Nov 30 - Slope Fields/Euler's Method

Slope fields are a way to visualize differential equations. The idea is simple - generate an array of points, calculate the slope at each point, and draw a short line segment through each point with the appropriate slope. The utility of slope fields is that they allow you to visualize solutions to differential equations (keeping in mind that in general, integrating to solve can be very difficult).

Euler's method is a process for finding an approximation for a y value given an initial condition. It is an iterative process and outside of occasionally dealing with obnoxious fractions, it should be fairly straightforward arithmetic. Tomorrow in class I'll give you a visual of what Euler's method is doing.

Thanks to Rachel for taking notes today.

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Nov 29 - Solutions to Differential Equations

Though we are in a new chapter (and starting a new section today), in fact the work that you are doing is wholly review. Each of the problems in the assignment tonight is either finding a derivative or finding an integral. Review.

I worked three examples in class today. The second really emphasizes that many differential equations don't have easily findable solutions and so verifying that a function is a solution is actually a viable method for solving diff eqns. The third example is motivated by simple harmonic motion and I showed the differential equation that is derived from a combination of Hooke's Law and Newton's Second Law. This is a nice application of the math, but you are not responsible for it in BC Calc.

Notes from today are posted below.

Nov 28 - Differential Equations

We started a short unit on solving differential equations today.

I worked two examples in class. The first represented the situation where

"the rate of growth of y is proportional to y"

This is the definition of exponential growth. It translates to

dy / dx = k*y which can be solved to yield y = Ce^(kx)

The second example exemplified the idea of a limiting value. We'll get more practice with these concepts.

Notes for today are posted below.

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Nov 18 - Review for Test 3

I worked out a couple problems from the review on the board. I'm hoping that notational issues are few and far between on the test on Monday.

The one concept I revisited today was using the FTOC to identify the derivative of a function and then using that derivative to describe the behavior (incr/decr, concave up/down) of the function. We've covered the idea of using the _graph_ of f '(x), the derivative of a function, to talk about f(x). This is that same concept.

Notes from today are posted below. As well, worked solutions to the test from last year and the review problems that I handed out Friday are posted in the Resources folder.

Nov 17 - Review Int/Diff with logs/exp

I worked a couple example problems in class today illustrating the power that you have as calculus students now. You have the tools to find the derivatives for most all regular functions in mathematics. !

The first example was y = x^x. It is a mix of the power function and the exponential function, but it's derivative is found by using logarithms. This idea of logarithmic differentiation is often used to find derivatives of "complicated" functions.

The other thing from class today was to emphasize that finding antiderivatives is a challenging exercise. One of the biggest mistakes I've seen students make in the past is using "bad" math - poor algebra skills typically. It gets easier with practice, but it does take some mental discipline.

Notes from today are posted below.

Nov 16 - Review (Integration)

Two main ideas from today.

A. You are officially responsible for knowing that the derivative of arctan(x) is 1 / (x^2 + 1). This implies that you need to recognize and use

integral of 1 / (x^2 + 1) = arctan(x)+ C

Expect to have to use this on homework, quizzes, and tests.

B. Your general plan of attack for an integration problem is as follows:

1. Do you recognize this as an "easy" antiderivative? For example if you see x^n, sec^2 x, or 1/x - you immediately recognize the antiderivative. Note that all 6 trig functions now fall in this category.

2. If not, then see if you can do algebra to rewrite the problem so that you do recognize it as an "easy" antiderivative. Doing algebra involves expanding (multiplying out), factoring, using trig IDs, log properties or exponential properties.

3. If not, then try a u-substitution to rewrite the problem so that you do recognize it as an "easy" antiderivative. You may have to try several different u-subs.

4. There is no 4. Try 1, 2, and 3 again.

I'll hand out a copy of the test from last year tomorrow in class.

Nov 15 - Integrating with e^x

Today we focused on integrating functions using our rule for e^x (the antiderivative of e^x is e^x + C). You will get a decent amount of practice over the next several days. Sometimes it can get confusing as to what you should use for substitution, but you will get the hang of it.

I also introduced the notion of finding areas that are "unbounded." This is in my mind one of the coolest aspects of integration - it emphasizes that each time you evaluate an integral, you are finding an infinite sum.

Thanks to Sumant for today's notes.

Nov 14 - Integration with 1/x

Today introduced a new and important integration idea. Because we defined ln x to be the antiderivative of 1/t as x goes from 1 to x, we can generalize that to be

Integral of 1/x dx = ln |x| + C (this is correct!!!!)

This will appear in a variety of ways, often following a u-substitution. The main confusion to AVOID is to recognize that the integration rule is very specific - it has to be 1/x. The rule can not be generalized to

Integral of 1/f(x) dx = ln |f(x)| + C; (this is not correct!!!!)

The only f(x) in the denominator that yields a ln(x) in the antiderivative is a linear function.

We will talk about integrating with e^x tomorrow, and then the rest of the week will be spent practicing integration and differentiation. There is a test next Monday.

Thanks to Madeline for her notes from today.


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Nov 11 - Exponential Functions

Yesterday I applied the Fundamental Theorem of Calculus to "assign" an antiderivative to y = 1/x. When we studied the properties of the antiderivative, it became apparent that it was a logarithm, thus ln(x) is defined to be the integral of 1/t on the interval 1 to x.

Today we use this definition of ln(x) to arrive at a definition of the number e. We will define e as the bound of the region that yields an area between y = 1/x, y = 0 (the x-axis), x = 1 and x = e equal to 1. Thus e is the number such that ln(e) = 1.

The most significant consequence of this is that it turns out if y = e^x, then dy/dx = e^x. It's weird and it's wonderful. And it brings a whole new feel to finding derivatives.

(Note too that it means that e^x is also it's own _antiderivative_, but we'll save that for next Tuesday...)

Notes from today are posted below.
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Nov 10 - Natural logarithm

We begin working with logarithmic functions today, starting with differentiation. The highlight of class today is seeing the "new" definition of ln(x). We give it an integral definition and see how log properties are consequences of calculus ideas. It's a brand new world!

We will rarely, if ever, worry about logs with other bases. Generally, we use the base change formula to turn another based logarithm into ln x. For example, log x = ln(x) / ln(10).

Notes from today are below. Tonight's assignment will be mostly practice finding derivatives that involve ln(x).