Average Rate of Change
The formula used to compute the average rate of change of a function f(x) on the closed interval [a, b].
What is f(b) - f(a)⁄b - a?
Geometrically, the instantaneous rate of change at x = a is represented by the slope of this line.
What is a tangent line?
The limit of a difference quotient as h → 0 that defines f'(x).
What is lim h→0
f(x+h) - f(x)⁄h?
Gottfried Wilhelm Leibniz introduced this derivative notation using differentials.
What is dy⁄dx?
To estimate f'(3) from a table of values, you pick the two surrounding x-values and compute this ratio.
What is the average rate of change (or slope of the secant line)?
Geometrically, the average rate of change corresponds to the slope of this line connecting two points on a curve.
What is a secant line?
To transition mathematically from a secant line slope to a tangent line slope, you must take this Calculus operation as Δx → 0.
What is a limit?
The alternative limit definition used specifically to find the derivative of f(x) at a single point x = a.
What is limx→a
f(x) - f(a)⁄x - a?
Joseph-Louis Lagrange introduced this standard tick-mark notation for derivatives.
What is f'(x)?
Given t = [0, 2, 4] and s(t) = [10, 18, 30], this is the best estimate for s'(2) using a symmetric difference quotient.
What is 5?
If an object's position is s(t) = t2 + 2t meters, this is its average velocity from t = 1 to t = 3 seconds.
What is 6 m/s?
True or False: If a secant line between x = 1 and x = 5 has a slope of 4, the instantaneous rate of change at x = 3 must also equal 4.
What is False?
Using the definition of the derivative, this limit expression represents f'(2) for f(x) = x3
What is limh→0 ((2+h)3 - 8)⁄h?
If y = g(x), write the operator notation that tells you to "take the derivative with respect to x" of g(x).
What is d/dx[g(x)]?
or g'(x)
Given x = [1, 3, 5] and f(x) = [4, 10, 22], this interval provides the estimate for f'(2).
What is [1, 3]?