Why is the change in y different than expected when using derivatives?

  • Context: High School 
  • Thread starter Thread starter Line
  • Start date Start date
  • Tags Tags
    Derivatives
Click For Summary

Discussion Overview

The discussion revolves around the understanding of derivatives and their application in estimating changes in the function y = x². Participants explore the discrepancies observed when using derivatives to predict changes in y for specific increments in x.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that derivatives and differentials apply to changes in x and y values, presenting a specific example with the equation y = x² and its corresponding values.
  • Another participant explains that a derivative represents an instantaneous rate of change, suggesting that changing x by 1 is too large to accurately reflect the behavior of the function at that point.
  • A question is raised about the applicability of derivatives, with a participant expressing confusion over their use for predicting changes in y when x is altered significantly.
  • Further clarification is provided that while derivatives can be used to approximate changes, their accuracy diminishes as the distance from the initial point increases.
  • One participant proposes an experiment to increment x by smaller values (0.1 and 0.01) to compare results with the definition of a derivative, hinting at the importance of limits in this context.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the application of derivatives, with some confusion remaining about their limitations and the conditions under which they provide accurate predictions. No consensus is reached on the best approach to resolve the discrepancies noted.

Contextual Notes

Participants highlight that the derivative is an instantaneous rate of change, which may not hold true for larger increments in x. The discussion does not resolve the mathematical steps or assumptions involved in the application of derivatives.

Line
Messages
216
Reaction score
0
I'm trying to understand something. Derrivatives are basically the same as differentials. The both apply to the chnange in XY values.

Now I have an eqaition y=x*x x*x=x sqaured.

So that being here are my XY values

X y
_______
1 1
2 4
3 9
4 16
5 25


So that being dv=2x . But if you plug that in x the change is different.

WIth that eqauition if x=1 the change in y should be 2. So if x changes by 1. But is chnage x by 1 y becomes 4.

If x=2 the change in y should be 4. But if we change x by 1 y=9 not 8.

What's going on?
 
Physics news on Phys.org
Because a derivative is an instantaneous rate of change, and changing x by 1 is a large enough change for the tangent line at the point to not be very close to the original function anymore.
 
So what is the derivative for? You can't just plug it in at any point and get the change for y?
 
Line said:
So what is the derivative for? You can't just plug it in at any point and get the change for y?

You can, BUT it is an instantaneous rate of change that is only exact at that point. If you try to use that derivative to find other points on the curve it will fail as the deviation from your original x increases, in other words let's say you have

y = f(x)

then
y' = f'(x)

and the differential is

dy = f'(x)dx

this allows you to approximate the original function by knowing its derivative and an initial point. But the further you go from the initial point, as dx increases, the approximation will become less valid because a derivative is an instantaneous rate of change and will be exact only at that point.
 
So that being here are my XY values

X y
_______
1 1
2 4
3 9
4 16
5 25


So that being dv=2x . But if you plug that in x the change is different.

WIth that eqauition if x=1 the change in y should be 2. So if x changes by 1. But is chnage x by 1 y becomes 4.

If x=2 the change in y should be 4. But if we change x by 1 y=9 not 8.

What's going on?
Let's experiment! What if, instead of increasing all of your x's by 1, you increased them by 0.1? So your table would look like:


X y
_______
1 ?
1.1 ?
1.2 ?
1.3 ?
1.4 ?


What if you did it by 0.01?


How do the results compare to the definition of a derivative? (You know, that stuff with limits!)

p.s. if y = x², then dy = 2x dx. (Of course, y' = 2x)
 

Similar threads

High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
High School Mass estimate only through mass rate of change
  • · Replies 4 ·
Replies
4
Views
2K
MHB Change of variables/ Transformations
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Why f is no longer a function of x and y ?
  • · Replies 32 ·
2
Replies
32
Views
4K
High School Confusion on Implicit Differentiation
  • · Replies 16 ·
Replies
16
Views
3K
High School Question about Limits and the Derivative
  • · Replies 29 ·
Replies
29
Views
4K
High School How does this derivative work? And why the speed of light remains?
  • · Replies 7 ·
Replies
7
Views
924
High School Second derivative differential equations in terms of y?
  • · Replies 4 ·
Replies
4
Views
2K