Section 2.8 Local Linear Approximation Differential
If \(y = f(x)\) we define the differential of \(y\) to be \(dy = f'(x)dx\text{.}\)
The best approximation of a curve at any point is given by its tangent line.
Note 2.8.2.
TERMS:
increment: change in function values, (ex: \(\Delta y = f(x_0 + \Delta x) - f(x_0)\))
differential: change in value of tangent line, (ex: \(dy = f'(x_0)dx\))
locally linear: a function that is differentiable at some point \(x_0\text{,}\) is said to be locally linear at \(x_0\) (since the graph "looks like" its tangent line)
Recall that at a point \((x_0, f(x_0))\) on a function \(y = f(x)\text{,}\) the slope of \(f(x)\) is given by the derivative at this point, \(f'(x_0) = \dfrac{df}{dx} \rvert _{x = x_{0}}\text{.}\)
The derivative at a point gives the slope of the tangent line. Near \(x = x_0\text{,}\) \(f(x) \approx f_{\tan}(x) = f(x_0) + f'(x_0)(x - x_0)\text{.}\)
Example 2.8.1.
Find the tangent line of \(f(x) = \sqrt{x}\) at \(x_0 = 4\text{:}\)
\(\displaystyle f'(x_0) = \dfrac{d}{dx} \left( x_0^{\frac{1}{2}} \right) = \dfrac{1}{2\sqrt{x_{0}}}\)
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slope: \(f'(4) = \dfrac{1}{2\sqrt{4}} = \dfrac{1}{4}\)
point: \((x_0, f(x_0)) = (4,f(4)) = (4,2)\)
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\(f_{\tan}(x) = f(x_0) + f'(x_0)(x - x_0)\)
\(f_{\tan}(x) = 2 + \dfrac{1}{4}(x - 4)\)
CLAIM.
Near \(x_0 = 4\text{,}\) it is reasonable to approximate \(f(x) = \sqrt{x}\) by \(f_{\tan}(x) = 2 + \dfrac{1}{4}(x - 4)\)