Equation of the Tangent Line? (Derivatives)

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SUMMARY

The equation of the tangent line is defined as y = f(a) + f'(a)(x - a), where f(a) represents the function's value at point a and f'(a) denotes the slope of the function at that point. The term (x - a) accounts for the horizontal distance from the point of tangency, allowing the slope to be applied correctly over any distance from a. This formulation is essential for understanding how the tangent line behaves at points other than the origin, distinguishing it from the standard line equation y = mx + b.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically derivatives.
  • Familiarity with the equation of a line in slope-intercept form (y = mx + b).
  • Knowledge of function notation and evaluation (f(a)).
  • Concept of limits and continuity in functions.
NEXT STEPS
  • Study the concept of derivatives in depth, focusing on their geometric interpretation.
  • Learn about the application of the tangent line in real-world scenarios, such as physics and engineering.
  • Explore the relationship between differentiation and linear approximation.
  • Investigate higher-order derivatives and their significance in curve analysis.
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Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of derivatives and their applications in analyzing functions.

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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