivative changes sign.

This gives us enough to come up with a decent graph. If you don't feel like you have enough information to go on, plug and chug a few points.

We did two long examples on Thursday. Friday, we took a quiz. Monday, we'll move on to the Mean Value Theorem (the third of our "value" theorems).

I don't know how I did it, but I'm off by one example in each class ... I've done one more optimization example in the 7:50 class than I have in the 10:50 class. I think the 10:50 class asked more questions regarding the lab that's due on Monday. I'll have to fix things up this week.

Between last Thursday and Monday, we graphed a fairly non-standard example using the first and second derivatives to fill in some information on our graphs. We've also done two (or three) long optimization examples. One involved a farmer and his fencing, one involved the manufacturing of a metal can, and the last involved a dilemma involving rowing and running. (See in-class handout.)

I will try to catch up the 10:50 class on the rowing-running example on Wednesday and in both classes we will do some more graphing of functions we never thought we could graph by hand. See you then.

On Wednesday in Calc I, we started looking at what the second derivative says about f. As it turns out, the second derivative tells us about the way the graph of f bends as it passes through an interval. A function f is concave up if it is curved like the outside of a right-side-up bowl -- its graph sits atop all its tangents. On the other hand, a function f is concave down if it is curved like the outside of an upside-down bowl -- its graph sits beneath its tangents. What we found is that f is concave up on an interval if and only if for every x in that interval and that f is concave down on an interval if and only if for every x in the interval. The points where the second derivative changes sign are called inflection points.

We then noticed that at a local maximum where the graph passes through the maximum point smoothly, that the graph is concave down at that point. Similarly, if our graph passes smoothly through a local minimum point, then the graph is concave up at that point. This led us to the fact that we could possibly compute the second derivative at a critical point and classify the critical point as a local maximum or minimum based on the result. To wit:

Second Derivative Test: Let be continuous at a critical point c:

• If and , then we have a local minimum at c.
• If and , then we have a local maximum at c.

If doesn't exist, or if doesn't exist, or if , then we must revert to the First Derivative Test.

We looked at a "kitchen sink" example, where we could use the Second Derivative Test at one critical point, but not another, and we came up with a full graph. We'll do a lot more of this in Section 4.5.

On Thursday, we'll start looking at optimization problems.

Monday in Calc I, we started examining what derivatives could actually tell us about a function. The long-term outlook will be to (1) draw better graphs than we used to in our Precalculus classes and (2) use the techniques to solve applied problems.

We started by looking at what could tell us about the original function f. We quickly came to the idea that if for all x in an interval, then the function f is increasing on that interval, and if for all x on an interval, than f is decreasing on that interval. (A function that is solely increasing or decreasing on an interval is said to be monotonic on that interval.)

We looked at an example or two of how to determines intervals of increase or decrease. (Such intervals are always open, by the way.) We noticed along the way that the critical points acted as boundaries for the intervals of increase and decrease, and also that if changed sign while passing through the critical point, that critical point could be classified as either a local maximum or a local minimum. This was the essence of the First Derivative Test. (Note that if doesn't change sign while passing through the critical point, then the critical point can be rejected as a possible local extremum.)

On Friday in 121, we continued our discussion of ways to find absolute and local extreme points. Our biggest problem was that we had lots and lots of points to sift through. But we found a trend on Monday that local maxima and minima turned up where the graph of had a "turnaround" of some kind, either a smooth one or a sharp turn. The smooth ones, where are called stationary points, and the sharp turns, or "cusps," occur at singular points, where doesn't exist. Together these two types of points make up the critical points for f. The critical points are the candidates for locations of local maxima and minima. In fact, if a function has a local maximum or minimum at c, then c must have been a critical point. (Note that all critical points must be in the domain of f.)

We then looked at a couple of examples. We saw that in the case where can be written as a fraction, the critical point question becomes one of when the numerator is 0, and when the denominator is 0.

We then applied this technique to the Extreme Value Theorem and came up with a procedure to find the absolute maxima and minima of our function f on a closed interval [a,b]:

1. Find the values of f at the critical points inside the interval [a,b].
2. Find the values of f at a and b.
3. The largest of the above values is the absolute maximum value on [a,b]; the smallest is the absolute minimum value.

We'll discover next week how to classify critical points as maxima, minima, or none of the above.

My blogging time has been eaten up by a variety of things lately, but I'll see if I can summarize two concepts in one entry!

We looked at the idea of using the tangent line to a function curve at a point of interest as a tool for approximating the function for points near the point of interest. The idea of the "point of interest," by the way, is that we know everything that goes on at that point exactly. Let . The line tangent to this curve at the point is given in point-slope form as:



If we solve this equation for y, we get what's called the linearization of f at c:



L(x) is an approximation to the actual value of f at a value x near a, and this gives us the linear approximation for f near a:



We did a couple of examples where we could estimate function values really easily -- yes, easily because all we were evaluating was a linear equation.

Another way to phrase the same process is through differentials. Differentials in a variable can be thought of as "a small change" in that variable. If y depends on x, what we want to know is how a small change in x affects y. We will define some actual changes first. Let be the actual change in x and be the actual change in y. If , perhaps the function f is quite hard to calculate in general. Let's assume we know the exact value of f at x, say. The actual change in y is

,

may be difficult to calculate exactly. Let's define the differential of x, . This differential is an independent variable; we can set it to any value. We will define the differential of y to be



This differential is a good approximation of and usually much easier to calculate. (See Figure 2 on p.180.) We can also use dy to come up with a good approximation for f near x, since



We did a quick example on error propagation. The basic idea is that if we make an error in measurement of a quantity (as we're likely to do because we're human), that error will go forward if we use our measurement to calculate another quantity, such as area or volume. We can estimate the maximum error in our calculated quantity by using differentials as above.

We then began looking at ways to determine various high and low points for a function. We defined absolute (global) maxima and minima and examined some graphs. Absolute extrema represented overall high and low points graphically. Not all graphs have either one, but we did determine one situation that guarantees the presence of absolute max's and min's.

Extreme Value Theorem: If f is a continuous function on a closed interval [a, b], then f attains an absolute maximum and an absolute minimum value somewhere on [a, b].

We'll look at criteria for discovering local and absolute extreme values in the cases where a graph might be difficult or impossible. See you next week.

Monday in Calc I, we introduced our last new types of functions for which we needed to find a derivative. These were the exponential and logarithmic functions. Here's a slightly different approach from the order we used in class.

We began with the natural logarithm function. We had to go back to first principles -- definition of the derivative -- to find that



We saw several examples of how to use this new rule in conjunction with existing rules. The Chain Rule version of this was the most common variation. If u is any function of x, then we can write



If we were looking at logarithms in another base b with , the situation wasn't too difficult. By the change-of-base formula, we have that



That expression ln b is constant, so when we take the derivative we can just move it aside. Thus



To get the derivative of the exponential function, we use the inverse relationship between the exponential and natural logarithm functions, as well as a little implicit differentiation:



That is, !!! (This is the only function, aside from 0, that is its own derivative).

In its Chain Rule version, again assuming u is a function of x, we have



We looked at a couple of examples of this. For exponentials in other bases, we used the fact that to write



We then talked about logarithmic differentiation. The idea behind logarithmic differentiation is this: taking the logarithm of a quantity generally allows us to take apart the quantity, splitting it into its simple pieces. Then we take the derivative of both sides. Implicit differentiation gives us that and thus a way to get the derivative alone. The procedure is:



and since we know what y is, we have our derivative.

Wednesday and Thursday, we talked in lecture about related rates problems. The idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.

It is impossible to cover the entire spectrum of possible problems. We need to remember (or have access to) a lot of geometry formulas to be successful at these. Many of the most common ones can be found in the stiff page near the back of your book.

A basic strategy (stolen from Stewart's 3rd edition) is the following:

1. Read the problem carefully.
2. Draw a diagram if possible.
3. Introduce notation. Assign symbols to all quantities that are functions of time.
4. Express the given information and the required rate in terms of derivatives.
5. Write an equation that relates the vatious quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables.
6. Use the Chain Rule to differentiate both sides of the equation with respect to t.
7. Substitute the given information into the resulting equation and solve for the unknown rate.

A common error is to substitute the given information too soon. If the variable related to the information changes its value during the problem, you must wait until the end, after you take the derivative, to substitute the value.

I will post on Wednesday's lecture after I say a few more things on Thursday!

Monday in 121 we introduced a kind of expansion of the derivative-taking we've been doing. This differentiation works to give us the instantaneous rate of change in the case that our curve is not the graph of a function, but of an equation containing x's and y's. Since y is not given as an explicit formula involving x, we say that y is implicitly understood to depend on x (but we can't say exactly how). Finding the derivative in this case is called implicit differentiation.

The technique itself is not difficult: We take the derivative across the entire relational equation. Every time we touch an x-term, we take the derivative as usual. Every time we touch a y-term, we have to tack on a due to the Chain Rule. Getting the hang of this seemed within most folks' grasp -- we often just have to decide when and where to invoke the Product, Chain, or Quotient Rules. The difficult part can often be the algebra involved in solving the resulting equation for .

We had an example on an equation of a circle to show that the new method and old method agreed; we also did an example on the Folium of Descartes, where there was no way to find the needed tangent line slope via the old methods.

Tomorrow we work on applications from implicit derivatives.

Friday in Calc I, we started our discussion of higher derivatives by treating them as a mechanical exercise, something we did just because we could, and we introduced notation.

We provided motivation by appealing to an application. If is the position for an object at time t, then one of the motivations for finding the derivative was that it represented the velocity . The acceleration is then defined to be the instantaneous rate of change of the velocity. That is,

.

Using Leibniz notation can often help us keep track of units. We can write

.

So, for instance, if s is in meters, and t is in seconds, the notation for v clues us in that v is in m/sec, and we can think of a as being in (m/sec)/sec or the equivalent m/sec².

We did a fairly quick example involving position, velocity, and acceleration. We then moved on to generic higher derivatives. We can notate the nth derivative as



We saw that for s, a position function, the third derivative is called the jerk, the instantaneous rate of change of acceleration. We also saw that for f, an nth degree polynomial, if we take more than n derivatives, we get a value of 0. (That doesn't happen in general.)

Usually to compute, say, the 256th derivative of a function, we have to compute the 255 derivatives that came before. Sometimes, we can get lucky and get an explicit formula for the nth derivative in general. We saw this with .

Monday we talk about how to find rates of change along graphs that aren't graphs of functions. See you then.

(My apologies to Sam Cooke. Look it up.)

Thursday, we talked about a way of taking derivatives of compositions of functions. Suppose where f and g are differentiable functions. Then the composed function F is differentiable and



That is we take the derivative of the "outer" function f and multiplying by the derivative of the "inner" function g. This is known as the Chain Rule.

For example, if we want to take the derivative of , we can think of the outer function as being "something to the 100th power." So the derivative should start off with "100 times the same something to the 99th power." We then multiply by the derivative of the inner function; that is, . So we have that . (In the special case where the outer function is a power function, the Chain Rule is often called the General Power Rule: .) It certainly beats writing 100 rows of Pascal's Triangle in this case!

We looked at another example or two of the Chain Rule. Remember what Prof. LaVarnway says: "Not a day goes by that you don't use the Chain Rule!"

On all of these the difficulty comes from (1) remembering the rules and (2) finishing the algebra in a satisfactory manner. We've been lax about algebra up until this point, but we will now tighten up for the following reason.

If you do your homework, you often gauge how well you do by how often you agree with the answer in the back of the book (assuming that the right answer is there, which it most assuredly is not sometimes). Many times, we have the correct answer, but we don't know it because the book's author has performed an algebraic trick or two to get the answer as compact as possible. Therefore, I'm going to ask that you work on the algebraic aspects of your work as much as possible.

We'll learn about higher order derivatives and their meaning on Friday and take a quiz. See you then.

Wednesday in Calc I, we worked out formulas for the derivatives of the six trigonometric functions.

This lecture was an exercise in how and when to work from first principles. The trigonometric functions are not algebraic, so we can't get anything new from using, say, the power rule. We had to pick one of the two base trigonometric functions (in this case, we picked ) and worked from the definition of the derivative. After a few minutes -- using identities we saw in section 1.8 and limits we saw in section 2.5 -- we quickly got our derivative.

But once we saw how to do this, we could quickly get a derivative for cos x.

For the other four trig functions, we noted that they are all defined as quotients of sines and/or cosines. Therefore, we didn't have to go back to the actual definition of the derivative to work out the needed formulas, but to the Quotient Rule. We did a full calculation for and noted the others would be the same. What we found was:



These formulas can be used in the midst of the other rules we already know, like the Product Rule and Quotient Rule.

On Thursday, we'll begin talking about the Chain Rule -- not a day will go by for the rest of your lives when you won't use the Chain Rule. See you then.

Today we talked more about derivatives. Now that we're getting to "shortcuts," we should be able to differentiate a lot of different functions.

We added three fairly simple rules to our toolbox.



The fun began when we looked at the derivatives of products and quotients. From very simple examples (e.g. ), we saw that the derivative of a product is not the product of the derivatives, and the derivative of a quotient is not the quotient of the derivatives. What we got instead was:



These can be tough to memorize, but the most difficult part of using them can be the algebra in simplifying.

We'll look Wednesday at derivatives of trigonometric functions. We'll have to go back to the definition of the derivative once more for this. You'll live. See you Wednesday.

This week has been brutal for finding time to blog for the class. I hope next week will be better.

We had noticed that the expression



kept appearing in various examples as a solution. We gave this expression its own notation, , and called it the derivative of f at a.

Computationally, this limit (if it exists) takes a lot of algebra to compute -- sometimes it can be quite difficult to do so. Conceptually, we can interpret the derivative at a point as either the slope of the tangent to the curve at a, or the instantaneous rate of change of f at a. For instance if f represents a position function for an object, the derivative at a would give the value of the object's instantaneous velocity there.

We also noted that the derivative of f at a results generally in a function of a, so we can think of the derivative as a function in its own right:



At long last, Friday was the day that our calculus "veterans" have waited for ... the revealing of the Power Rule.

We first noted that any function that is differentiable at a point a is also continuous at that point. However, the converse is not true: the function is continuous at 0 but not differentiable there. A function will not be differentiable at a particular point if there is a sharp corner there (as in the previous example), a discontinuity of any kind, or a vertical tangent line (e.g. at x = 0).

We took a moment to examine alternative notations for the derivative that our book uses frequently.

We then began developing formulas for derivatives of common functions. Using the definition of the derivative, we immediately saw that the derivative of a constant function is zero. Using our knowledge of linear functions, we saw that .

We then looked at several examples of derivatives of power functions. With the help of Pascal's Triangle and some slick factoring, we found that we could state the



The population rejoiced!

Let's see ... this week on Monday, we began exploring Mathematica and on Friday, we took our fourth quiz.

We'll figure out some other derivative rules on Monday. See you then.

Wednesday in Calc I, we introduced two problems that seemed to have nothing in common, but whose solutions used exactly the same method.

The tangent problem asked the question of how we can compute the slope of a curve at a particular point, since the slope changes from point to point (unlike with a line). We can reduce the problem to finding the slope of a particular line -- a tangent line. This is a line that (at least locally) only touches a curve at our point of interest. BUT ... we only know one point on the tangent line, and we generally need two points to determine a line's slope.

Another problem is one of velocity. Can we determine the velocity of an object at a particular instant? Normally we think of velocity in terms of the formula: (change in displacement) divided by (change in time). The problem is that time doesn't change during an instant.

It turns out we can use limits in both cases to sneak up on an answer. (Limits are a great way to sidle up to an answer when our usual "full-frontal assault" technique won't work.) Given a function f and a value of interest, a, we can figure out both the slope of the tangent line at x = a and the instantaneous velocity by way of the expression



By itself, the quotient represents both , the slope of the secant line between the x-values a and a + h and the average velocity over the time interval . It's by making the difference in the x-values -- or the length of the time interval we're measuring -- go to 0 that we achieve our goal.

We looked at a couple of examples for the tangent problem (). I then commended to you the job of reading up on examples of the velocity problem.

I was supposed to have talked about the Intermediate Value Theorem ... remembered in the 10:50 class, but forgot in the 7:50 class. I'll write about it when I've updated everyone.

See you at the test on Friday.

Monday we talked about the idea of continuity. A function f is continuous at a point c if



This entails more than we might think. For this equation to hold, we need (1) the limit on the left-hand side of the equation to exist; (2) the function f to be defined at c and; (3) both results to equal each other.

A function that is not continuous at c is discontinuous at c. There are a few types of discontinuities, which we classify based on how seriously the function is "broken" at the point of interest. The function has a vertical asymptote at x = 0; the discontinuity there is essential (that is, unfixable). Another unfixable discontinuity is a jump discontinuity. The greatest integer function has a jump at every integer and has discontinuities there as well.

There are, however, discontinuities that can be fixed. These are called removable singularities. Graphically, these show up as open holes, places where the function would take on the indeterminate value 0/0. For example, the function



is discontinuous at x = 2 since f is undefined there. However, the limit as exists:



So we can "fix" our function f (i.e., remove the singularity) by re-defining:



On Wednesday, we'll talk about the Intermediate Value Theorem -- one of three "value theorems" we'll encounter -- and move on toward derivatives. See you then.

On Friday, we went over sections 2.5 and 2.6, which discussed limits involving the transcendental functions: trigonometric functions and exponential and logarithmic functions.

The good news was that all of these functions, as long as we stay away from any "bad" x-values, work with our "plug-in" method for limits. We learned about two special limits involving the sine and cosine functions as we approach x = 0. We also looked at the limits at plus and minus infinity for exponential functions, as well as the limit at infinity for logarithmic functions (assuming the base is greater than 1). Logarithmic functions have a vertical asymptote at x = 0, so they have a limit approaching 0 from the right.

We took a look at the most commonly used base for exponential and logarithmic functions, the natural base e.



We found that

,

a fact which we sort of learned back in the days when we looked at compound interest problems.

We'll talk about continuity on Monday.

Thursday in Calc I, we answered lots of homework questions on limits. We then took a brief look at limits involving the trigonometric functions. The good news was that all of these functions, as long as we stay away from any "bad" x-values, work with our "plug-in" method for limits. We learned about two special limits involving the sine and cosine functions as we approach x = 0. Then we took the quiz.

Tomorrow, we'll talk about exponential and logarithmic functions with regard to limits. We'll see you then.

Wednesday in Calc I, we fully explored limits involving infinity. First up was the concept of a limit at infinity. If "the limit of f (x) exists as x goes to infinity equals L", this means that we can make the value of f (x) as close to L as we like by taking x sufficiently large. (A similar definition exists for "the limit of f (x) exists as x goes to negative infinity equals L," with the change being that we end with "... taking x sufficiently large negative.") We saw that any line y = L for which L was such a limit was a horizontal asymptote.

We looked at examples involving algebraic functions, using the trick of dividing all terms in our function by the highest power of x in the denominator of the function as well as the fact that the limit of 1/x ^r = 0 for any positive r as well as for any negative r for which x ^r exists.

We then examined function limits whose values are infinity or negative infinity. Visually, the associated x-values correspond to vertical asymptotes on the graph of the function.

On Thursday, we'll talk about limits involving trigonometric functions and start dealing with limits of exponential and logarithm functions. Plus we'll take the third quiz.

We'll see you then.

Today we answered a lot of homework regarding the delta-epsilon proofs for limits, as well as some regarding how to work with limits of functions involving the indeterminate 0/0 (factor, or expand, followed by cancelling).

We looked at the Squeeze Theorem (sometimes called the Sandwich Theorem, or Pinching Theorem) as a way of getting limits when the "product rule" for limits won't work. We then introduced the idea of infinite limits and limits at infinity.

We'll continue the discussion on Wednesday.

This will be a short entry due to pressing time constraints!

Today we talked as briefly as possible about the formal definition of a limit. (Deltas and epsilons ...) We went over the proof of why the limit as x approaches 3 of 4x - 5 equals 7, and how the deltas and epsilons quantified the idea of "closeness." Then we took the second quiz.

On Monday, we'll finally get to those shortcuts on limits. See you then.

This will be short due to pressing time constraints!

Today in 121, we looked at the ways limits can screw up. These include the presence of jumps in the graph, vertical asymptotes, and too much oscillation. We could also have a limit not exist if the left-hand and right-hand limits both exist at our point but don't agree. Then we began talking (as briefly as possible) about the formal definition of a limit.

Tomorrow we will finish our look at the formal definition of limit and prove the existence of a couple of easy limits. Then we'll start talking about "limit shortcuts and rules." You'll have earned it by then.

We spent today talking more about limits. We showed how to make tables of function values as we approach a particular x-value of interest. We noted that for the polynomial, we could have just substituted the x-value of interest and gotten the right limit value. It turns out that this approach always works for limits involving polynomial functions.

We can often try the same technique on limits for rational functions, as long as we stay away from "bad" x's. If we approach an x-value that makes the denominator zero, our only hope is to factor the numerator and denominator of our rational function and hope for some cancellation. If we cannot both factor and cancel, the limit will either not exist or be infinite.

Thursday, we'll talk about the ways limits can screw up and the formal definition of limit. We'll also prove the existence of a couple of easy limits. See you then.

Today was the end of the "review period." We answered some questions on the homework from Section 1.6 and then began talking about the difference between algebraic functions and transcendental functions. Two types of transcendental functions are the exponential and the logarithmic functions. These are discussed in Section 1.7. We reviewed the relevant properties of each.

We then briefly recalled the trigonometric functions. We recalled the SOHCAHTOA version of sine, cosine, and tangent, and then expanded to the unit circle. Using the circle, we were able to show the periodic properties of sine and cosine.

On Wednesday, we move into Chapter 2, talking about limits of functions.

Today we kept the lecture short and sweet due to the quiz. We reviewed the idea of inverse functions, functions that "reverse" the action of a given function. We recalled that the functions with an inverse were exactly the one-to-one functions. Just as a graph is the graph of a function if it passed the vertical line test, a function is one-to-one if and only if its graph passes the horizontal line test. (Those are the tests that say a graph is the graph of a function (a one-to-one function) if no vertical (horizontal) line touches the graph in more than one point.)

We also recalled how to find a formula for the inverse of a one-to-one function. (It can't always be done due to the algebra difficulty.) We also recalled how to graph an inverse, even if we don't have a formula in hand.

Then we took the quiz. Friday, we'll talk about exponentials, logarithms, and maybe even trig functions.

Today in Calculus I, we talked about how to make new functions from old ones. We did this in two ways. Graphically, we recalled all the ways we can transform a graph through shifting, stretching, and squashing. We then talked about how adding, subtracting, multiplying, dividing, or composing two functions works and how to find the domains of such combined functions.

Thursday, we'll talk about inverse functions, and recall exponential and logarithmic functions.

Monday in Calculus I we went over the syllabus, homework schedule, grading policies, and expectations for the semester.

We then talked about a list of "Things I Think You Know," which included the topics from Sections 1.1 through 1.5 of our book. If you haven't looked those sections over to check for knowledge gaps please do so before the end of add/drop.

The goal for Wednesday will be to go over the material in Section 1.6 on algebra of functions, and possibly start Section 1.7 on exponentials and logarithms.

See you then!