Test 1 Review Math 308-301 & 302 Summer
2009
General Information
Test 1 will be given on Wednesday, 6/24/09, during our usual class
time. It will cover the following material from Nagle, Saff and
Snider: Chapter 1, sections 1.2, 1.4; Chapter 2, 2.1-2.3; Chapter 3,
sections 3.1-3.3, 3.5; and Chapter 4, 4.2, 4.3, 4.5. On Monday (6/22),
my office hours will be 1:40-4:15 pm.
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Calculators. You may use scientific calculators to do numerical
calculations logs, exponentials, and so on. You
may not use any calculator that has the capability storing
material, or of doing algebra or calculus.
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Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
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Methods of integration and trig formulas. Be able to integrals
we've done in class and for homework. Integrals such as
∫ ekt cos(ωt)dt
will be provided. You should know and be able to apply the trig
formulas as well.
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Structure. There will be 7 to 10 questions, some with multiple
parts. The problems will be similar to ones done for homework, and as
examples in class and in the text.
Topics covered
First Order ODEs
- Solutions & IVP (1.2)
- Euler's method (1.4)
- Separable equations (2.2)
- Linear 1st order ODEs (2.3)
- Solution via integrating factors
- Solving the IVP
Applications of First Order ODEs
-
Falling bodies Air friction is modeled by
Ffric = −bv. Be able to find limiting
velocities. (2.1)
- Radioactive decay (2.3)
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Mixing problems Set these up using amounts, applying
dx/dt = rate in − rate out. If the problem requires
concentration, get it from this equation: concentration =
amount/volume. (3.2)
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Newton's law of cooling (3.3)
- Find k (parameter identification) and then find T(t).
- time-of-death
- coffee-cooling
- Temperature in building
- thermostat IVP problems, modeling outside temperature
- transient solution
- asymptotic temperature (transient has died away), phase angle
- Circuits RL and RC circuits (3.5)
- VR = IR
- VC = Q/C and I = dQ/dt
- VL = LdI/dt
Second Order ODEs
- Linear operators (4.2)
- Definition
- "D" notation
- Linear differential equations in operator notation
- Linear combinations
- Superposition principle (Theorem 1, pg. 161)
- Existence and uniqueness of IVP, L[y]=g, y(x0)
= y0, y'(x0) = y′0 (Theorem 2, pg. 163)
- Homogenous linear ODEs (4.3,4.5)
- Fundamental sets, IVPs
- Wronskians
- Linear independence and linear dependence
- Fundamental sets for the constant coefficient case, with the
auxiliary equation having real, distinct roots (4.5)
Updated 6/19/09.