The Formula of the Tangent Line
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Let's assume that we have a function $f(x)$ and its derivative <Sign in to see all the formulas>. Our goal here is to find the formula for the tangent line at any arbitrary point $x_0$.
Recall that the tangent line is the straight line whose slope is <Sign in to see all the formulas> that takes on the value <Sign in to see all the formulas> at $x_0$. See the figure below.
Since the tangent line is a straight line it has the formula <Sign in to see all the formulas>, our goal is to find the value of $m$ and $b$.
From the definition of the tangent line we know that <Sign in to see all the formulas> so the hard part is determining $b$. But we know that at $x_0$ the tangent line has the value <Sign in to see all the formulas>. Therefore, we know that at $x_0$, <Sign in to see all the formulas>. So now doing some algebra we can quickly find $b$:
<Sign in to see all the formulas>
Since <Sign in to see all the formulas> we see that
<Sign in to see all the formulas>
Now we can plug both $m$ and $b$ into the formula <Sign in to see all the formulas> to get the formula for the tangent line:
<Sign in to see all the formulas>
Finally, with a little algebra we get our final form of the formula for the tangent line:
<Sign in to see all the formulas> |
