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Math 180

Math 180. 3.11 – Linearization and Differentials. We can approximate functions with tangent lines when close to the point of tangency. Since linear functions are relatively easy to work with, this can really help simplify calculations.

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Math 180

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  1. Math 180 3.11 – Linearization and Differentials

  2. We can approximate functions with tangent lines when close to the point of tangency. Since linear functions are relatively easy to work with, this can really help simplify calculations.

  3. Suppose we want to find the tangent line at . Then we have a point on the line, and the slope of the line, so using point-slope form, we get:

  4. Using function notation, we get what’s called the ____________of at : Note that when is close to , .

  5. Using function notation, we get what’s called the ____________of at : Note that when is close to , . linearization

  6. Using function notation, we get what’s called the ____________of at : Note that when is close to , . linearization

  7. Ex 1. Find the linearization of at .

  8. Linearization help approximate a function, but when we want to approximate the change in a function, we use what are called ____________.

  9. Linearization help approximate a function, but when we want to approximate the change in a function, we use what are called ____________. differentials

  10. For , the differential is an independent variable, and the differential is defined as:

  11. Ex 2. Find if . Then find when and . Now find the true change of the function, . Now calculate the approximation error, .

  12. Ex 2. Find if . Then find when and . Now find the true change of the function, . Now calculate the approximation error, .

  13. Ex 2. Find if . Then find when and . Now find the true change of the function, . Now calculate the approximation error, .

  14. Ex 3. Write a differential formula that estimates the change in the lateral surface area of a right circular cone when the radius changes from to and the height does not change.

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