Derivative of e^x

Today we talked about the derivative of exponential functions (e^x). We found out that the derivative of e^x is the same as the function (e^x). We then did examples to apply the different properties of exponential functions and derivatives.

The Derivative of ln(x)!!

Today, we calculated that the derivative of ln(x) is (1/x), and applied it to several problems, incorporating logarithmic ideas AND the chain and product rules!

Classnotes- 11/02/10

Today in class we discussed how to solve complicated intervals by replacing inside functions with "u".

More Reimann Sums

Today in class, We learned how to find the left and right hand sums by using the area of the graph of a function. We also learned some rules for definite integrals and did some examples to use what we learned.

Riemann Sums

Today in class we learned how to do Riemann sums with infinite subintervals. When doing so, you should use the Right Hand Sum because it is simpler. Note: the limit as n approaches infinity of the Riemann sum with infinite subintervals is the same as the integral.

Antiderivatives

Today in math we discussed how to find antiderivatives in differential equations. We then took it a step forward and found the differential equation at a given point, and finally re-introduced Distance f(x), Velocity f'(x), and Acceleration f''(x).

Start of Ch. 4 (10-19-10)

Today we started ch. 4. We talked about antiderivitaves or anti-deferation. Mr. Vishack mentioned that we are going to go through this unit backwards in relation to how we went through deferential equations. He said that first we are going to learn the rules of antiderivitaves and then learn the concept from where the rules come from.

2nd Derivative Test

After returning two old quizzes and giving today's quiz on concavity, I worked two example problems. The first illustrated the use of the 2nd derivative test. It's quite straightforward, but what students sometimes forget is that the 2nd derivative test is used to find relative extrema. Previously, we have used the 1st derivative test; now we have a second method for finding relative minimums and maximums.

The second example for today emphasizes the relationship between the 1st derivative, the 2nd derivative and the graphs of functions. This will be our focus of study for much of the next week.

Tomorrow's quiz will be a review quiz - it will cover some topic that a previous quiz was over.

Notes from today are attached below.

Minimums and Maximums

Today we learned about maximums and minimums. Important points:

-The absolute maximum/ minimum could be the end points of the interval, but the relative minimum/maximum could not (absolute: closed interval, relative: open interval).

-When finding the intervals where a function is increasing/decreasing, make sure the number line is drawn as shown in the notes.

-The relative minimum/maximum can be found when you draw the arrows on the number line, so you can see where the graph would be at its highest and lowest points.

Critical Numbers and Increasing/Decreasing Intervals

We didn't get to maximum/minimum or relative maximum/minimum today; that'll be tomorrow.

Basically, a critical number is a defined point (c,f(c)) on f(x) where f'(c) is zero or f'(c) is undefined. AKA when a function goes from increasing to decreasing or vice versa.

An interval [a,b] is increasing if f'(x)>0 for all x in (a,b)
An interval [a,b] is decreasing if f'(x)<0 for all x in (a,b)

The critical number can be used to find the increasing and decreasing intervals of a function. Make a number line, mark all critical numbers, and plug one number from each section created into the derivative equation to determine increasing and decreasing intervals.

September 23 - Proof by contradiction and MVT

As we learned yesterday, the Mean Value Theorem is essentially Rolle's Theorem generalized so the two points (a,b) do not have to output the same y-value. We did not have a quiz today, but we will on Monday involving the Mean Value Theorem in an "interesting" way. Last night's homework applied the MVT to finding the c value guaranteed by the theorem and to a proof (p.176 #79). This proof, however, involved more algebra than it did calculus.

In class yesterday, we were first introduced to Proof by Contradiction with an example. This method of proof requires that in order to prove a statement, you must show that everything besides that statement is not possible. The difficulty in these type proofs is determining how to approach it and making a supposition to disprove.

Today in class we worked another example of a Proof by Contradiction. The homework tonight includes 3 such proofs.

Notes from today are below.

September 22 - The Mean Value Theorem

The Mean Value Theorem is a generalization of Rolle's Theorem (covered yesterday). In Rolle's Theorem, the function had to have the same y coordinate at the endpoints of the interval. The Mean Value Theorem addresses a function over any interval and essentially says that the average (mean) rate of change of a function over some interval will equal the instantaneous rate of change of the function somewhere in the interval.

You will see some generic problems finding the c value guaranteed by the MVT. We will also work some proofs using the MVT. One mathematical method of proof that you will need to use is called "Proof by Contradiction" and I worked an example of this in class today.

Notes from today are below.

September 21 - Rolle's Theorem

We begin Chapter 3 with Rolle's Theorem. This is the first of several theorems that we will talk about that describe the behavior of what I have previously called "nice" functions. That is, we are better defining "nice."

Rolle's Theorem is actually a silly sort of theorem. Generalized, it becomes a more powerful and important statement about functions; alas, poor Rolle didn't generalize and so didn't get the greater glory that could have been his. (Lagrange would eventually prove the Mean Value Theorem - we will cover this tomorrow.)

Another new concept that we covered today is the idea of differentiability. We have previously talked about when a function is continuous. Now you should be able to describe when a function is differentiable. Intuitively, if a function has a "sharp point" (absolute value, x^(2/3)), then it is not differentiable at that sharp point. Additionally, if a function is not continuous at some x = c, then it is also not differentiable at x = c.

Notes from today are below.

September 8 - The Chain Rule

The Chain Rule is the rule for the composition of functions - that is one function is "inside" another, ie f(x) = a(b(x)). The Chain Rule is an all encompassing rule - you have already been using it though it hasn't been apparent.

The thought process for using the Chain Rule is to think

"The derivative of f(x) will be the derivative of the outside function times the derivative of the inside function."

The trick is that when you differentiate the "outside function," you don't change the function on the inside. That probably doesn't make sense to read, but ultimately, with practice, I think you will come to understand.

Notes from today are shown below and include several worked examples.

September 7 - Product and Quotient Rules continued

As far as the derivative rules go, there wasn't anything new.

A consequence of now having the quotient rule is that we can now differentiate all six trig functions. You are now responsible for each of those derivatives:

d(sin x)/dx = cos x
d(tan x)/dx = (sec x)^2
d(sec x)/dx = sec x tan x

d(cos x)/dx = -sin x
d(cot x)/dx = -(csc x)^2
d(csc x)/dx = -csc x cot x

There is a nice pattern to the derivatives that should make it a little easier to memorize them. Better to have them learned sooner than later.

September 2 - The Product and Quotient Rules

Today we introduced two more rules for finding derivatives of functions. Previously we have seen (seemingly silly) rules for finding the derivatives of sums or differences. The derivative of a function that was defined as a sum (f(x) = a(x) + b(x)) is simply the sum of the derivatives (f'(x) = a'(x) + b'(x)). Similarly for differences.

It turns out the patterns for products and quotients are a little more complicated. I am opting not to show you where the formulas come from, but you are certainly encouraged to look up the proofs in your text.

Notes from class today include the Product Rule, the Quotient Rule and several worked examples.

August 31 - The derivative continued

Today introduced two new concepts - one is fairly minor but the other is more significant.

The first concept is notational. Students know that if I'm referring to a function f(x), then its derivative is written f ' (x). New for today is the idea that if I'm referring to a function y, then its derivative is written dy/dx. I do not care for the notation y', though note that your book will use it on occasion. I think it's sloppy and doesn't impart as much information as is needed.

The second concept is the derivative of sin x. We worked out the derivative using the definition as well as a trig ID, some algebra and our known trig limits. Once sin x has been differentiated, the others follow quite nicely. For now, you only need to know:

f(x) = sin x, then f ' (x) = cos x

and

f(x) = cos x, then f ' (x) = –sin x

The proof of the derivative is shown in my notes below.

August 30 - The derivative

So while today should be fairly momentous as we begin talking about the derivative, it is in fact a review of concepts previously seen at the end of last semester in pre-calculus.

The most important concern is that students remember that the derivative is defined as the limit of the slope between two points. That's it. It's the limit of the difference quotient as the delta x approaches zero.

We began today to study the methods for finding the derivative of a given function. I will continually refer to these methods as "shortcuts" since they do not involve limits at all. I want you to do your best though to always keep in mind the fact that limits are in play however.

The definitions of the derivative, the initial shortcut rules and a couple of examples are included in my notes below.

August 26 - Unbounded behavior

It's very exciting I think in math class to get to use the infinity sign ∞.

Infinity is a cool, confusing, complicated idea. In many ways, capturing what is meant by both the infinite (very large) and infinitesimal (very small) is what calculus is all about.

Today we talked about using ∞ and -∞ to describe when limits do not exist. Often you can but sometimes you can't. Any time direct substitution leads to 1/0, infinity is in play. The hardest part for students is determining whether the expression will be positive or negative.

Notes from today are below.

August 25 - One Sided Limits, Continuity

After a second day of practicing evaluating limits, today focused on two smaller (but still important) ideas.

The first idea is that of the one-sided limit. The limit normally asks you to consider the behavior of a function for x values that are a little less than and a little greater than some value c. Sometimes this results in a limit not existing because of different left/right behavior.

Evaluating a one-sided limit is just like evaluating a normal limit. However, in the instance where behavior is different on the left and right sides of x = c, the one-sided limit simply asks for the particular side. If it is lim x -> c+, then you describe the right side behavior. If it is lim x -> c-, then you describe the left side behavior.

Continuity is an important idea in calculus because our main tools are only guaranteed to work when we have a continuous function. (In fact, at some point in the near future, we will worry about the notion of differentiability. This includes continuity but is in fact more restrictive).

Continuity is defined in calculus for a function at a point. Any math student can intuitively look at the graph of a function and determine whether it is continuous or not. If it has any holes, breaks, or gaps in it, it isn't continuous at that point. You will mainly be concerned at being able to cite the reason why the function is not continuous.

Notes from today are below.

August 23 - Limit sin x / x as x->0

We spent some time talking about the true/false problems from the homework on Friday. I think that the issue there is getting comfortable with If –, then – statements perhaps more than the actual limits concepts involved.

The main topic for today was the limit as x->0 of sin(x) / x. See the diagram below:


Angle theta is drawn in standard position on the unit circle. Thus the dashed red line has length sin(theta) (since the hypotenuse is 1) and the thick green arc has length theta (Arc Length = Radius * Central Angle and the radius = 1). Clearly the arc has greater length than the triangle side, but as the angle theta gets smaller, the side of the triangle and the length of the arc get closer to being the same length. Or alternatively, if their lengths are growing similar, the ratio of the lengths is approaching 1. Thus an argument that as theta->0, sin(theta) / theta -> 1

We worked some examples to illustrate how to work problems with trig limits. Those are copied below:

August 20 - Limits continued

We started class today with a quiz on solving trig equations.

I covered a number of topics related to limits today and I'm hoping that most if not all of this was review. We worked some examples about limit properties (basically limits behave in a pleasing, intuitive way) and then talked about algebraic techniques for evaluating limits.

As a general rule, the first step in evaluating a limit is to do direct substitution. If this yields a value, then often that is the limit (nice functions). More often however, direct substitution gives a result of 0/0 which is an example of an indeterminate expression. For the student, this implies there is more work to do in order to evaluate the limit or to conclude that the limit doesn't exist.

We worked two examples - one showing the factor/cancel algebra and the other showing the multiplying by the conjugate algebra.

Finally, I reminded students that there are two trig limits with which they should be familiar. They are:

lim ((sin x) / x) = 1
x->0

and

lim (1 - cos x) / x = 0
x->0

We will talk about proofs of these on Monday.

There will be a limits quiz on Monday.

Aug 19 - Limits introduction

We talked about limits today in a very informal way. The main idea that you need to remember is that the statement

lim f(x) = L
x->c

means that "for x values that are close to c, the y values are close to L."

What "close to" means is not well defined here. There is analysis that we will talk about that makes more specific the intuitive idea of "close to," but really for our purposes in calculus, the intuition should be fine.

Given the graph of y = f(x) here (click for an enlargement) :


Evaluate the limit of f(x) for each of the following:
x->0, the limit is 1
x->-3, the limit is 3
x->1, the limit is 2. this limit problem is somewhat bothersome b/c of the presence of the point at (1, 1). understand that the limit only refers to points "near" x = 1, not necessarily "at" x = 1.
x->-1, the limit does not exist due to different left/right behavior

Note that if you were asked for f(1), you could say f(1) = 1. If you were asked for f(-3), you would say that f(-3) is undefined.

Recall that there are three reasons for which a limit does not exist

• different left/right behavior
• unbounded behavior
• oscillating behavior

We will investigate these more tomorrow.