Math 5B Midterm 2

if f is concave up on [a,b] for average value, what is f avg

f avg < f(b) + f(a) / 2

if f is concave down on [a,b], then what is f avg

f avg > f(b) + f(a) /2

what does it mean if y>/= 0 in a logistic differential equation

population cannot be negative

k in logistic differential equations

speed of population growth

c in logistic differential equation

carrying capacity

steps of partial fraction differentiation

1. separate variables

2. take integral of both sides (use partial fraction decomp)

3. take e of both sides

4. drop absolute value and reciprocal

5. to find particular solution = plug in given values to find the constants

axis of a slope field

x and y

axis of phase plot

x - vertical

y - horizontal

if g’(y) is negative, then y hat is

locally stable

if g’(y) is positive, then y hat is

unstable

if g’(y) i= 0, then y hat is

inconclusive

harvesting parameter

h measures how much harvesting of the population is going on

• different parameters result in different differential equations

what happens to equilibria when h is positive

inverse relationship

• as harvesting increases, the stable equil will decrease

finding equilibria and stability with h

plug in the h value into the y’ equation to find equilibria

• determine stability based on the slope field

identifying the equilibria given the p and bifurcation plot

draw a vertical line thru the parameter and it should intersect with the y’

bifurcation plot

horizontal = h

vertical = N

bifurcation

where the equil switches from unstable to stable

creating bifurcation plots

• plot the equilibria on bifurcation plot

• use local stability criterion to determine stability