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7 terms

# Calc Test 3

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Critical Number
let f be defined at c. If f '(c)=0 or if f is not differentiable at c, then c is a critical number of f
The First Derivative Test
Let c be a critical number of a function f which is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows
1. If f '(x) changes from negative to positive at c then f has a relative minimum at (c,f(c))
2. If f '(x) changes from positive to negative at c then f had a relative maximum at (c,f(c))
3. If f '(x) is both positive or both negative around c then f(c) is neither a relative minimum or relative maximum.
Point of Inflection
If (c,f(c)) is a point of inflection of f, than either f "(c)=0 or f "(c) does not exist. Concavity must change at c though.
Second Derivative Test
Let f be a function such that f '(c)=0 and the second derivative of f exists on the open interval containing c
1. If f "(c)>0 then f has a relative minimum at (c,f(c))
2. If f "(c)<0 then f has a relative maximum at (c,f(c))
3. If f "(c)=0 then the test fails. f may have a relative maximum or a relative minimum or neither. In such cases, use the First derivative Test
Rolle's Theorem
If
1. f(x) is continuous on the interval [a,b]
2. f(x) is differentiable on the interval (a,b)
3. f(a) = f(b)
then Rolle's Theorem can be applied
Mean Value Theorem
If
1. f(x) is continuous on the interval [a,b]
2. f(x) is differentiable on the interval (a,b)
then the statement f '(c) = [f(b) - f(a)] / [b-a] is true
11 step curve sketch
Intercepts
Symmetry
Domain and Range
Continuous
Vertical asymptotes
Differentiability
Extrema
Concavity
Points of Inflection
Horizontal Asymptotes
Slant Asymptotes