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Applications of Differentiation - Coggle Diagram
Applications of Differentiation
Minimum
Absolute (or global)
Relative (or local)
If there is an
open
interval containing c on which f(c) is a minimum, then f(c) is a relative minimum. So, there's a relative minimum at (c,f(c)).
Theorem 3.2: Relative Extrema Occur Only at Critical Numbers
If f has a relative min or rel. max at x=c, then c is a critical number of f.
NOTE: the converse is not true; the critical numbers of a function need not produce relative extrema.
Definition: f(c) is a minimum of f on the interval I when f(c) is less than or equal to f(x) for all x in the interval
Maximum
Absolute (or global)
Relative (or local)
If there is an
open
interval containing c on which f(c) is a maximum, then f(c) is a maximum minimum. So, there's a relative minimum at (c,f(c)).
Definition: f(c) is a maximum of f on the interval I when f(c) is greater than or equal to f(x) for all x in the interval
Possible Intervals
[a,b]
[a,b)
(a,b]
(a,b)
(May be finite or infinite)
Theorem 3.1: The Extreme Value Theorem.
Conditions for Theorem: If f is
continuous
on a closed interval [a,b].
Conclusion of Theorem: Then f has both a minimum AND a maximum on the interval.
Critical Number
Definition: 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.
Guidelines for finding extrema on a closed interval
Find the critical numbers of f in (a,b)
Evaluate f at each critical number in (a,b)
Evaluate f at each endpoint of [a,b]
The least of these outputs is the minimum and the greatest is the maximum
Theorem 3.3: Rolles Theorem
Theorem 3.4: Mean Value Theorem
Statement
Conditions
Conclusion
Interpretations
What it means in terms of rate of change
What it means geometrically
Statement
Conditions
Conclusion
Theorem 3.6: The First Derivative Test (to determine relative extrema)
Concavity
Definition:
Theorem: 3.7 Test for Concavity
Point of Inflection
Definition:
Theorem 3.8: Points of Inflection (how to determine them)
Theorem 3.9: Second Derivative Test (to determine relative extrema)
Limits at Infinity (end behavior of function on an infinite interval)
Definition
Horizontal Asymptote
Note: functions that are not rational may have different left and right H.A.'s
Ex: (3x-2)/(sqr(2x^2+1))
Theorem 3.10: Limits at Infinity
Basically 1/x goes to zero as x goes to pos or neg infinity. The statement is more general for c/(x^r) with c any real and r any pos rational
Infinite Limits at Infinity
Strategies to evaluate
Discuss end behavior of function
Use long division to rewrite improper rational function as a sum of a polynomial and a rational function