MVT and Verify Solutions
The significance of the value of C as described by the Mean Value Theorem is...
C is the x-value at which the instantaneous rate of change is equal to the average rate of change over an interval.
In respect to the discussion of differential equations, a slope field is...
A visual representation of all the possible particular solutions to a differential equation.
The first step to solving a differential equation is to...
Isolate the functions of different variables on each side of the equation.
In exponential growth equations, k is the...
Growth rate constant
What must you never forget?
+C
What does Rolle's Theorem stipulate (what is unique about this application)?
Rolle's Theorem is a specific application of the MVT in which the average rate of change is equal to 0 (the y-values of the limits are equal).
If all the slopes in each column of a slope field are the same, then the differential equation...
Does not include a y-variable.
It can be ascertained from a general solution that a C value...
Exists.
In exponential growth equations, the value of C is always the...
The initial amount of the thing being modeled.
The first step in finding the original equation from its parametric representation is to...
Solve both given equations for t!
The most common units for acceleration are...
Meters per second squared --- m/s2
What must be given along with the initial condition by a problem to be solved with Euler's Method?
The step size or the number of equal steps.
What is necessary in order to find the C-value of a particular solution?
An initial condition or some other (x,y) pair.
In logistic growth equations, the limit as t approaches infinity of P(t) is also known as...
M, or the carrying capacity.
The second language Ms. Smith speaks...
On a test, you see an equation in the form of:
y'+y'''=0 , y=4xcosx
You are tasked with verifying this solution.
The first step to solve would be to...
Find what y' and y''' are equal to using the given function of x.
Is it helpful to separate and integrate in Euler's Method problems?
No. Why would you do this to yourself?
In respect to general solutions, C=K=A=B...
To what extent is this statement true?
True. The constant C can be represented by any capital letter.
___ must be included in the initial condition of a logistic growth equation for it to be solvable...
The population at time t=0 and the population at some other time t.
The function that is in the denominator of the second derivative of a parametric equation.
dx/dt
If a equals b, but f(a) does not equal f(b), does MVT still apply?
Yessir; Like Rolle's Theorem, but you are searching for where the derivative is indeterminate.
If all the slopes in a slope field are different, then what can be said about its differential equation?
The differential equation includes both an x- and a y-variable.
When you can't separate and integrate a differential equation on the AP exam, you must...
Cry! jk, you won't see anything you can't solve unless you don't need to solve it (i.e. slope fields).
The constant k in logistic growth equations is always negative because...
The denominator is getting smaller / approaching 1 as the value of t increases (so that the maximum population is equal to the carrying capacity).
The process by which the first derivative of a parametric equation is found...
Find dy/dt and dx/dt. Substitute into form dy/dx.