Shape of Graphs Flashcards
How do you find the critical numbers?
set derivative equal to zero and solve for x
How do you find the absolute min & max of a function?
Plug critical numbers and end points into f(x)
Smallest result is abs min, largest is abs max
What are the Rolle’s Theorem Conditions?
- f(x) is continuous on closed interval
- f(x) is differentiable on open interval
- f(a)=f(b)
What does it mean if Rolle’s Theorem conditions are satified?
There exists at least 1 c in the open interval such that f’(c)=0
What do you do once all Rolle’s Theorem conditions are satisfied?
solve for critical numbers
What are the MVT conditions?
- f(x) is continuous on closed interval
- f(x) is differentiable on open interval
What does it mean if MVT conditions are satisfied?
There is a least 1 c on closed interval such that
f’(c) = (f(b)-f(a))/b-a
What do you do once verifying MVT conditions are satisfied?
set derivative = (f(b)-f(a))/b-a
and get x values (critical numbers)
What method(s) can be used to find local max & local min
1st derivative test
2nd derivative test
1st derivative test
Create intervals with critical numbers and test values in each using derivative
What do you plug into for sign analysis in 1st derivative test?
f’(x)
How do you find increasing and decreasing intervals?
Find critical numbers and perform sign analysis using original equation
2nd derivative test
plug c values into f’‘(x) to find local max & min
How do you find inflection poitns?
set top (and bottom if applicable) of f’‘(x)= 0
How do you find where the function is concave up/down?
create intervals with inflection points and plug test values into f’‘(x)
