Chapter 3.1-3.2

Question 1

What is the definition of derivative?

Answer 1

instantaneous rate of change

slope of tangent line

Question 2

what equation do you use if it is a derivative of function?

Answer 2

lim f(x+h) -f(x) / h “as h approaches 0”

Question 3

what equation do you use to find the derivative of a point?

Answer 3

2 options:

1. lim f(x+h) -f(x) / h “as h approaches 0”
2. lim f(x)-f(c)/ (x-c) “as x approaches C

Question 4

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?

Answer 4

at the local minimum or local maximum the derivative graph will be at zero on the y axis

Question 5

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?

Answer 5

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)

Question 6

what is the definition of differentiability?

Answer 6

able to get a derivative

Question 7

if the left and right hand derivatives are different then does a derivative exist at what ever x value they ask you about?

Answer 7

NO

Question 8

what are the three types of continuous graphs that don’t have differentiability?

Answer 8

1. A corner
2. A cusp
3. A vertical tangent

Question 9

how might f’(a) fail to exist?

Answer 9

1. a discontinuity

2. either corner, cusp, or vertical tangent

Question 10

how do you know if a function is a corner?

Answer 10

left and right derivatives equal numbers

Question 11

how do you know if a function is a cusp?

Answer 11

left and right equal infinity or negative infinity. if one side is infinity then the other side is negative infinity

Question 12

how do you know if a function is a vertical tangent?

Answer 12

infinity or negative infinity on both left and right sides

Question 13

how do you know if a function is discontinuous?

Answer 13

one side equals a number and the other side equals negative infinity or infinity

Question 14

what is local linearity?

Answer 14

if you zoom in on a point to see if it looks like a line… if it does then it is differentiable

Question 15

if a function is differentiable is it always continuous?

Answer 15

yes!

Question 16

if a function is continuous is it always differentiable?

Answer 16

No!

Question 17

what is another word for vertical asymptote?

Answer 17

infinite discontinuity

Question 18

when you find the derivative from the left or right what do you use?

Answer 18

lim f(x+h) - f(x)/ h “as h approaches zero from the left/right”

Question 19

when we say as h approaches zero from the left what does that mean?

Answer 19

we want a small number because we wants what is closest to zero but it will still be negative

Question 20

What is the symmetric difference quotient?

Answer 20

(f(a +h) - f(a-h)) / ((a+h) - (a-h))…“where h is close to zero”
-calculator uses h= .001

Question 21

what does it mean to find the numerical derivative of f(x) at x= #

Answer 21

-just taking the slope of a # + .001 and # - .001

Question 22

when a calculator finds the symmetric difference quotient should it always be trusted? what can you do instead?

Answer 22

No!

• you can graph it (on computer or mentally)
• pay attention to asymptotes or holes

Question 23

what is it asking for when it says “what is the relationship between f and f’”

Answer 23

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?

Question 24

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?

Answer 24

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