AP Calculus AB - Integrals and Word Problems

Integrals and Word Problems-65Q:How do you take the derivative of an integral? How do you take an integral of a derivative?

Integrals and Word Problems-65A: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 Word Problems-66Q:What is the blanket statement for the definite integral?

Integrals and Word Problems-66A: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 Word Problems-67Q:How do you find the units of an integral? How do find the units of the derivative?

Integrals and Word Problems-67A: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 Word Problems-68Q: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 Word Problems-68A: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 Word Problems-69Q: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 Word Problems-69A: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 Word Problems-70Q:Mathematically, what is the difference between how much you have of something and how much you gain of
something?

Integrals and Word Problems-70A: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 Word Problems-71Q:What does the definite integral (ₐ∫ᵇ f(x) dx) stand for on a graph? What does it stand for in a word problem?

Integrals and Word Problems-71A: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 Word Problems-72Q: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 Word Problems-72A:If a problem asks for what happens at a specific time, simply plug that number into the necessary equation.

Integrals and Word Problems-73Q: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 Word Problems-73A: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 Word Problems-74Q: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 Word Problems-74A:If a problem asks what times (multiple times) something happens you must do a sign chart for the necessary equation.