Derivatives
Antiderivatives
In and Out
Theorems
Rosie
100
The derivative is e2x + 1.
What is the derivative of 0.5e2x + x?
100
The antiderivative is ln (x + 2) + C.
What is the antiderivative of
100
In the first 10 minutes, a tank has water flowing into it at a constant rate of 10 gallons/min and water leaking out of it at a rate of f(t) = t2 + 1. This happens when the time is 3 minutes.
At what time are the rates equal?
100
If f is continuous on (a, b) and f(a) < y < f(b) then there exists a C in (a, b) where f(c) = y.
What is the Intermediate Value Theorem (IVT)?
100
My Walks With Rosie
Graphs of Kruger's walks with his dog, Rosie, which are used to illustrate calculus concepts.
200
The derivative is
x^2/(3x^3+1)
What is the derivative of
ln(3x^3+1)/9?
200
The antiderivative is
-1/(x+2)+C
What is the antiderivative of
1/(x + 2)^2?
200
In the first 10 minutes, a tank has water flowing into it at a constant rate of 10 gallons/min and water leaking out of it at a rate of f(t) = t2 + 1, since that tank contained 244 gallons of water in it initially. 262 gallons must be the answer.
During the first 10 minutes, what was the maximum number of gallons of water in the tank?
200
If f is continuous on [a, b] then an absolute maximum or absolute minimum exists either at x = a, x = b, x = c where c is a critical number of f.
What is the Extreme Value Theorem (EVT)?
200
In my graph, My Walk With Rosie, v(t) = sin t in (0, 2π) as pictured above, t = π.
At what time does Kruger and Rosie turn around and move in the opposite direction.
300
The derivative is sec2(x)etan x.
What is the derivative of etan x ?
300
The antiderivative is
lnsqrt(x+2)+C
What is the antiderivative
1(2x+4)?
300
In the first 10 minutes, a tank has water flowing into it at a constant rate of 10 gallons/min and water leaking out of it at a rate of f(t) = t2 + 1, since that tank contained 244 gallons of water in it initially. The amount is 2/3 gallons.
How much water was in the tank after 10 minutes?
300
If f is differentiable and continuous on (a, b) there exists a C in (a, b) where
f^ '(c)=(f(b)-f(a))/(b-a)
What is the Mean Value Theorem (MVT)?
300
In the My Walk With Rosie graph pictured above, v(t) > 0 in (0, π) and (π, 2π).
On what intervals in My Walk With Rosie, is our distance increasing?
400
The derivative is 0.
What is the derivative of C?
400
The antiderivative is ln (cos x) + C?
What is the antiderivative of -tan x?
400
A(t)=244+int_0^10 10-x^2-1 dx
What is the function that computes the amount of water in the tank at any time t in [0, 10]?
400
Sharp Points or vertical inflection points
What can continuous and non-differentiable spots look like on a graph?
400
In the My Walk With Rosie graph pictured above, v(t) = sin t in (0, 2π), the displacement is 0.
What is 
500
The derivative is
What is the derivative of
1/(x^2+9)^3?
500
The antiderivative is ln (tan x) + C?
What is the antiderivative of sec x csc x?
500
An accumulative function
What type of function is
244+int_0^t 10-x^2-1dx?
500
On Sunday when I was driving back from Plainview, there must have been at least one moment in time where the speed of my car was equal to my average speed.
What is example of the Mean Value Theorem (MVT)?
500
In the My Walk With Rosie graph pictured above, v(t) = sin t in (0, 2π), the total distance is 2π.
What is