Derivative Flashcards
What is a derivative?
slope of a tangent line
what does a positive derivative mean?
f(x) has a positive rate of change, therefore increasing
what does a negative derivative mean?
f(x) has a negative rate of change, therefore decreasing
what is f(x) at x0?
the instantaneous rate of change
what is the fundamental theorem on existence
each trajectory has it’s own set path
how do we represent when r is inversely proportional to c?
r/c
what is the reason for sketching vector fields?
to compute a specified number of change vectors
what does the numeration in the carrying capacity fraction mean
the population of the species
what does the denominator in the carrying capacity fraction mean
the limited amount of resources
explain what f(x1)-f(x0)/x1-x0 means
f(x1)-f(x0) represents the rate of change over the interval of
determinism
when you choose a certain vector in the vector field, the path of a trajectory is already pre-determined and there is only one way forward and one way backward
why can’t trajectories touch
there can only be one vector for each point in the vector field, therefore the trajectories cannot cross going into different directions
when do you square a variable in a differential equation?
when something is proportional to the same variable and make sure that it is mentioned that they are
logistic growth
when a growth is limited, such as carrying capacity
A′= 0.1A −0.01A^2 −0.5AB
what is 0.1A-0.01A^2 equal to?
0.1A(1-A/10)
what does it mean if the trajectory doesn’t touch the axes?
the population will not go to zero
what does it mean when a change vector is tangent to the trajectories?
It means that the trajectory is following the path of the change vectors and that the vectors are defining the movement of the trajectory.
average rate of change equation
f(x1) - f(x0)/x1-x0
what is the instantaneous rate of change equation that includes delta x
f(x+deltax)-f(x)/deltax
what happens if derivative is zero
it means that the derivative is constant
what is average rate of change
the rate of change over time
what is instantaneous rate of change
the rate of change over a certain interval of time
how to graph a derivative
you graph the slope of the line
what does the tangent line represent
the instantaneous slope
what does the secant line represent
the average slope
per capita meaning
multiply by the state variable
proportional meaning
equal to another variable, therefore you have to include that variable into the input
what is a parameter
a constant
what is a periodic trajectory
when the trajectory repeats itself and repeats the same behavior continuously over time
how do you make an approximation more accurate
smaller time step, shorter segments
