Application Of Differentiation - Ch 4 Flashcards
(27 cards)
Is f(-1) = 37 a local maximum? Is it a absolute maximum?
No, local cannot be at an endpoint. However this is the absolute maximum
What are the two conditions for the extreme value theorem?
Continuous and on closed interval
What are the min and max values on these graphs?
Can you locate local extremes by finding where derivative equals zero?
No. X^3 has derivative 0 at x=0, but it’s not an extreme.
However it is true that if there is a local extreme, and the derivative exists, then derivative is zero. Formats theorem.
Can there be a local extreme if the derivative does not exist?
Yes, y=|x|. Local min at 0, but derives no exists
What’s a critical number?
A number c in the domain of f such that either f’(c) = 0 or does not exist
How to find absolute max and min values of continuous function of a closed interval
Find f value at critical numbers
Find f value at endpoints
Largest and smallest are the extremes
What’s the mean value theorem
Significance is that there is some number in the middle where the instantaneous rate of change equals the average rate of change
How can derivatives determine if graph is concave upward or downward?
Con cavity test
How do you find points of inflection
Second derivative changes sign
How use derivatives to find local max or min
Second derivative test to determine if local min or max
What’s l’hospitals rule
Conditions: f and g are differentiator and g’ not equal zero
The limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Useful when limit is an indeterminate form eg 0/0 or infinity/infinity
Pic
Are these indeterminate:
1^(infinity)
0^(infinity)
Yes
No (its zero)
Are these indeterminate?
0^0
infinity^0
1^infinty
yes
infinity is a concept its not a number
pic
Get (inifinity * 0) but LHR only applies to quotient, so need to rewrite as a fraction
rewrite as fraction so can use LHR.
log each side
can move log into limit bc log is continuous
