Clarification on the output of partial derivatives

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SUMMARY

The discussion clarifies a misunderstanding regarding the interpretation of partial derivatives in the context of the function Z = x² + xy + y². The user correctly calculated the partial derivative @z/@x at the point (0.2, 0.3) as 0.7, indicating the slope of the tangent line. However, the confusion arose from a statement in a Khan Academy video suggesting that every increase in X results in a constant increase of 0.7 in Z. This statement is inaccurate as it oversimplifies the behavior of the nonlinear function, which requires evaluation at each point for precise changes.

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  • Learn how to evaluate nonlinear functions at specific points
  • Explore the implications of partial derivatives on function behavior
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Students and educators in mathematics, particularly those studying calculus, as well as anyone seeking to deepen their understanding of partial derivatives and their implications in nonlinear functions.

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1. In the Khan academy video I watched on partial derivatives, I understand absolutely everything except for the last 20 seconds which confused me.

http://www.youtube.com/watch?v=1CMDS4-PKKQ

Using the formula:

Z = x² + xy + y²

@z/@x = 2x +y

x=0.2, y=0.3

2(.2) + .3 = .7

What I know this means: At an x value of 0.2 and a y constant of 0.3, the slope of the tangent line will be 0.7.

At the end of the Khan video though, he stated:

"So everytime X increases, 1, Z will increase by 0.7"

Did I miss something or is this a mistake? What is in the video is a curve that is constantly changing so the slope will always be different. The only way to say that every increase in X will result in 0.7 is if we are dealing with a planar surface correct?
 
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Yes, your interpretation is correct. He misspoke and should have said z increases approximately 0.7 units from its value at (0.2, 0.3) if x increases from 0.2 to 1.2 using a linear approximation from that point. The linear approximation deviates from the curve; the only way to get the actual increase is to evaluate the nonlinear function z at each point.
 
Thanks. I was frustrated thinking that I didn't understand some crucial general concept but now am good.
 

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