Theorums
A function, f(x), is continuous across a closed interval [a,b], it will include every value between f(a) and f(b) within that range.
What is the intermediate value theorem?
1) limit f(x) as x approaches c exists
2) limit f(x) as it approaches c-= limit f(x) as it approaches c+
3) f(c) = limit f(x) as x approaches c
what is a continuous limit?
the slope of a function found by using 2 points
(y2 - y1) / (x2 - x1)
What is average rate of change?
what is an integral?
a branch of mathematics that studies change, particularly instantaneous rate of change, and how things change over time
What is calculus?
A continuous function on a closed interval will reach both an absolute maximum and minimum value within that interval.
What is the extreme value theorem?
- f(c) is undefined (DNE)
- f(c) does not equal limit f(x) as x approaches c
what is a removable, hole discontinuity?
the slope (derivative) of a function at a point
What is instantaneous rate of change?
f'(x) = 8x
what is f(x) = 4x2 + c
*Average value question*
f(2) = 4
f(6) = 16
what is f'(x) = 3?
For a function that is continuous on a closed interval, [a,b], and differentiable on an open interval (a,b), within the closed interval there is a point where the function is instantaneous slope equals its average rate of change f'(c).
What is the mean value theorem?
limit f(x) as it approaches c- does not equal limit f(x) as it approaches c+
what is a non-removable, jump discontinuity?
f(x) = x2 + 4x + 78
what is f'(x) = 2x + 4?
f'(x) = sin(x) - 9x2 + 2
what is f(x) = cos(x) - 3x3 + 2x + c
*find the equation of the tangent line*
f(x) = 3x2 + 4x
g(3) = 7
what is y = 22x - 59?
If a function f is continuous on the closed interval [a,b], and differentiable on (a,b), and if f(a) = f(b), then there exists at least one point c in the open interval such that f'(c) = 0.
What is Rolle's theorum
- limit f(x) as it approaches infinity+ = +/- infinity
- limit f(x) as it approaches infinity- = +/- infinity
what is a non-removable, infinite discontinuity?
f(x) = x3 - 7x2 + 25x - 39
what is f'(x) = 3x2 - 14x + 25
the integral of a function, f'(x), from [4,10] = 5
f(4) = 3
what is f(10) = 8?
*find the area*
There is a function f(x) = x2 - 6x + 9 [0,3]
base is perpendicular to the x-axis
base is a square
What is 48.6?
the integral of f'(x) from [a,b] is equal to f(b) - f(a)
what is the fundamental theorem of calculus?
"I've reached my limit!"
What did the frustrated AP Calculus AB student yell?
f(x) = sin(2x) - cos(x2) + tan2(x)
what is f'(x) = 2cos(2x) + 2xcos(x)sin(x) + 2sec2(x)tan(x)
Find c:
f'(x) = 6x + 2
f(2) = 17
What is c = 1?
*find the volume*
There is a function f(x) = x2 - 6x + 9 [0,3]
Rotated about the x-axis
What is 152.68?