8. Linearization and differentials

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Linear Approximation and Differentials

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Presentation transcript:

8. Linearization and differentials

Linearization Can approximate the value of a function by “linearizing” it (using the tangent line as an approximation) If the curve is concave up, the approximation will be an underestimate If the curve is concave down, the approximation will be an overestimate

steps Find the equation of the tangent line at the original point Plug in the new value of x into “x” in the equation of the tangent line This will give the approximation of the new y value Can also find the change in y ∆f = f’(a)∆x (this is called the differential)

Example 1 Estimate the 4th root of 17. Is it an over or under estimate?

Example 2 Approximate 3.015. Determine if it is an over or under estimate.

Example 3 - Differentials A machined spherical bearing was measured with a caliper. The bearing’s radius was found to be 2.3 inches with a possible error no greater than 0.0001 inches. What is the maximum possible error in the volume of the spherical bearing if we use this measurement for the radius?

Example 4 The position of an object is . Estimate the distance traveled over the time interval [3,3.025]