Chapter 3.1-3.2 Flashcards
What is the definition of derivative?
instantaneous rate of change
slope of tangent line
what equation do you use if it is a derivative of function?
lim f(x+h) -f(x) / h “as h approaches 0”
what equation do you use to find the derivative of a point?
2 options:
- lim f(x+h) -f(x) / h “as h approaches 0”
- lim f(x)-f(c)/ (x-c) “as x approaches C
consider an equation f(x)…now if this equation has a local minimum or a local maximum what does that tell you about the derivative of that function?
at the local minimum or local maximum the derivative graph will be at zero on the y axis
consider an equation f(x)…now if as you see its graph and you see that the slope of the tangent line is negative what does that tell you about the derivative graph? what if the slope of the tangent line is positive?
if the slop of the tangent line of f(x) is negative then the derivative graph will be in the negative y values (or below the x axis)….if the slope of the tangent line of f(x) is negative then the derivative graph will be in the positive y values (or above the x axis)
what is the definition of differentiability?
able to get a derivative
if the left and right hand derivatives are different then does a derivative exist at what ever x value they ask you about?
NO
what are the three types of continuous graphs that don’t have differentiability?
- A corner
- A cusp
- A vertical tangent
how might f’(a) fail to exist?
- a discontinuity
2. either corner, cusp, or vertical tangent
how do you know if a function is a corner?
left and right derivatives equal numbers
how do you know if a function is a cusp?
left and right equal infinity or negative infinity. if one side is infinity then the other side is negative infinity
how do you know if a function is a vertical tangent?
infinity or negative infinity on both left and right sides
how do you know if a function is discontinuous?
one side equals a number and the other side equals negative infinity or infinity
what is local linearity?
if you zoom in on a point to see if it looks like a line… if it does then it is differentiable
if a function is differentiable is it always continuous?
yes!
if a function is continuous is it always differentiable?
No!
what is another word for vertical asymptote?
infinite discontinuity
when you find the derivative from the left or right what do you use?
lim f(x+h) - f(x)/ h “as h approaches zero from the left/right”
when we say as h approaches zero from the left what does that mean?
we want a small number because we wants what is closest to zero but it will still be negative
What is the symmetric difference quotient?
(f(a +h) - f(a-h)) / ((a+h) - (a-h))…“where h is close to zero”
-calculator uses h= .001
what does it mean to find the numerical derivative of f(x) at x= #
-just taking the slope of a # + .001 and # - .001
when a calculator finds the symmetric difference quotient should it always be trusted? what can you do instead?
No!
- you can graph it (on computer or mentally)
- pay attention to asymptotes or holes
what is it asking for when it says “what is the relationship between f and f’”
it is asking as f is increasing what is f’ doing? and as f decreases what is f’ doing? and as f has a local min or max what is f’ doing?
When given a graph and it says it is not differentiable at x= # and it says to compare left and right hand derivatives what do you do?
you set up the left hand derivative with the equations used on the left hand… lim f( x+h) - f(x) / h “as h approaches zero from the left/right” then you look at the graph and notice where the graph from the left/right can go so far. the number you get (an x value) is plugged in for x in this equation and it should get you your result. ONLY IF GIVEN A GRAPH AND IT SAYS COMPARE LEFT/RIGHT HAND DERIVATIVES TO PROVE IT IS NOT DIFFERENTIABLE
