AP Calculus BC - Integrals and Differential Equations

Integrals and Differential Equations
52Q:
What are the steps you should follow when trying to find integrals?

Integrals and Differential Equations
52A:
The steps to integration are:
1. u-sub
2. arcs?
3. simplify

Integrals and Differential Equations
53Q:
What are the steps to finding a particular solution of a differential equation?

Integrals and Differential Equations
53A:
1. get y’s with dy’s and x’s with dx’s
2. integrate
3. put the +C with x’s
4. solve for y (if you e^( ) the plus C, it becomes times C) (you are done here for general solutions)
5. plug in the initial condition to solve for the C 6. find the domain

Integrals and Differential Equations
54Q:
How do you know that a problem is asking you to find the particular solution?

Integrals and Differential Equations
54A:
A problem is asking you to find the particular solution if a differential equation is given (
dy/dx = equation) and it asks for what the original equation or solution was.

Integrals and Differential Equations
55Q:
(BC only)
What is the equation for a logistic differential equation?

What are the different ways that they can ask for you to find the value of A?

Integrals and Differential Equations
55A:
A logistic differential equation is always of the form:
dp/dt = kP(A-P)
or
dp/dt = kP(1-A/P).
If a logistic differential equation asks for the maximum population, carrying capacity, or limit as t approaches infinity, then you are finding y=A.

Integrals and Differential Equations
56Q:
(BC only)
How do you find the y value of the point of inflection/y-value where the maximum rate occurs?

How do you find the maximum rate?

Integrals and Differential Equations
56A:
The point of inflection / maxumum rate happens at y=A/2. You find the maxumum rate by plugging in A/2 into the differential equation.

Integrals and Differential Equations
57Q:
(BC only)
When is the logistic equation increasing?

When is the logistic equation concave up and concave down?

Integrals and Differential Equations
57A:
A logistic equation is increasing for all values as long as the initial condition is below A. If not, it is always decreasing. The logisitic equation is concave up for 0 to A/2 and concave down for A/2 to A.

Integrals and Differential Equations
58Q:
(BC only)
What are the steps to using Euler’s Method?

Integrals and Differential Equations
58A:
Steps to Euler's Method:
1. Plug the initial condition into dy/dx
2. Multiply this answer by the step size (usually labeled either h or ∆x)
3. add this to the previous height
4. this is your new initial condition
repeat until you get to the x-value asking for (hint: if you are going backwards then your h is negative)

Integrals and Differential Equations
59Q:
(BC only)
How do you find an improper integral?

Integrals and Differential Equations
59A:
The steps to improper integration:
1. Take the integral
2. replace infinity with b
3. Take the limit as b approaches infinity of the fundamental theorem of calculus

Integrals and Differential Equations
60Q:
How do you take the derivative of an integral?

How do you take an integral of a derivative?

Integrals and Differential Equations
60A:
When you take the derivative of a definite integral you must use the 2nd fundamental theorem of calculus.
d/dx[ₐ∫ᵇ f(x)dx]= b'•f(b) - a'•f(a)

When you take the definite integral of a derivative you must use the 1st fundamental theorem of calculus.
ₐ∫ᵇ f '(x)dx]= f(b) - f(a)

Integrals and Differential Equations
61Q:
What is the blanket statement for the definite integral?

Integrals and Differential Equations
61A:
The blanket statement is: “[Answer] [units] is the amount of [possessive noun] that the [proper noun] has [gained/lost] from t = a to t = b.”

Integrals and Differential Equations
62Q:
How do you find the units of an integral?

How do find the units of the derivative?

Integrals and Differential Equations
62A:
To find the units of an integral you must multiply the units of the equation times the units of the x. To find the units of the derivative you must divide the units of the equation by the units of the x.

Integrals and Differential Equations
63Q:
What does the average value stand for on a graph?
What does average value stand for in a word problem?
How do you know that a question is asking you to find the average value?
What is the formula for the average value?

Integrals and Differential Equations
63A:
Average value gives you the average height of the graph over that interval.
Average value gives you the average amount of the words of the equation for a word problem.
Find the average value if it asks for the average words of the equation.
1/(b-a) ⋅ₐ∫ᵇ f(x)dx

Integrals and Differential Equations
64Q:
What does the average rate of change stand for on a graph?
What does average rate of change stand for in a word problem?
How do you know that a question is asking you to find the average rate of change?
What is the formula for average rate of change?

Integrals and Differential Equations
64A:
The average rate of change is the average slope from one point on a graph to another.
The average rate of change is the average change in the possessive noun in a word problem.
Find the average rate of change if it asks for the average rate of the words of the equation.
avg rate = (y₂-y₁)/(x₂-x₁)

Integrals and Differential Equations
65Q:
Mathematically, what is the difference between how much you have of something and how much you gain of something?

Integrals and Differential Equations
65A:
To find how much of something you have you must find the initial amount plus the integral of gain minus the integral of the loss. To find how much you gain or loss of something you just take the integral of the rate of gain or rate of loss.

Integrals and Differential Equations
66Q:
What does the definite integral (ₐ∫ᵇ f(x)dx) mean on a graph?
What does it stand for in a word problem?

Integrals and Differential Equations
66A:
The definite integral stands for the area under the curve for a graph. The definite integral stand for how much was gained or lost in a word problem.

Integrals and Differential Equations
67Q:
In calculus word problems, if a problem asks what happens to the amount at a specific time (example: “at time t = 2 is the particle on the right or left”), what do you do?

Integrals and Differential Equations
67A:
If a problem asks for what happens at a specific time, simply plug that number into the necessary equation.

Integrals and Differential Equations
68Q:
In calculus word problems, if a problem asks “at what time” (as in a single time) something happens (example: “at what time is the particle at rest”), what do you do?

Integrals and Differential Equations
68A:
If a problem asks for what time (single time) something happens, then you must set the necessary equation equal to the necessary number (example: velocity equals zero) and then solve the equation for the time.

Integrals and Differential Equations
69Q:
In calculus word problems, if a problem asks “at what times” (as in a multiple times) something happens (example: “at what times is the particle moving to the right”), what do you do?

Integrals and Differential Equations
69A:
If a problem asks what times (multiple times) something happens you must do a sign chart for the necessary equation.