On a graph
The derivative at a point.
What is the slope of the tangent line?
lim_(h->0) {3x + h - 3x}/{h}
What is 3?
For example, (x^2 + 2^x)' = 2x + 2^x ln 2
What is the sum rule?
The second derivative of the position function
What is acceleration?
The absolute value of the antiderivative of acceleration
What is speed?
It's what the sign of the first derivative reveals.
What is the direction?
or
What is increasing or decreasing?
lim_(h->0) {(x + h)^2 - x^2}/{h}
What is
2x
For example, d/dx x lnx = ln x + 1
What is the product rule?
P'(5) , where P(t) gives the population of mice in Oscar's kitchen, t days after December 1st, 2025.
What is the rate of change in mice per day on December 5th 2025.
The function whose first derivative is cos(x)
What is sin(x) ?
int_a^b f(x)\ dx
What is the net signed area between the curve and the horizontal axis between a and b ?
lim_(h->0) {sin(x+h) - sin(x)}/{h}
What is cos(x) ?
For example, d/dx y^2 = 2y dy/dx
What is the chain rule?
\int_1^4 v(t)\ dt, where v(t) is the velocity of a rocket, in feet per second, t seconds after launch.
What is the distance traveled by the rocket between 1 and 4 seconds?
The property of the first derivative that determines whether a function is increasing or decreasing
What is the sign?
An interval on which f''(x) > 0
What is concave up?
lim_(h->0) {(x+h)e^(x + h) - xe^x}/{h}
What is e^x + xe^x?
The main derivative rule used to find the derivative of tan(x)
What is the quotient rule?
The width w that minimizes cost of a fence surrounding a rectangular field with cost function
C(w) = 10w + 250/w
What is w=5 ?
A left or right Riemann one can be used to approximate a definite integral.
What is a sum?
A point where f'(x) = 0 and f''(x) < 0 .
What is a local maximum?
(using the second derivative test)
lim_(h->0) {cos^2(x+h) - cos^2(x)}/{h}
What is -2cos(x)sin(x) ?
For example,
lim_{x to 3} {x^2 - 4x + 3}/{3cos(x-3) - x} = lim_{x to 3}{2x-4}{-3sin(x-3)-1}
What is L'Hopital's rule?
The speed at which the top of a 5 foot ladder slides down a wall when the base of the ladder is 3 feet from the wall and sliding away from the wall at 1 foot per second.
What is 3/4 feet per second?
Triangles used to describe how shadows cast by moving skateboarders change related to the speed of the skateboarder
What are similar?