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4.4 Linearization and Differentials

4.4 Linearization and Differentials. Linearization. Linearization uses the concept that when you zoom in to the graph at a point the y-values of the curve are very close to the y-values of the tangent line. (3, f(3). 3. Linearization. We can approximate

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4.4 Linearization and Differentials

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  1. 4.4 Linearization and Differentials

  2. Linearization • Linearization uses the concept that when you zoom in to the graph at a point the y-values of the curve are very close to the y-values of the tangent line (3, f(3) 3

  3. Linearization • We can approximate without a calculator by using the tangent line to f(x) at (3, f(3)) which is (3, f(3) 3

  4. Linearization • We can approximate without a calculator by using the tangent line Approximate value Actual value

  5. Linearization Note that the closer x is to 3 the better the approximation

  6. is the standard linear approximation of f at a. To find the linearization of f(x) at the point (a, f(a)) Start with the point/slope equation: linearization of f at a The linearization is the equation of the tangent line, and you can use the old formulas if you like.

  7. Important linearizations for x near zero:

  8. Example • Find the linearization L(x) of f(x) at x = a. • Compare f(a + 0.1) to L(a + 0.1)

  9. Example • Find the linearization of at x = 0. Use it to approximate:

  10. Example • Find the linearization of y = cosx at x = 0. Use it to approximate cos(0.5).

  11. Example • Choose a linearization with center not at a but nearby so that the derivative will be easy to evaluate.

  12. Example • Use linearization to find the value of at x = 0.12 to the nearest hundredth.

  13. Hw: p. 229/1-5,11,13,14,37

  14. Differentials: When we first started to talk about derivatives, we said that becomes when the change in x and change in y become very small. dy can be considered a very small change in y. dx can be considered a very small change in x.

  15. Let be a differentiable function. The differential is an independent variable. The differential is: dy is dependent on x and dx

  16. Examples: find the differential

  17. Examples: a) Find dyb) Evaluate dy for the given values of x and dx • y = x2 – 4x ; x = 3, dx = 0.5 • y = xlnx; x = 1, dx = 0.1

  18. Differentials can be used to estimate the change in y when x = a changes to x = a + dx True ChangeEstimated Change

  19. Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in r very small change in A (approximate change in area)

  20. (approximate change in area) Compare to actual change: New area: Old area:

  21. Example • The earth’s radius is 3959 + 0.1 miles. What effect does a tolerance of + 0.1 miles have on the estimate of the earth’s surface area?

  22. Example • Estimate the volume of material in a cylindrical shell with a height of 15m, radius 10m and shell thickness 0.2m.

  23. HW: p.229/19-26,31-35,38,39,42

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