A point where f’(x) = 0 or is undefined; Used to find a maximum or minimum value.
What is a Critical Point?
100
When ƒ' is positive
ƒ is increasing
100
The absolute minimums and maximums on the graph of f (x) = (x^2) - 8x + 4
What is a minimum at 4?
100
ƒ(x) = x^2 - 5x + 7, [-1,3]
What is x=1?
100
The limit as x approaches 0 of (3x - x) / x
What is 2?
200
The highest maximum or lowest minimum values on a graph in a given interval (Can be endpoints on a close interval).
What are Absolute Extrema?
200
When ƒ'' and ƒ' are negative
ƒ is decreasing and concave down
200
The absolute minimums and maximums on the graph of f (x) = (x^2)(e^x)
What is a minimum at 2 and a maximum at -2?
200
ƒ(x) = x^3 - 6x^2 + 9x + 2, [0,4]
What is x=0.845,3.155?
200
The limit as x approaches 2 of (5x^2 - 10x) / (x - 2)
What is 10?
300
This happens when an interval in the second derivative is negative.
What is Concave Down?
300
The slope of the secant line equals the slope of the tangent line at point c in this Theorem.
What is Mean Value Theorem?
300
The absolute minimums and maximums on the graph of f (x) = ln((x^2) + 1)
What is a minimum at 0?
300
ƒ(x) = x^2 + sqrt(x + 2), [-2,2]
What is x=0.077?
300
The limit as x approaches infinity of (x^2 + 3x + 1) / (x lnx)
What is infinity?
400
Used when both the top and bottom of a fraction equal 0 or ∞ when the limit approaches 0.
What is L'Hopital's Rule?
400
1) ƒ(x) is continuous on [a, b] and
2) the first derivative exists on the interval (a, b) and
3) ƒ(a) = ƒ(b)
then there is at least one number x = c in (a, b) such that ƒ'(c) = 0.
What is Rolle's Theorem?
400
The inflection point(s) of the graph of f (x) = (2x^3) - (3x^2) - 4
What is 1/2?
400
ƒ(x) = (x^2)e^x, [-5,1]
What is 0.166?
400
The limit as x approaches infinity of xe^-3x
What is 0?
500
(y - 2) = 5(x - 3) at (3,2) where dy/dx = 5
What is a tangent line approximation?
500
It exists at ƒ(x) when ƒ'(x) is zero and ƒ''(x) is negative
What is a relative maximum?
500
The inflection point(s) of f (x) = (x^3) - (12x^2) + x + 2
What is 2?
500
ƒ(x) = sin(2x) + cos(x), [0,π]
-- (π is pi)
What is x=0.770,2.371?
500
The limit as x approaches infinity of
(arctan3x - π/2) / (arctan7x - π/2)
-- (π is pi)