Help with differentials

[Daddymomo1]

Find Delta y and dy for the values indicated.
y = 12.8/x, for x = -3, for Delta x = -0.003
So far, this is what I managed to solve

 

[topsquark]

Find Delta y and dy for the values indicated.
y = 12.8/x, for x = -3, for Delta x = -0.003
So far, this is what I managed to solve
View attachment 34368

Why do you keep bringing an "f" into it?
[imath]y(x) = \dfrac{12.8}{x}[/imath]

[imath]y(-3) = \dfrac{12.8}{-3}[/imath]

[imath]y + \Delta y = \dfrac{12.8}{x + \Delta x}[/imath]

And then dy = y'(x) dx.

-Dan

 

[blamocur]

Find Delta y and dy for the values indicated.
y = 12.8/x, for x = -3, for Delta x = -0.003
So far, this is what I managed to solve
View attachment 34368

You computation of [imath]f(x+\Delta x)[/imath] is wrong

 

[Steven G]

\(\displaystyle Since\ f(x) \neq x\ it\ follows\ that\ f(x+\Delta x) \neq (x+\Delta x)\)

 

[Daddymomo1]

What is the difference between "delta y" and "dy" ?

By dy I mean differential dy. My bad.

 

[Daddymomo1]

Why do you keep bringing an "f" into it?
[imath]y(x) = \dfrac{12.8}{x}[/imath]

[imath]y(-3) = \dfrac{12.8}{-3}[/imath]

[imath]y + \Delta y = \dfrac{12.8}{x + \Delta x}[/imath]

And then dy = y'(x) dx.

-Dan

is y’ = -12.8/x^2 ?

dy is the limiting value of "delta y"

Ah okay, didn’t know that.

\(\displaystyle Since\ f(x) \neq x\ it\ follows\ that\ f(x+\Delta x) \neq (x+\Delta x)\)

So if there an f I need to plug in 12/x?

I fixed my computation for f(x+deltax) and got this,
But I still feel like something isn’t right

I did it again this time using 4 decimal places instead of 3 and my answers for delta y and dy are almost the same. Here is my full solution:
Can someone check it for me or point out if there is something wrong? Thanks.
Edit: In my answer for dy, it’s -4.266666667x10^(-3). I accidentally wrote x^(-3).

 

[Steven G]

dy/dx = f'(x) so ( a little hand waving) dy = f'(x)dx
dy(a) = f'(a)dx

 

[blamocur]

I did it again this time using 4 decimal places instead of 3 and my answers for delta y and dy are almost the same. Here is my full solution: View attachment 34374
Can someone check it for me or point out if there is something wrong? Thanks.
Edit: In my answer for dy, it’s -4.266666667x10^(-3). I accidentally wrote x^(-3).

Looks close Your results for [imath]dy[/imath] and [imath]Delta y[/imath] have different numbers of significant digits, so the difference between them looks larger than it actually is.

 

[Daddymomo1]

Alright thank you all for the help, I appreciate it!

 

[Steven G]

I did it again this time using 4 decimal places instead of 3 and my answers for delta y and dy are almost the same. Here is my full solution: View attachment 34374

On the left side, 2nd line from the last, you wrote -3^2 and replaced it with 9. Well -3^2 = -9, not 9!
Not sure where -3^2 came from.

 

[Daddymomo1]

On the left side, 2nd line from the last, you wrote -3^2 and replaced it with 9. Well -3^2 = -9, not 9!
Not sure where -3^2 came from.

x is -3, so I substituted it. I didn’t put 9 anywhere.

 

[Steven G]

x is -3, so I substituted it. I didn’t put 9 anywhere.

OK, I'll go with that. Then what happened to the -3^2? Where is it?

(12/-3^2)(-.003) is a positive number. How did you get a negative number? I'll answer that for you. You replaced -3^2 with 9 and then computed (12/9)(-.003) and got -4.266666667x10^(-3). Your error is that -3^2 = -9, not 9. Now go back and fix it correctly.

 

[Daddymomo1]

OK, I'll go with that. Then what happened to the -3^2? Where is it?

(12/-3^2)(-.003) is a positive number. How did you get a negative number? I'll answer that for you. You replaced -3^2 with 9 and then computed (12/9)(-.003) and got -4.266666667x10^(-3). Your error is that -3^2 = -9, not 9. Now go back and fix it correctly.

Its (-12.8/-3^2)(-.003) and I got that answer.

 

[Steven G]

I never said that you got the wrong answer. You made two mistakes that cancelled one another out. In the end you made two mistakes--whether they cancelled one another out does not undo your two mistakes.

Its (-12.8/-3^2)(-.003) and I got that answer. How come the 12.8 suddenly appears as -12.8???

I will still maintain that (12.8/-3^2)(-.003) is a positive value. After all, 12.8 is positive, -3^2 is negative and -.003 is negative. Now (+/-)(-) = +.

You should know that -3^2 and (-3)^2 do not give the same result!
Just consider 12-3^2 = 12 - 9 and not 12+9. Why is that? Because -3^2 is NOT 9

 

[topsquark]

Its (-12.8/-3^2)(-.003) and I got that answer.

Translation: You needed to write (-12.8/(-3)^2)(-.003)

-Dan

 

[Daddymomo1]

I never said that you got the wrong answer. You made two mistakes that cancelled one another out. In the end you made two mistakes--whether they cancelled one another out does not undo your two mistakes.

Its (-12.8/-3^2)(-.003) and I got that answer. How come the 12.8 suddenly appears as -12.8???

I will still maintain that (12.8/-3^2)(-.003) is a positive value. After all, 12.8 is positive, -3^2 is negative and -.003 is negative. Now (+/-)(-) = +.

You should know that -3^2 and (-3)^2 do not give the same result!
Just consider 12-3^2 = 12 - 9 and not 12+9. Why is that? Because -3^2 is NOT 9

Translation: You needed to write (-12.8/(-3)^2)(-.003)

-Dan

Ohh I see! I’m sorry, I didn’t know that (-12.8/-3^2) is not the same as (-12.8/(-3)^2). Not everything makes sense hahaha. Thank you all so much for all the help and clarifications. I appreciate it!