Why does differentiation find the approximate value?

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Discussion Overview

The discussion revolves around the concept of differentiation and its application in finding approximate values, specifically in the context of calculating the cube root of numbers. Participants explore the reasoning behind why differentiation yields approximate rather than exact values and the conceptual understanding of derivatives.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions why using the formula dy=8+dy/dx*(8.03-8.00) provides an approximate value for the cube root of 8.03 instead of an exact value.
  • Another participant asks for a conceptual understanding of the derivative at the point 8.
  • A response explains that the derivative represents the slope of the tangent line to the curve y = x^{1/3}, suggesting that the tangent line provides values that are close but not identical to those on the curve.
  • It is noted that the tangent line at (8, 2) lies above the curve, indicating that the approximate values derived from the tangent line will be slightly larger than the actual values on the curve.
  • Some participants express concerns about their understanding of calculus, indicating that they have learned primarily through formulas without grasping the underlying concepts.
  • One participant highlights the importance of explanatory text in textbooks, suggesting that a lack of conceptual understanding may stem from an overemphasis on formulas.
  • A later reply discusses the idea that the tangent line resembles the curve near a specific point, using the example of the sine function to illustrate how tangent lines can provide rough estimates for function values.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concept of differentiation and its implications for approximation. There is no consensus on the best approach to learning calculus or the adequacy of textbooks in conveying these concepts.

Contextual Notes

Some participants indicate gaps in their understanding of calculus concepts, suggesting that their learning has been formulaic rather than conceptual. This may reflect limitations in their educational materials or personal study approaches.

Aichuk
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Like imagine I have to find the cube root of 8.03. So I cube 8 and use the dy=8+dy/dx*(8.03-8.00) formula. But why is this finding the value of cube root of 8.03 and why is this value approximate instead of exact?
 
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What is the derivative at 8, conceptually?
 
1/12
 
Aichuk said:
Like imagine I have to find the cube root of 8.03. So I cube 8 and use the dy=8+dy/dx*(8.03-8.00) formula. But why is this finding the value of cube root of 8.03 and why is this value approximate instead of exact?
Look at the graph of ##y = x^{1/3}##. The derivative, dy/dx, gives the slope of the tangent line to this curve. The formula you show gives you the y values on the tangent line, which is close to, but slightly different from the y values on the curve.

Since the tangent line at (8, 2) lies above the curve ##y = x^{1/3}## , the approximate values will be a little larger than the values on the curve.
 
Number Nine said:
What is the derivative at 8, conceptually?
Aichuk said:
1/12
The value of the derivative isn't what Number Nine was asking. He was asking about the meaning of the derivative at that point.
 
Your replies made me realize all the holes in my calculus knowledge. I've been taught calculus in school just through formulas so it looks like my concepts are very weak. I have no idea what to do about it because my textbook also only uses formulas.
 
Aichuk said:
Your replies made me realize all the holes in my calculus knowledge. I've been taught calculus in school just through formulas so it looks like my concepts are very weak. I have no idea what to do about it because my textbook also only uses formulas.
Most calculus texts contain formulas, but they usually contain explanatory text as well. Are you saying that your textbook doesn't have explanations to go with the formulas? Sometimes students focus on the formulas and ignore the surrounding text.
 
Basically it comes from this really simple idea: "The tangent line to ##x_0## resembles the curve near ##x_0##". For instance the curve defined by ##y=\sin x## resembles the tangent line to it at ##0## near zero:

LGMGc.jpg


For example it is really hard to determine ##\sin(0.1\,\rm rad)## without using a calculator. But since the tangent line has a simple form, namely ##y=mx+b##, one can easily exploit the fact that the tangent line resembles the curve of ##\sin x## near ##0## since ##0.1## is approximately zero to find a rough estimate for ##\sin(0.1\,\rm rad)##.
 
Mark44 said:
Most calculus texts contain formulas, but they usually contain explanatory text as well. Are you saying that your textbook doesn't have explanations to go with the formulas? Sometimes students focus on the formulas and ignore the surrounding text.

My textbook contains examples of HOW to use, but not WHY
 
  • #10
Aichuk said:
My textbook contains examples of HOW to use, but not WHY
Assuming each section of your textbook ends with a set of problems, maybe these problems are the WHY the formulas are used.
 

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