Theorems Flashcards
Mean Value Theorem (MVT): If f(x) is ____ on [a, b] and ___ on (a, b), then there exists a number c in (a, b) such that ____
continuous, differentiable, f’(c) = f(b) - f(a) / b - a
Application: This gives you the secant line, or the average slope of the entire function with the intervals determining the points for it
Extreme Value Theorem (EVT): If f is _____ on [a, b], then f has both a _____ and a _____.
continuous, absolute maximum, absolute minimum
Intermediate Value Theorem (IVT): If f is ____ on [a, b], and f(a) ___ f(b), then there is a value c in (a, b) such that f(c) ____.
continuous, does not equal, = k
2nd Derivative Test: if f’(c) = ___, and f’‘(x) > ___, then (c, f(c)) is a _______. If f’(c) = ____, and f’‘(x) < ____, then (c, f(c)) is a _____.
0, 0, relative minimum, 0, 0, relative maximum
What are Critical Numbers (two things) about f’(x):
f’(x) is undefined or equals 0
Rolle’s Theorem (Special case of MVT): if f is ____ on [a, b] and ___ on (a, b), and f(a) ___ f(b) there is at least one number c in which ______.
continuous, differentiable, =, f’(c)=0
Concave Upward: f’‘(x) > ____ and f’(x) is _____
0, increasing
Concave Downward: f’‘(x) < ____ and f’(x) is ____
0, decreasing
Particle Position: Moving right when v(t) ____, moving left when v(t) ____.
> 0, < 0
Particle Speed: Speeding up when v(t) ____ and a(t) ____. Slowing down when v(t) ____ and a(t) ___ OR VICE VERSA.
Speeding Up: > 0, > 0, OR < 0 < 0. (They must be the same)
Slowing Down: >0, < 0. (They must be opposite)
Candidates Test: Find ___, then set equal to __ OR ___. Plug in found ____ AND ___ into ___. Then determine ___ based on y values.
f’(x), 0, find where it’s undefined, x values, endpoints, f(x), extrema
First Derivative Test for Relative Minimum (on interval [a, b]): if f’(c) = ___ OR ____, f’(x) ___ from (a, f’(c)), and f’(x) ___ from (f’(c), b).
0, is undefined, < 0, > 0,
First Derivative Test for Relative Maximum (on interval [a, b]): if f’(c) = ___ OR ____, f’(x) ___ from (a, f’(c)), and f’(x) ___ from (f’(c), b).
0, is undefined, > 0, < 0,