7.6 #4,12,15,19,47
7.7 #1,2,5,6,19,30,33,78
Tuesday, April 29, 2008
Monday, April 28, 2008
Monday April 28
More partial fractions.
Long division of polynomials (i.e. the usual long division algorithm, but with polynomials).
Factoring the denominator using complex roots.
We haven't yet addressed the possibility that the denominator has repeated factors; that will be next time.
Long division of polynomials (i.e. the usual long division algorithm, but with polynomials).
Factoring the denominator using complex roots.
We haven't yet addressed the possibility that the denominator has repeated factors; that will be next time.
Friday, April 25, 2008
Friday April 25
Section 7.5. Hyperbolic trig functions.
We talked a bit about why the word "hyperbolic" is there. Both circles and hyperbolae are conic sections.
Section 7.6. Partial fractions.
So far we did a couple of examples. The basic trick, to figure out the coefficient on some term, is to multiply through by that denominator THEN set the once-denominator equal to 0. All other terms die, exposing the one you're trying to compute.
We talked a bit about why the word "hyperbolic" is there. Both circles and hyperbolae are conic sections.
Section 7.6. Partial fractions.
So far we did a couple of examples. The basic trick, to figure out the coefficient on some term, is to multiply through by that denominator THEN set the once-denominator equal to 0. All other terms die, exposing the one you're trying to compute.
Midterm #1 breakdown
In the extremely unlikely event that your final letter grade were to be determined from only the midterm #1 grade, it would be as follows:
80-100 A
60-80 B
45-60 C
30-45 D
<30 F
We will not be figuring out your final grade using this letter, but rather the raw number, so don't worry if e.g. you got 79 rather than 80.
80-100 A
60-80 B
45-60 C
30-45 D
<30 F
We will not be figuring out your final grade using this letter, but rather the raw number, so don't worry if e.g. you got 79 rather than 80.
Wednesday, April 23, 2008
HW #3 due Wednesday April 30
6.3 #9,15,23,25
7.2 #2,3,7,20,36,45,66
Read 7.3 and looking heavenward, reflect internally upon the value of complex exponentials.
Integrate exp(2x) sin(x).
7.2 #2,3,7,20,36,45,66
Read 7.3 and looking heavenward, reflect internally upon the value of complex exponentials.
Integrate exp(2x) sin(x).
Wednesday April 23
Last time I introduced complex numbers, and how to think about multiplying them: they scale and rotate.
This time we studied pure rotations (i.e. scaling by 1), and showed that if there's any justice, the complex number z such that multiplying by z implements rotation by theta, should be z = exp(i theta).
This gave us formulae for sin and cos in terms of complex exponentials, which is good because the latter are much easier to work with.
This time we studied pure rotations (i.e. scaling by 1), and showed that if there's any justice, the complex number z such that multiplying by z implements rotation by theta, should be z = exp(i theta).
This gave us formulae for sin and cos in terms of complex exponentials, which is good because the latter are much easier to work with.
Tuesday, April 22, 2008
Office hours Thursday
I am traveling and will be back tomorrow; the Tuesday office hour is moved to Thursday.
Answers to the first midterm will be up soon.
Answers to the first midterm will be up soon.
Friday, April 18, 2008
Tiny mistake in practice midterm
In question #9, the integral over t should have a dt, not a dx. It's fixed now.
I'll put up answers tomorrow.
I'll put up answers tomorrow.
Wednesday, April 16, 2008
Monday April 14
We talked about linear density along a rod, and mass as the integral of density.
Using the linear density, you can compute not just the total mass (integral of rho(x) dx),
but also the rotational inertia (integral of x rho(x) dx).
We also talked about population density. If the density per square mile is only a function of the distance from the origin, then total population = integralr=0r=D rho(r) 2 pi r dr, where the extra factor of 2 pi r accounts for the length of the circle at radius r.
Then we talked about change in energy as
integral of (force(t) times speed(t) times cos(angle between force and velocity) dt).
So if the force is pushing the same direction as the moving object, it gains energy;
if it's the opposite direction, it loses energy;
if it's perpendicular (like whirling something around your head on a string) it stays the same.
Using the linear density, you can compute not just the total mass (integral of rho(x) dx),
but also the rotational inertia (integral of x rho(x) dx).
We also talked about population density. If the density per square mile is only a function of the distance from the origin, then total population = integralr=0r=D rho(r) 2 pi r dr, where the extra factor of 2 pi r accounts for the length of the circle at radius r.
Then we talked about change in energy as
integral of (force(t) times speed(t) times cos(angle between force and velocity) dt).
So if the force is pushing the same direction as the moving object, it gains energy;
if it's the opposite direction, it loses energy;
if it's perpendicular (like whirling something around your head on a string) it stays the same.
Practice midterm #1 (which is next Monday)
It's here, in PDF.
If this gives you a sense that you'd better know your sine from cosine -- you're right! You can draw them on your cheat sheet, of course.
If this gives you a sense that you'd better know your sine from cosine -- you're right! You can draw them on your cheat sheet, of course.
The case of the missing TA
While I don't have the complete story at this time on why the 2-4 PM TA was missing in action, a replacement has been obtained. He is Michael Scullard, and will indeed be at section tomorrow. I apologize deeply to the people left floundering in those sections, and hope and believe that things will run smoothly from here on out.
Friday, April 11, 2008
Friday April 11
We computed the volume of ball, using the area of a disk.
Then we tried to compute it using the area of a sphere.
Since we (pretended we) didn't know that area, we worked backwards, differentiating the volume formula w.r.t. radius to get the area of the sphere.
We computed the volume of a cube by slicing it into equal squares. Pretty silly.
Then we computed it again, by slicing at 45 degrees, into rectangles all of the same height. Sillier.
Then I described what the slices look like lengthwise: triangles, then hexagons, then triangles again, and left it to the masochists to confirm that the volume is still 1.
Definition: the average value of a function. Statement of the Mean Value Theorem in this context.
Then we tried to compute it using the area of a sphere.
Since we (pretended we) didn't know that area, we worked backwards, differentiating the volume formula w.r.t. radius to get the area of the sphere.
We computed the volume of a cube by slicing it into equal squares. Pretty silly.
Then we computed it again, by slicing at 45 degrees, into rectangles all of the same height. Sillier.
Then I described what the slices look like lengthwise: triangles, then hexagons, then triangles again, and left it to the masochists to confirm that the volume is still 1.
Definition: the average value of a function. Statement of the Mean Value Theorem in this context.
Thursday, April 10, 2008
Wednesday April 9
We did a number of examples of substitution. Basic principle: look deep into your complicated function for the smallest substitution you can make.
For example, when integrating 1 / x (ln x)1/2, let u = ln x not (ln x)1/2 or anything more complicated.
6.1 Area between two curves.
6.2 Volume = integral of areas
We set up the integral for computing the volume of a sphere.
For example, when integrating 1 / x (ln x)1/2, let u = ln x not (ln x)1/2 or anything more complicated.
6.1 Area between two curves.
6.2 Volume = integral of areas
We set up the integral for computing the volume of a sphere.
Tuesday, April 08, 2008
HW #2 due Monday April 14
Section 5.6: Turn in: 26, 38, 40, 60, 90; Additional: 13, 17, 19, 25, 85
Section 6.1: Turn in: 4, 30, 32, 50; Additional: 17, 37, 41
Section 6.2: Turn in: 6, 10, 52, 56; Additional: 7, 9, 13, 57
Section 6.1: Turn in: 4, 30, 32, 50; Additional: 17, 37, 41
Section 6.2: Turn in: 6, 10, 52, 56; Additional: 7, 9, 13, 57
Friday April 4 and Monday April 7
4/4.
5.2. When a function has a different formula on different intervals, you're better off integrating it separately on the different intervals and adding them together.
5.3,4. The Fundamental Theorem of Calculus, in two versions:
v2. if you integrate something continuous then differentiate, you get back the function you started with.
v1. if you differentiate something differentiable then integrate, you almost get back the function you started with; you get it up to an additive constant.
An example of an integral to compute total food eaten by a pet hamster and lion.
An example with the Baby Name Wizard of computing the number of Allens between the ages of 30 and 40, as an integral. I drew the graphs of Beulah, Brianna, Adolf, and Monica.
4/7.
Definite integrals vs. indefinite. Definite integrals are numbers; indefinite integrals are lazily written (by everyone) as functions, when really they're "functions up to an additive constant". So make sure you include that "+ C" when indefinitely integrating something.
5.6. Recall that the chain rule of differentiation (f(g(x)))' produces a product f'(g(x)) g'(x). So when faced with an integral of a product, see if you can recognize the product as being the derivative of a composite function.
Why is the dx so important to remember in integrals? Because you need to change it, too, when you do change of variable. Otherwise you get the wrong answer, as I demonstrated.
5.2. When a function has a different formula on different intervals, you're better off integrating it separately on the different intervals and adding them together.
5.3,4. The Fundamental Theorem of Calculus, in two versions:
v2. if you integrate something continuous then differentiate, you get back the function you started with.
v1. if you differentiate something differentiable then integrate, you almost get back the function you started with; you get it up to an additive constant.
An example of an integral to compute total food eaten by a pet hamster and lion.
An example with the Baby Name Wizard of computing the number of Allens between the ages of 30 and 40, as an integral. I drew the graphs of Beulah, Brianna, Adolf, and Monica.
4/7.
Definite integrals vs. indefinite. Definite integrals are numbers; indefinite integrals are lazily written (by everyone) as functions, when really they're "functions up to an additive constant". So make sure you include that "+ C" when indefinitely integrating something.
5.6. Recall that the chain rule of differentiation (f(g(x)))' produces a product f'(g(x)) g'(x). So when faced with an integral of a product, see if you can recognize the product as being the derivative of a composite function.
Why is the dx so important to remember in integrals? Because you need to change it, too, when you do change of variable. Otherwise you get the wrong answer, as I demonstrated.
Friday, April 04, 2008
Shortening HW #1
As we've gotten off to a slow start I'm moving the 5.6 questions to next week's homework. Don't turn them in this week. What remains:
HOMEWORK 1: Due Monday, April 7
* Section 5.2: Turn in: 14, 76; Additional: 13, 53
* Section 5.3: Turn in: 10, 20, 32, 46; Additional: 11, 19
* Section 5.4: Turn in: 4, 10, 18, 28; Additional: 19, 23
* Section 5.5: Turn in: 8, 14; Additional: 9
HOMEWORK 1: Due Monday, April 7
* Section 5.2: Turn in: 14, 76; Additional: 13, 53
* Section 5.3: Turn in: 10, 20, 32, 46; Additional: 11, 19
* Section 5.4: Turn in: 4, 10, 18, 28; Additional: 19, 23
* Section 5.5: Turn in: 8, 14; Additional: 9
Wednesday, April 02, 2008
PLEASE PLEASE PLEASE
Ask questions in class! I said it on Monday, and I guess I should have said it again today.
I didn't even notice that I'd used epsilon and "is an element of", two different mathematical symbols that look very similar, and that there might be confusion. If you have a question of the form "what is so-and-so / what does so-and-so mean?" it (1) is surely a question somebody else has in this 200-strong class and (2) is very easy to answer quickly, no trouble at all!
There were maybe 5 questions in class today. I'd be happier with 50, even -- especially -- if they're as simple as the above. Even if they're complicated, go for it.
I didn't even notice that I'd used epsilon and "is an element of", two different mathematical symbols that look very similar, and that there might be confusion. If you have a question of the form "what is so-and-so / what does so-and-so mean?" it (1) is surely a question somebody else has in this 200-strong class and (2) is very easy to answer quickly, no trouble at all!
There were maybe 5 questions in class today. I'd be happier with 50, even -- especially -- if they're as simple as the above. Even if they're complicated, go for it.
Wednesday April 2
Though in general we'll be computing integrals in terms of their interpretation as areas, I gave the definition as a limit of Riemann sums. Since the limit in question is a limit over refinements of partitions (not the simple sort of limit of real numbers one uses to compute derivatives), I talked about limits in general for a while first.
Most substantively going forward, I stated a few of the nice properties integrals have, like the integral of a sum being the sum of the integrals, the integral of C*f being C * the integral of f (for C a constant), and that one can break the interval being integrated over into two pieces and add them up.
This motivates the definition we'll use for integrating "from a to b" when b < a, namely the negative of integrating from b to a.
Most substantively going forward, I stated a few of the nice properties integrals have, like the integral of a sum being the sum of the integrals, the integral of C*f being C * the integral of f (for C a constant), and that one can break the interval being integrated over into two pieces and add them up.
This motivates the definition we'll use for integrating "from a to b" when b < a, namely the negative of integrating from b to a.
Tuesday, April 01, 2008
Monday March 31
We discussed briefly the topics we'll cover: integration, applications thereof, techniques for integrating, differential equations. I mentioned infinite series, but we didn't get to talk about them yet.
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