Flashcards in Chapter 3.1-3.2 Deck (24)

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1

## What is the definition of derivative?

###
instantaneous rate of change

slope of tangent line

2

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

### lim f(x+h) -f(x) / h "as h approaches 0"

3

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

###
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

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?

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

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?

### 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)

6

## what is the definition of differentiability?

### able to get a derivative

7

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

### NO

8

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

###
1. A corner

2. A cusp

3. A vertical tangent

9

## how might f'(a) fail to exist?

###
1. a discontinuity

2. either corner, cusp, or vertical tangent

10

## how do you know if a function is a corner?

### left and right derivatives equal numbers

11

## 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

12

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

### infinity or negative infinity on both left and right sides

13

## how do you know if a function is discontinuous?

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

14

## 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

15

## if a function is differentiable is it always continuous?

### yes!

16

## if a function is continuous is it always differentiable?

### No!

17

## what is another word for vertical asymptote?

### infinite discontinuity

18

## 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"

19

## 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

20

## 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

21

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

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

22

## 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

23

## 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?

24