3.10 - Linear Approximation and Differentials

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3.10 - Linear Approximation and Differentials
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Linear (or Tangent Line) Approximations
For values close to a,
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Linear Approximation Example
1. Determine the linearization (another name for
linear approximation) of f (x) ln x at a 1.
2. Find the linear approximation of the function
and use it to approximate the real numbers
and
Hint To determine the x-value to substitute into
L(x), simply set the original function equal to
what you are estimating and solve for x.
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Differentials
Up to now, weve thought of dy/dx as notation for
a derivative. We can think of dx and dy as
separate quantities called differentials.
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Differentials
We can now think of dy / dx as a ratio of two
quantities and separate them. So for a given
change in x (dx) we can calculate a change in y
(dy).
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Differentials Examples
3. Find the differential dy and evaluate dy for
the given values of x and dx.
4. The radius of a circular disk is given as 24
cm with a maximum error in measurement of 0.2 cm.
(a) Use differentials to estimate the maximum
error in the calculated area of the disk. (b)
What is the relative errors (dA / A)