Topics to be included in Midterm 1
I. Integrals
1. Indefinite integrals (antiderivative) F(t)
2. Definite integrals ab f (t)dt:
- relation to indefinite integrals through the fundamental theorem of
calculus (the algorithm for computing definite integrals)
- summation property
- Integrals and areas (integral = area under the curve)
- average = (total or integral)/(length of interval)
- mass=integral of density
R
3. Improper integrals of type 1 (infinite limits of integration)
- to compute such integral use definition
Z
∞
f (t)dt = lim (F (T ) − F (a))
T →∞
a
If the limit is finite, the integral converges,
if the limit is infinite the integral diverges
- if the integral cannot be computed, the convergence can be analyzed
by approximating the integrand by its leading behavior at infinity, and
deducing the convergence from definition, as above (alternatively, this
last
step can be replaced by the rule:
R∞ 1
1 xp dx converges if p > 1 and diverges otherwise)
4. Improper integrals of type 2 (the integrand becomes infinite at one of
the limits of integration)
- to compute such integral use definition
Z
0
1
f (t)dt = lim(F (1) − F ())
→0
If the limit is finite, the integral converges,
if the limit is infinite the integral diverges
- if the integral cannot be computed, the convergence can be analyzed
by approximating the integrand by its leading behavior at 0, and deducing the convergence from definition, as above (alternatively, this
last
step can be replaced by the rule:
R1 1
0 xp dx converges if p < 1 and diverges otherwise)
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II. Solving systems of ODEs
1. Example of autonomous differential equation models: Population growth,
Newton’s law of cooling, diffusion across the membrane, selection
2. Graphical analysis of 1d autonomous diff. equations - phase line, direction field, equlibria, long-term behavior and time course of representative solutions
- analytical computation of equilibria and their stability
3. Solving ODEs with separation of variables
4. 2d models: competition, Newton’s law of cooling
5. Graphical analysis of 2d systems
- phase plane, nullclines, equilibria, direction field, time course and
long-term behavior of solutions
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