Question

Calculate the average rate of change for the function f(x) = x4 + 3x3 − 5x2 + 2x − 2, from x = −1 to x = 1.

Answer 1

Molly, would you please type out the formula for "average rate of change" and explain what makes up the numerator?

Answer 2

A(x)=f(b)-f(a)/b-a the numerator would be the function wouldn't it?

Answer 3

Should read: A(x)= [f(b)-f(a)]/(b-a)

Answer 4

That's what I had.

Answer 5

@mathmale

Answer 6

Molly, would you please explain in your own words what f(b) and f(a) mean (referring to the expression by DWC)?

Answer 7

Hint: f(x) = x4 + 3x3 − 5x2 + 2x − 2. Where do a and b come from?

Answer 8

f(b) is f(3) and f(a) is f(1)

Answer 9

Molly, please go back and review the original problem statement. Maybe you have a typo?

Answer 10

Molly, mine is not quite what you had, though you probably thought it was. Note my brackets and parentheses. f(b)/b - a is not f(b)/(b-a)

Answer 11

Molly: Your "f(b) is f(3) and f(a) is f(1)" needs revision. Hint: a=-1 and b=1. Where did those come from? Not trying to waste your time, but these preliminaries are essential in finding the correct answer to the problem you've posted.

Answer 12

I agree completely with DWC here:

"douglaswinslowcooper

Should read: A(x)= [f(b)-f(a)]/(b-a)"

Answer 13

Im so lost. :/ Okay so that formula has nothing to do with the a b and c that are in quadratic formulas?

Answer 14

His A(x) is the ratio

"how much y changes"
-------------------
"how much x changes"

Answer 15

No, nothing whatsoever to do with the quadratic formula.

Molly, can you think of a rate of change you encounter every time y ou drive a car?

Answer 16

Going up a hill?

Answer 17

Hint: your speedometer indicates speed in miles per hour, or, to paraphrase that,
distance covered
speed = ------------------------------ = average rate of change
time required to go that distance

Answer 18

DWC's formula indicates exactly that.

f(b)-f(a) represents how much your function changes as x goes from -1 to +1

b-a represents how much x changed

Answer 19

Since this seems new to you, I'll give you step by step instructions this time:

Evaluate f(x) = x4 + 3x3 − 5x2 + 2x − 2 at x=1 (first) and at x=-1 (second).

Answer 20

So itd be [f(-1)-f(1)]/-2

Answer 21

f(-1) would be -1^4+3(-1)^3-2(-1) right?

Answer 22

You're getting there! But order is important: f(1) - f(-1), and 1-(-1).
f(1) - f(-1)
So it'd be ------------------
2

You'll need to evaluate the function f(x) at both x=1 and x=-1.

How would you go about doing that?

Answer 23

substitute both 1 and -1 for x in the problem?

Answer 24

Yes, exactly. But as an aside: I used synthetic division to evaluate the function at -1 and 1.
What method would you normally use?

Answer 25

same. its a lot easier than long division, lol

Answer 26

f(x)= x4 + 3x3 − 5x2 + 2x − 2
ought to be written as:
f(x)=x^4 + 3x^3 - 5x^2 + 2x - 2.

If x=1,
f(1)= (1)^4 + 3(1)^3 + 5(1)^2 + 2(1) - 2 = ???

Answer 27

Please use synthetic division to evaluate f(1).

Answer 28

f(1) is 53

Answer 29

I'm going to type out my result, so that you have something with which to compare yours:

Answer 30

its 1x^3+4x^2+9x+7

Answer 31

Here's what I typed earlier:

If x=1,
f(1)= (1)^4 + 3(1)^3 + 5(1)^2 + 2(1) - 2 = ???

Note how I began with the FOURTH power of x term, whereas you began with the THIRD power of x. The original equation you typed in is of the fourth order (4th power of x is the highest power of x).

Answer 32

yeah but aren't you supposed to take a power away from the original equation?

Answer 33

No, not at all. We could discuss that if you like, but my immediate response is "no, we do ot take a power away from the original equation when evaluating the function at some particular x value."

Answer 34

I 'd thought you were going to evaluate f(1) using synthetic division.

I did that a few minutes ago and got the result f(1)=-1.

Answer 35

1 | 1 3 -5 2 -2
1 4 -1 1
1 4 -1 1 -1<= value of f(1)

Answer 36

Oh I got 11 because I did positive 5 instead of negative and forgot the positive two. Let me try and figure out the problem I think I got it now hold on.

Answer 37

Okay the answer is positive 5 correct?

Answer 38

I'll give you some time. Please find f(1), f(-1), f(1)-f(-1), and 1-(-1).

Then evaluate

f(1)-f(-1)
-----------
2-(-1)

Answer 39

Here's what I did earliler, using synth. div.:

1 | 1 3 -5 2 -2
1 4 -1 1
1 4 -1 1 -1<= value of f(1)

so f(1) = -1.

Have you any way of sharing your work with me, as by uploading an image of your work? I really need to see what you're doing before I can respond meaningfully.

Answer 40

I got it right its positive 5. Thank you so much for your help I honestly understand it now! :)

Answer 41

Very happy to work with you! See you again on OpenStudy. MM