Define local linearity .

Local linearity is the property that the graph of a differentiable function looks more and more like a straight line the further you zoom in on a point. It is what allows the tangent at that point to stand in for the function nearby.

Why replace a function by its tangent line at all?

Because a linear function is far simpler to compute with than most others. The trade is accuracy, so the approximation is only worth using close to the point where the tangent touches the curve.

AP®MathsCollege BoardCalculus BCFlashcardsUnit 4: Contextual Applications of DifferentiationLinearization

Linearization (College Board AP® Calculus BC): Flashcards

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FrontApproximating Values of a Function
Define local linearity.

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Define local linearity.
Local linearity is the property that the graph of a differentiable function looks more and more like a straight line the further you zoom in on a point.
It is what allows the tangent at that point to stand in for the function nearby.
Fill in the two missing terms in the linearization of $f$ at $x = a$ :
$L (x) =______+______(x - a)$
The completed function is $L (x) = f (a) + f^{'} (a) (x - a)$ .
Both ingredients come from the single point $x = a$ : the function's value there, and its derivative there.
Use the tangent to $y = \sqrt{x}$ at $x = 64$ to approximate $\sqrt{65}$ .
The approximation is $8.0625$ .
Since $f (64) = 8$ and $f^{'} (64) = \frac{1}{16}$ , the linearization is $L (x) = 8 + \frac{1}{16} (x - 64)$ , so $L (65) = 8 + \frac{1}{16}$ .
True or False?
A tangent line approximation is equally accurate anywhere along the tangent.
False.
It is most accurate close to the point of tangency and gets steadily worse further away.
For $y = x^{3} - 4 x + 3$ at $(2, 3)$ the tangent is out by $0.0006$ at $x = 2.01$ but by more than $0.5$ at $x = 1.7$ .
Fill in the two missing words linking concavity to the accuracy of a tangent approximation:
where the graph is concave up the tangent gives an $______$ of the true value, and where it is concave down the tangent gives an $______$ instead.
The completed rule is: where the graph is concave up the tangent gives an underestimate of the true value, and where it is concave down the tangent gives an overestimate instead.
Concave up means $f^{''} (x) > 0$ and the curve bends away above the tangent; concave down means $f^{''} (x) < 0$ and it bends away below.
True or False?
The linearization $L (x)$ and the equation of the tangent at the same point are the same line.
True.
They are one line written two ways: $y - f (a) = f^{'} (a) (x - a)$ rearranged gives $L (x)$ exactly.
The name changes with what the line is being used for, not with the mathematics.
Without computing $\sqrt{65}$ , decide whether the tangent approximation at $x = 64$ is an over or an underestimate.
An overestimate.
The second derivative of $\sqrt{x}$ is $- \frac{1}{4} x^{- \frac{3}{2}}$ , which is negative for every $x > 0$ , so the graph is concave down and the tangent lies above it.
Why replace a function by its tangent line at all?
Because a linear function is far simpler to compute with than most others.
The trade is accuracy, so the approximation is only worth using close to the point where the tangent touches the curve.

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