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3.10 - Linear Approximation and Differentials

3.10 - Linear Approximation and Differentials. Linear (or Tangent Line) Approximations. For values close to a ,. Linear Approximation – Example. 1. Determine the linearization (another name for linear approximation) of f ( x ) = ln x at a = 1.

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3.10 - Linear Approximation and Differentials

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  1. 3.10 - Linear Approximation and Differentials

  2. Linear (or Tangent Line) Approximations For values close to a,

  3. 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.

  4. Differentials Up to now, we’ve thought of dy/dx as notation for a derivative. We can think of dx and dy as separate quantities called differentials.

  5. 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).

  6. 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)

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