Linearization and Newton’s Method 4.5 Linearization and Newton’s Method

What you’ll learn about Linear Approximation Newton’s Method Differentials Estimating Change with Differentials Absolute, Relative, and Percent Change Sensitivity to Change Essential Questions How does approximation techniques work in engineering and science?

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What you’ll learn about Linear Approximation Newton’s Method Differentials Estimating Change with Differentials Absolute, Relative, and Percent Change Sensitivity to Change Essential Questions How does approximation techniques work in engineering and science?

Linearization Example Finding a Linearization 1.

Procedure for Newton’s Method Guess a first approximation to a solution of the equation f(x) = 0. A graph of y = f (x) may help. Use the first approximation to get a second, the second to get a third, and so on, using the formula

Using Newton’s Method Use Newton’s method to solve Graph it to get a good approximation. Therefore the solution appears to be – 0.3221853546.

Differentials Example Finding the Differential dy Find the differential dy and evaluate dy for the given value of x and dx.

Differential Estimate of Change Three types of changes that can be found. Absolute, Relative and Percent Change Absolute True Estimated Relative and Percent Change True Estimate Relative Percent .

Differential Estimate of Change

Example Estimating Change with Differentials The radius of a circle increases from a = 5 m to 5.1 m. Use dA to estimate the increase in the circle’s area A.

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