10.3 #5,7,10,14 but you can use whatever test you like
#41,47
10.5 #1,5,6,24
10.7 #5,13,21,79
Thursday, May 29, 2008
Wednesday, May 28, 2008
Wednesday May 28
Some criteria for convergence of infinite series:
If it "absolutely converges", it converges.
Comparison theorem; if a series with positive terms is bounded above by a convergent series, then it too converges.
Integral test; if our series comes from a decreasing positive function, then the series converges if and only if the improper integral converges. We actually proved this!
If it "absolutely converges", it converges.
Comparison theorem; if a series with positive terms is bounded above by a convergent series, then it too converges.
Integral test; if our series comes from a decreasing positive function, then the series converges if and only if the improper integral converges. We actually proved this!
Monday, May 19, 2008
Practice problems for midterm #2 (this Friday)
7.2 # 13,23,29,37,49
7.6 # 17,22,24
7.7 # 15,28,34
9.1 # 9,20,33,43
9.2 # 6
9.5 # 9,13
7.6 # 17,22,24
7.7 # 15,28,34
9.1 # 9,20,33,43
9.2 # 6
9.5 # 9,13
Thursday, May 15, 2008
Monday and Wednesday May 12 & 14
5/12
9.2 Exponential growth and decay. Newton's law of cooling.
Nibbling on basil example of exponential growth (possibly negative).
9.4 What the logistic equation means, above carrying capacity.
9.5 First-order linear homogeneous equations.
A derivation of the integrating factor, based on the idea that one solves the homogeneous first and uses it as a stepping-stone.
5/14
Rederivation of the general solution.
Applied to a nonmotivated example from the book.
Then we thought about filling up a bathtub, initially half-full of cold water, with warm water, while it's draining (more slowly than it fills). During this process the water draining becomes warmer and warmer. What's the temperature at the time the bath is full?
This turned out to be a linear inhomogeneous first-order DE.
10.1 Infinite sequences. The definition of "this sequence converges to x", with epsilons and large Ns.
9.2 Exponential growth and decay. Newton's law of cooling.
Nibbling on basil example of exponential growth (possibly negative).
9.4 What the logistic equation means, above carrying capacity.
9.5 First-order linear homogeneous equations.
A derivation of the integrating factor, based on the idea that one solves the homogeneous first and uses it as a stepping-stone.
5/14
Rederivation of the general solution.
Applied to a nonmotivated example from the book.
Then we thought about filling up a bathtub, initially half-full of cold water, with warm water, while it's draining (more slowly than it fills). During this process the water draining becomes warmer and warmer. What's the temperature at the time the bath is full?
This turned out to be a linear inhomogeneous first-order DE.
10.1 Infinite sequences. The definition of "this sequence converges to x", with epsilons and large Ns.
Wednesday, May 14, 2008
Friday, May 09, 2008
Wednesday & Friday May 7 & 9
9.1 Differential equations.
Separability.
Straight lines through the origin.
Circles around the origin.
Initial value problems.
Growth of a cell.
The logistic equation.
Linear homogeneous & inhomogeneous.
The spring equation (2nd order homogeneous).
The spring equation in presence of gravity (inhomogeneous).
Separability.
Straight lines through the origin.
Circles around the origin.
Initial value problems.
Growth of a cell.
The logistic equation.
Linear homogeneous & inhomogeneous.
The spring equation (2nd order homogeneous).
The spring equation in presence of gravity (inhomogeneous).
Tuesday, May 06, 2008
Monday May 5
Comparison theorem for improper integrals.
With proper integrals, we don't have to worry about whether the answer is actually finite; it is. With improper ones sometimes we can't calculate them exactly, but we still want to know the answer is finite. The comparison theorem lets us do that; it's much more important than in the proper-integral situation.
9.1 Diff-eqs.
We just started this, and solved y' = y/x (the solutions are straight lines through the origin, which makes sense).
With proper integrals, we don't have to worry about whether the answer is actually finite; it is. With improper ones sometimes we can't calculate them exactly, but we still want to know the answer is finite. The comparison theorem lets us do that; it's much more important than in the proper-integral situation.
9.1 Diff-eqs.
We just started this, and solved y' = y/x (the solutions are straight lines through the origin, which makes sense).
Friday, May 02, 2008
Friday May 2
I pointed out that step 1 of the partial fractions expansion algorithm -- long division of the numerator by the denominator, leaving a remainder term -- can be thought of as "peeling off the terms that blow up at x = infinity". Which can be done almost the same way as we do the rest of the algorithm; divide by the highest power of x, then look at the limit as x->infinity.
Not that there's much reason to do it that way; I was just sayin'.
Then we did improper integrals. I spent a long time on one example: integral0picot(x) dx. This is improper at both ends, and there's no best way to evaluate it, in that different approaches give different answers. So one should declare this to have no answer.
Not that there's much reason to do it that way; I was just sayin'.
Then we did improper integrals. I spent a long time on one example: integral0picot(x) dx. This is improper at both ends, and there's no best way to evaluate it, in that different approaches give different answers. So one should declare this to have no answer.
Wednesday April 2
We saw the general rule for doing partial fraction expansion, i.e. what to do if the polynomial in the denominator has repeated roots.
Then we touched upon improper integrals.
Then we touched upon improper integrals.
Thursday, May 01, 2008
Subscribe to:
Posts (Atom)