Equation of the Tangent Line? (Derivatives)

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Discussion Overview

The discussion revolves around understanding the equation of the tangent line in calculus, specifically the formula y = f(a) + f'(a)(x - a). Participants explore the components of the equation, including the significance of the terms and the reasoning behind the structure of the equation.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the term (x - a) in the tangent line equation, questioning why it is not simply x.
  • Another participant explains that the slope f'(a) represents how much the line rises when moving from point a, and that (x - a) accounts for the distance from a to x.
  • A later reply suggests that in the equation y = mx + b, the term b represents the y-intercept when x = 0, and that the tangent line's equation allows for movement around the point a instead of starting at x = 0.
  • Another participant confirms this understanding by agreeing with the explanation provided about the y-intercept and the flexibility of the tangent line's starting point.
  • One participant points out a mathematical verification that shows how the equation holds true at the point x = a, reinforcing the correctness of the tangent line formula.

Areas of Agreement / Disagreement

Participants generally agree on the structure of the tangent line equation and its components, but there is some initial confusion regarding the term (x - a). The discussion reflects a process of clarification rather than a definitive resolution of all questions.

Contextual Notes

Some assumptions about the understanding of linear equations and derivatives may be implicit in the discussion. The participants do not fully explore the implications of the tangent line equation beyond its immediate application.

Who May Find This Useful

This discussion may be useful for students learning about derivatives and the concept of tangent lines in calculus, as well as those seeking clarification on the components of linear equations.

Velo
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So, I can't wrap around my head of why the Equation of the Tangent Line is:
[M]y = f(a) + f'(a)(x - a)[/M]
I get it that it's the equation of a line, and so it should be something like [M]y = mx + b[/M]. I also understand why f(a) = b (since it's a point in that line) and why f'(a) = m (since it's the slope), but where did the (x-a) come from? Shouldn't it just be x?
Thanks for the help in advanced :T
 
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Velo said:
So, I can't wrap around my head of why the Equation of the Tangent Line is:
[M]y = f(a) + f'(a)(x - a)[/M]
I get it that it's the equation of a line, and so it should be something like [M]y = mx + b[/M]. I also understand why f(a) = b (since it's a point in that line) and why f'(a) = m (since it's the slope), but where did the (x-a) come from? Shouldn't it just be x?
Thanks for the help in advanced :T

Hi Velo!

The slope of a line is how much it goes up when we move 1 point to the right.
f'(a) is the slope at a.
We're not going 1 point to the right though, but as much as x is bigger than a, that is (x-a).
 
I like Serena said:
Hi Velo!

The slope of a line is how much it goes up when we move 1 point to the right.
f'(a) is the slope at a.
We're not going 1 point to the right though, but as much as x is bigger than a, that is (x-a).

Oh, I think I got it now... So in the equation [M]y = mx + b[/M], b is the y when [M]x = 0[/M] in that equation, correct? :o And then, since our starting point in the tangent line's equation doesn't actually have to be[M]x = 0[/M], we move around that point instead?
 
Velo said:
Oh, I think I got it now... So in the equation [M]y = mx + b[/M], b is the y when [M]x = 0[/M] in that equation, correct? :o And then, since our starting point in the tangent line's equation doesn't actually have to be[M]x = 0[/M], we move around that point instead?

Yep. (Nod)
 
Thanks a lot :3 Was really struggling with this for some reason, even though it was actually pretty simple >..<
 
You could have checked that y= f'(a)x+ f(a), at x= a, is y= f'(a)a+ f(a), NOT f(a). With y= f'(a)(x- a)+ f(a) when x= a. y= f'(a)(a- a)+ f(a)= f'(a)(0)+ f(a)= f(a).
 

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