Midterm Flashcards
Definition of Continuity
A function in continuous if
- f(c) is defined
- lim x–>c f(x) exists
- f(c) = lim x–>c f(x)
IVT
Since f(c) is continuous on interval [a, b] and _ < c < _, then there is at least one value where f(c)=0
Critical Number
value such that f’(c)=0 or f’(c) is undefined
Rolle’s Theorem
Let f be a function that is continuous on [a,b] —a closed interval—, differenciable on (a, b) —an open interval—, and f(a) = f(b), then there is at least one value c on (a,b) such that f’(c)=0
Mean Value Theorem
Let f be a function that is continuous on [a,b] —a closed interval—, differenciable on (a, b) —an open interval—, and f(a) = f(b), then there’s at least one number c in (a,b) such that f’(c) = (f(b) - f(a)) / (b - a)
Extreme Values Theorem
A function that is continuous may have extreme values (max or min)
If the derivative is ____, the function is increasing.
positive
If the derivative is ____, the function is decreasing.
negative
If the derivative changes from positive to negative at x=c, then x=c is a ____.
local maximum
If the derivative changes from negative to positive at x=c, then x=c is a ____.
local minimum
If the 2nd derivative is positive on an interval I, then the graph of f(x) is concave ___ on interval I.
up
If the 2nd derivative is negative on an interval I, then the graph of f(x) is concave ___ on interval I.
down
If the graph of f(x) lies above the tangent lines on interval I, the graph is concave ___ on interval I.
up
If the graph of f(x) lies below the tangent lines on interval I, the graph is concave ___ on interval I.
down
The derivative of a position function is ___.
velocity
