A. Evaluate the following:
1.
d 2x 2
e e
dx
2.
3.
4.
d
e
dx
3
5.
6.
x
2
d
log 2 3t
dt
d
ln 4e 4 x
dx
7.
8.
d e2x
dx 4 x
3
d
log 2 8 5 x
dx
d ex
ln
dx e
d log3 x
9
dx
B. Differentiate the follo
18.03 Muddy Card responses, April 21, 2006
1. LTI = Linear, Time Invariant. This is a property of an operator or a system. An operator (which is a rule L that converts one function of time to another
18.03 Muddy Card responses, April 14, 2006
1. A number of people brought up the point made at the end of Lecture 25, on April 12: how do we know what initial conditions yield the unit step or impulse
18.03 Lecture 26, April 14 Laplace Transform: basic properties; functions of a complex variable;
poles
diagrams; s-shift law.
[1] The Laplace transform connects two worlds:
-| The t domain |
| |
18.03 Class 25, April 12, 2006 Convolution [1] We learn about a system by studying it responses to various input signals. I claim that the weight function w(t) - the solution to p(D)x = delta(t) with
18.03 Class 24, April 10, 2006 Unit impulse and step responses [1] In real life one often encounters a system with unknown system parameters. If it's a spring/mass/dashpot system you may not know the
18.03 Class 23, April 7 Step and delta. Two additions to your mathematical modeling toolkit. - Step functions [Heaviside] - Delta functions [Dirac] [1] Model of on/off process: a light turns on; first
18.03 Class 29, Apr 24 Laplace Transform IV: The pole diagram
[1] I introduced the weight function = unit impulse response with the mantra that you know a system by how it responds, so if you let it r
18.03 Class 31, April 28, 2006 First order systems: Introduction [1] There are two fields in which rabbits are breeding like rabbits. Field 1 contains x(t) rabbits, field 2 contains y(t) rabbits. In b
18.03 Class 32, May 1 Eigenvalues and eigenvectors [1] Prologue on Linear Algebra. [a b ; c d] [x ; y] = x[a ; c] + y[b ; d] :
Recall
A matrix times a column vector is the linear combination of the co
18.03: Dierential Equations, Spring, 2006 Driving through the dashpot
The Mathlet Amplitude and Phase: Second order considers a spring/mass/dashpot system driven through the spring. If y (t) denotes t
18.03 Class 38, May 15 Nonlinear systems: Jacobian matrices [1] The Nonlinear Pendulum.
The bob of a pendulum is attached to a rod, so it can swing clear around the pivot. This system is determined by
18.03 Class 37, May 12 Introduction to general nonlinear autonomous systems. [1] Recall that an ODE is "autonomous" if and not on t: x' = g(x) x' depends only on x
For example, I know an island in the
18.03 Muddy Card responses, May 10, 2006
1. So what good are exponential matrices? It seems to me that they dont allow us to skip any steps: it looks like you still have to calculate eigenvalues and t
18.03 Class 36, May 10 Review of matrix exponential Inhomogeneous linear equations [1] Prelude on linear algebra: AB.
If A and B are matrices such that the number of columns in A is
the same as the
18.03 Class 35, May 8 The companion matrix and its phase portrait; The matrix exponential: initial value problems. [1] We spent a lot of time studying the second order equation x" + bx' + kx = 0
and
18.03 Class 34, May 5 Classification of Linear Phase Portraits The moral of today's lecture: Eigenvalues Rule (usually) A is
[1] Recall that the characteristic polynomial of a square matrix p_A(lamb
18.03 Class 33, May 3 Complex or repeated eigenvalues [1] The method for solving u' = Au that we devised on Monday is this:
(1) Write down the characteristic polynomial A) p_A(lambda) = det(A - lambda
18.03 Class 22, April 5 Fourier series and harmonic response [1] My muddy point from the last lecture: I claimed that the Fourier series for f(t) converges wherever $f$ is continuous. What does this r
18.03 Muddy Card responses, April 3, 2006
1. A number of people were confused by my derivation of the Fourier coecients of the function f (t), even, periodic, period 2 , with f (t) = 4 for 0 < t < /2
18.03 Class 21, April 3 Fun with Fourier series [1] If f(t) is any decent periodic of period 2pi, it has exactly one expression as (*)
f(t) = (a0/2) + a1 cos(t) + a2 cos(2t) + . + b1 sin(t) + b2 sin(2
18.03 Class 9, Feb 27, 2006 Review: Linear v Nonlinear [1] review of linear methods [2] Comment on special features of solutions of linear first order ODEs not shared by nonlinear equations. [1] First
18.03 Muddy Card responses, February 24, 2006
Thank you all for your frank responses. Ill try to answer some of the most common confusions. 1. A rst order ODE is autonomous if it has the form y = g (y
18.03 Class 8, Feb 24, 2006 Autonomous equations I'll use (t,y) today.
y' = F(t,y)
is the general first order equation y' = g(y) .
Autonomous ODE:
Eg [Natural growth/decay] Constant growth rate: so y'
18.03 Class 7, Feb 22, 2006 Applications of C:
Exponential and Sinusoidal input and output:
Euler: Re e^cfw_(a+bi)t Im e^cfw_(a+bi)t [1] Integration
e^cfw_2t cos(t) ?
= = e^cfw_at cos(bt) e^cfw
18.03 Class 6, Feb 21, 2006 Roots of Unity, Euler's formula, Sinusoidal functions
[1] Let Roots of unity
a>0. Since i^2 = -1 , (+- i sqrt(a)^2 = - a C.
C ,
:
Negative real numbers have square ro
18.03 Class 5, Feb 17, 2006 Complex Numbers, complex exponential Today, or at least 2006, is the 200th anniversary of the birth of complex numbers. In 1806 papers by Abb\'e Bul\'ee and by Jean-Robert
18.03 Class 4, Feb 15, 2006 First order linear equations: solutions. [1] Definition: A "linear ODE" is one that can be put in the "standard form" _
| |
| x' + p(t)x = q(t) | |_|
On Monday we looked