derivative basics
What is the derivative of f(x)=x2
f(x) = 2x
What test tells you if a function has a max or min by analyzing sign changes in f′(x)?
First Derivative Test
What does the second derivative tell us about the graph?
Concavity
what rule do you use to differentiate f(x)=(3x+2)4?
chain rule
If f′(x) changes from positive to negative at x=2, what does x=2 represent?
Local maximum
What does it mean if f′′(x)>0 at a critical point?
Concave up → local minimum
Find the derivative of f(x)=sin(x)⋅x2?
f′(x)=cos(x)⋅x2+sin(x)⋅2x
What does it mean if f′(x) doesn’t change sign at a critical point?
It is not a max or min
Use the second derivative to classify x=2x=2 for f(x)=x3−3x2+4
Find f′′(x), test sign at x=2
What is a critical point?
A point where f′(x)=0 or f′(x) is undefined
Apply the First Derivative Test to f(x)=x3. What's the result?
No local max or min
What does it mean if f′′(x)=0?
It may be a point of inflection
What does it mean if f′(x)>0 on an interval?
The function is increasing on that interval
If a function increases on (–∞, 1) and decreases on (1, ∞), what is at x=1?
Local maximum
If f′(2)=0 and f′′(2)<0=, what does this indicate?
Local maximum at x=2