Question

http://edugen.wiley.com/edugen/courses/crs4196/art/qb/qu/ch0/EAT_1227298660014_0_33367326491234830.gif

Thats the graph of f' and assume f is continuous and f(0)=0
Find f(3)

Answer 1

Well, what does the value of f' tell you about f?

Answer 2

its the derivative?

Answer 3

Sweethert, keep it in chat please. this is for mmonish's question.

Answer 4

Yes, but what does the derivative tell you? What does it mean to be the derivative?

Answer 5

i have no idea..HELP

Answer 6

What is the derivative of the function f(x) = 5x + 3?

Answer 7

5

Answer 8

slope?

Answer 9

I can't see your graph

Answer 10

Right. The derivative is the slope of the curve at any given x value.

Answer 11

So, if the derivative is a constant, what does that mean about the curve of the original function?

Answer 12

@myinninaya: i don't know y

@polpak: the slope is ?

Answer 13

The slope is a constant.

Answer 14

What does the graph of f(x) = 5x+3 look like?

Answer 15

a line

Answer 16

yes

Answer 17

Right, and you can see that if you take the derivative of a line you will get a constant value for the slope.

\(y = mx + b \implies y' = m\)

Answer 18

yea

Answer 19

So look at the graph of f'. What is f' from x=0 to x=2?

Answer 20

straight line...so slope is 0

Answer 21

so you see a horizontal line?

Answer 22

yea

Answer 23

No, the value of f' over that interval is constantly 1. f' = 1 on the interval from 0 to 2.

Answer 24

Which means that f must have a constant slope of 1 over that interval.

Answer 25

ok

Answer 26

So if f(0) = 0, what will f(2) be?

Answer 27

1?

Answer 28

No. Imagine a line with a slope of 1, that goes through the origin.

Answer 29

Where will that line cross the x=2 line?

Answer 30

1/2

Answer 31

Each time x increases by 1, the function increases by 1. That's what it means to have a slope of 1. \(\frac{\text{change in f(x)}}{\text{change in x}}\)

Answer 32

= 1

Answer 33

As much as x changes, the function changes.

Answer 34

So if f(0) = 0, f(2) = ?

Answer 35

1/2

Answer 36

\(\frac{\Delta f}{\Delta x} = 1.\)
\(\Delta x = 2 \implies \Delta f =?\)

Answer 37

How much did x change from 0 to 2?

Answer 38

2

Answer 39

The function must also have changed by the same amount if the ratio equals 1

Answer 40

So how much does the function change over that same interval?

Answer 41

2?

Answer 42

Yes

Answer 43

2/2 = 1 = the slope.

Answer 44

So now we know f(2). What do we know about the slope over the interval from x=2 to x=3?

Answer 45

its negative?

Answer 46

Negative what specifically?

Answer 47

slope

Answer 48

We have a graph of f'. The value of f' is the slope of f.

Answer 49

What is the value of f' from x=2 to x=3?

Answer 50

-1

Answer 51

What is the slope of f from x=2 to x = 3?

Answer 52

It's not a trick question =)

Answer 53

The value of f' is the slope of f.

Answer 54

-1

Answer 55

Right.

Answer 56

So if the slope of f is -1 from 2 to 3. And the
\(\frac{\text{Change of f}}{\text{Change of x}}\) = Slope = -1,

What is the change in x?
What is the change in f?

Answer 57

1

Answer 58

1 is which?

Answer 59

x

Answer 60

Right. So

\[\frac{\text{Change in f}}{\text{Change in x}} = -1\]
\[\frac{\text{Change in f}}{1} = -1\]
Change in f =?

Answer 61

-1

Answer 62

Right. So f(2) = 2, and from 2 to 3, the slope of f is -1, then the value of f will change by -1, what is f(3)?

Answer 63

-2?

Answer 64

Is any of this making sense?

Answer 65

No f changes by -1. It goes down 1.

Answer 66

f goes down 1 every time x goes up 1. That's what it means to have a -1 slope. Just like having a slope of 1 means that f goes up 1 each time x goes up 1.

Answer 67

-1

Answer 68

So if f(2) = 2, and x goes up 1, what will f do?

Answer 69

f(3)=3

Answer 70

Your answers are all over the place. I'm not sure what is confusing you about what I'm asking, but clearly something is out of sync. Do you understand what I'm saying about slope?

Answer 71

If I have a line g(x) = x. What is the slope?

Answer 72

1

Answer 73

Ok, and what is g(2) and g(4) and g(5)?

Answer 74

2,4,5

Answer 75

Ok. And on the interval from x=2 to x=4, how much does x increase?

Answer 76

2

Answer 77

And g(x) over that interval increases how much?

Answer 78

2?

Answer 79

Yes

Answer 80

Change in g = g(4)-g(2)

Answer 81

just like change in x was 4-2

Answer 82

So as x increases by 1 , g increases by 1. That is what it means to have a slope of 1.

Answer 83

If we have h(x) = 2x, What is the slope?

Answer 84

i hope you know that this is an antiderivative problem

Answer 85

I do.

Answer 86

But there are some small holes in your understanding that make solving these very easy. If I fill the holes you can answer all these in less than a minute.

Answer 87

oh cool

Answer 88

What is the slope of h?

Answer 89

2

Answer 90

So what is h(2), h(4), and h(5)?

Answer 91

4,8,10

Answer 92

So each time x increases by 1, h increases by how much?

Answer 93

2

Answer 94

And what is the derivative of h?

Answer 95

2

Answer 96

So if I tell you that j(0) = 1, and the derivative of j (also called j') = 5, Can you tell me what j(3) =?

Answer 97

(remember that the derivative is the slope)

Answer 98

15