Difference Quotient - average rate of change

[zebra1707]

Homework Statement

Original function f(x) = -x^2+5x-2

I have calculated the difference quotient as -2x-h+5

Then use difference quotient to calculate the average rate of change:

for the following x=2 and x=2+h

I need this to go on to the next part of the question and i wanted to ensure that I calculated this correctly

Homework Equations

f(x+h)-f(x)
h

The Attempt at a Solution

Change in x = 2+h-2 = h

Change in y = plugged f(2+h)-f(2) into the original function

= -h^2-4h-4+10+5h-2-4 = -h^2+h

Δy/Δx = (-h^2+h)/(h) = -h+1 or 1-h (not sure which format is correct)?

Plug answer back into to the difference quotient - answer is consistent with DQ formula

If this is correct I need to then answer the following question which is:

as h approaches 0 what numerical value does

f(2+h)-(2) approach
h

The question is where do I start?

Cheers

 

[tiny-tim]

hi zebra1707!

Δy/Δx = (-h^2+h)/(h) = -h+1 or 1-h

yes, and now the limit of that as h -> 0 is … ?

 

[zebra1707]

hi zebra1707!


yes, and now the limit of that as h -> 0 is … ?

Hi TT, that's where I am stuck, I am not sure as to the next step. Can I assume that the answer to the previous question is correct?

 

[tiny-tim]

to find the limit of a continuous function as h -> 0, you can always just put h = 0

(the only exception being that if that gives 0/0)

 

[Ray Vickson]

Homework Statement

Original function f(x) = -x^2+5x-2

I have calculated the difference quotient as -2x-h+5

Then use difference quotient to calculate the average rate of change:

for the following x=2 and x=2+h

I need this to go on to the next part of the question and i wanted to ensure that I calculated this correctly

Homework Equations

f(x+h)-f(x)
h

The Attempt at a Solution

Change in x = 2+h-2 = h

Change in y = plugged f(2+h)-f(2) into the original function

= -h^2-4h-4+10+5h-2-4 = -h^2+h

Δy/Δx = (-h^2+h)/(h) = -h+1 or 1-h (not sure which format is correct)?

Plug answer back into to the difference quotient - answer is consistent with DQ formula

If this is correct I need to then answer the following question which is:

as h approaches 0 what numerical value does

f(2+h)-(2) approach
h

The question is where do I start?

Cheers

Why did you bother to repeat the calculation of [f(2+h)-f(2)]/h? You already stated that you had calculated [f(x+h) - f(x)/h = -2x - h + 5. Why wouldn't you just put x = 2 in that formula?

As for whether to use 1-h or -h+1: well, what's the problem? They are exactly the same thing, written in different order.

Finally: surely you can see what happens to the value of 1-h as h gets smaller and smaller and smaller, eventually going to zero.

RGV

 

[zebra1707]

Ray

I appreciate your comments. However your reply is bordering on "well aren't you an idiot" I have used this forum for a few years now, and never once have I received a reply like this.

Not everyone fully understands calculus as you do - Just a simple " did you realize that you have repeated the calculation" and maybe an explanation as to where I was going wrong...

Sorry all I needed was a little guidance..

 

[Ray Vickson]

Ray

I appreciate your comments. However your reply is bordering on "well aren't you an idiot" I have used this forum for a few years now, and never once have I received a reply like this.

Not everyone fully understands calculus as you do - Just a simple " did you realize that you have repeated the calculation" and maybe an explanation as to where I was going wrong...

Sorry all I needed was a little guidance..

Well, the first question was asked in order to try to get you to "unblock" something. When I assisted students one-on-one in the office, they would often do just what you did, and I would ask them why they did that; usually that would lead eventually to a realization on their part that there is a relation between something like [f(x+h) - f(x)]/h and [f(1+h) - f(1)]/h, and that one really is just a special case of the other. Sometimes just asking the question leads to greater understanding. But, as I said, that was in a one-on-one situation in an office or a classroom.

The other statement about what happens to -1 + h in the limit h --> 0 could have been put another way, I admit. But the question still stands unchanged, and you really should try to answer it: what DOES happen to -1 + h as h gets smaller, and smaller, and smaller? Use some numbers if you have to. Draw a picture if you have to. Do whatever you feel most comfortable with, but try, try try to answer the question. That is how you learn---by overcoming difficulties.

RGV

 

[Mark44]

Why did you bother to repeat the calculation of [f(2+h)-f(2)]/h? You already stated that you had calculated [f(x+h) - f(x)/h = -2x - h + 5. Why wouldn't you just put x = 2 in that formula?

Ray

I appreciate your comments. However your reply is bordering on "well aren't you an idiot" I have used this forum for a few years now, and never once have I received a reply like this.

IMO, Ray's response was NOT bordering on "aren't you an idiot", so it seems to me that you are reading something into what he said that probably wasn't there.

 

[zebra1707]

Thank you Ray, I am a mature age student trying to nut things out for myself, I don't always have access to teachers, tutors etc - so its a challenge most of the time.

I appreciate the response.

Regards

 

[zebra1707]

This is still confusing me..

The equation represents the difference quotient
f(2+h)− f(2) /h

it represents the slope of the secant line through the points (2, f (2)) and
(2+h, f(2+h)) on the curve.

But i used this equation to find the DQ = -h+1 in the first place - feels like I am going around in circles

If you could point me in the direction of an example - that would be helpful

Regards

 

[Ray Vickson]

This is still confusing me..

The equation represents the difference quotient
f(2+h)− f(2) /h

it represents the slope of the secant line through the points (2, f (2)) and
(2+h, f(2+h)) on the curve.

But i used this equation to find the DQ = -h+1 in the first place - feels like I am going around in circles

If you could point me in the direction of an example - that would be helpful

Regards

Well, h can be positive or negative, but let's start with the positive case. For h = 0.1, DQ = -1 + .1 = -0.9. For h = 0.01, DQ = -0.99, and for h = 0.001, DQ = -0.999, etc. Now look at the case of negative h. For h = -0.1, DQ = -1.1. For h = -0.01, DQ = -1.01, etc. So, you can see what is happening as |h|---the absolute value of h----becomes smaller and smaller, going to zero.

Before, when I said h gets smaller and smaller, I really meant |h|, but if h > 0 it is the same thing anyway.

RGV

 

[Mentallic]

This is still confusing me..

The equation represents the difference quotient
f(2+h)− f(2) /h

it represents the slope of the secant line through the points (2, f (2)) and
(2+h, f(2+h)) on the curve.

But i used this equation to find the DQ = -h+1 in the first place - feels like I am going around in circles

If you could point me in the direction of an example - that would be helpful

Regards

Actually,

\frac{f(2+h)-f(2)}{h}

and

1-h

are equivalent in this example. If you use a different function, you'll probably get a different answer.

So you have that the secant between the point (2,f(2)) and (2+h, f(2+h)) have a gradient of 1-h, and now we want to do what calculus is all about and that is to find the tangent, which is a secant that has two points which are infinitesimally close to each other (basically in the same spot), so we want to let h=0.

So the moral of the story is, while both of those expressions above are equivalent, when we let h=0, one of them gives us the result 0/0 which tells us nothing, and the other gives us the answer 1-0=1.

 

[zebra1707]

Thank you Mentallic, greatly appreciated.