Equation of the Tangent Line? (Derivatives)

In summary: 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).
  • #1
Velo
17
0
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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  • #2
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).
 
  • #3
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? :eek: 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?
 
  • #4
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? :eek: 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)
 
  • #5
Thanks a lot :3 Was really struggling with this for some reason, even though it was actually pretty simple >..<
 
  • #6
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).
 

1. What is the equation of the tangent line?

The equation of the tangent line is a mathematical expression that represents the slope of a curve at a specific point. It can be written in the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.

2. How is the equation of the tangent line related to derivatives?

The equation of the tangent line is directly related to derivatives. In fact, the slope of the tangent line at a point is equal to the value of the derivative at that point. This means that by finding the derivative of a function at a specific point, we can determine the slope of the tangent line at that point.

3. What is the process for finding the equation of the tangent line?

The process for finding the equation of the tangent line involves finding the derivative of the function at the given point. This derivative will give us the slope of the tangent line. Then, we can plug in the x-value of the given point into the derivative to find the y-value. This point, along with the slope, can be used to write the equation of the tangent line in the form y = mx + b.

4. Can the equation of the tangent line be used to approximate the value of a function?

Yes, the equation of the tangent line can be used to approximate the value of a function at a specific point. This is because the tangent line is a good approximation of the curve at that point. By plugging in the x-value of the point into the equation of the tangent line, we can get an estimate for the y-value of the function at that point.

5. Are there any limitations to using the equation of the tangent line?

Yes, there are limitations to using the equation of the tangent line. This method can only provide an approximation of the function at a given point and may not accurately represent the behavior of the function as a whole. Additionally, it may not work for functions that are not continuous or differentiable at the given point.

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