Studied by 4 peopleCollins. *** are the more important ones.
Property of f:
x=a is a critical value (CV’s) of f
Fact about f’:
f’(a)=0 or f’(a) is undefined
Property of f:
f has a horizontal tangent line at x=a
Fact about f’:
f’(a)=0 (CV’s)
Property of f:
f has a relative maximum at x=a
Fact about f’:
f’ changes from positive to negative at x=a (number line w/ f’ on top and f on bottom)
Property of f:
f has a relative minimum at x=a
Fact about f’:
f’ changes from negative to positive at x=a (number line w/ f’ on top and f on bottom)
Property of f:
f is increasing at x=a
Fact about f’:
f’(a)>0 (number line w/ f’ on top and f on bottom)
***Property of f:
f has a point of inflection at x=a
***Fact about f’:
f’ changes from increasing to decreasing at x=a (number line w/ f’ on top a f on bottom)
***Property of f:
f has a point of inflection at x=a
***Fact about f’:
f’ changes from decreasing to increasing at x=a
(number line w/ f’ on top a f on bottom)
***Property of f:
f is concave up at x=a
***Fact about f’:
f’ is increasing a x=a
(number line w/ f’ on top a f on bottom)
***Property of f:
f is concave down at x=a
***Fact about f’:
f’ is decreasing at x=a
(number line w/ f’ on top a f on bottom)
***Property of f:
f has a point of inflection at x=a
***Fact about f”:
f” changes sign at x=a
(number line w/ f” on top a f on bottom)
***Property of f:
f is concave up at x=a
***Fact about f”:
f”(a)>0
(number line w/ f” on top a f on bottom)
***Property of f:
f is concave down at x=a
***Fact about f”:
f”(a)<0
(number line w/ f” on top a f on bottom)
Absolute maximums and minimums are found by
Making a table and plugging in CV’s and interval values into f(x)
f
s(t). Slope to find v(t).
f’
v(t). Or s’(t). Slope to find a(t). Area to find s(t).
f”
a(t). Or v’(t) or s”(t). Area to find v(t)
Analytically
Relative/local extrema (max/min) of f(x)
f’(x)=0 or f’(x) DNE —> CV’s of f(x). Get the CV’s from the derivative.
# line, w/ f’ on top and f on bottom and Cv’s on number line.
Analytically
Absolute/global extrema
f’(x)=0 or f’(x) DNE
Table with x (CV’s and end points), and f(x)
Analytically
Concavity/ Points of Inflection of f(x)
f’(x)=0 or f’(x) DNE. “candidates for P.O.I.”
# line ( w f” on top, f on bottom. CV’s on the # line)
Note
Flashcard