5-5 Day 2 Linearization and differentials

(change in linearization of f corresponds to a change dx) Estimating Change with Differentials  a + dx a We can approximate the change in a function using the change in the linearization f L @ a these are = * The differential df = f ' (x)dx  value of df at x = a is L (change in linearization of f corresponds to a change dx)

Def: Differential Estimate of Change Let f (x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is df = f ' (a)dx Ex 7) The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in the circle’s area A. Compare this estimate with the true change in A and find the approximation error. new: r = 10 dr = .1 old: error:

Calculating the Change in f As we move from x = a to x = a + dx … there are three different ways.   TRUE ESTIMATED Absolute Change Relative Change Percentage Change

dr = 1 abs. change = rel. change = Percent = 8.3% Ex 8) Inflating a bicycle tire changes its radius from 12 inches to 13 inches. Use differentials to estimate the absolute change, the relative change, and the percentage change in the perimeter of the tire. abs. change = rel. change = Percent = 8.3% Ex 9) Suppose the earth were a perfect sphere and we determined its radius to be 39590.1 miles. What effect would the tolerance 0.1 mi have on our estimate of the earth’s surface area?

Ex 10) How accurately should we measure the radius r of a sphere to calculate the surface area S = 4r2 within 1% of its true value? want How accurate? Radius must have an error of  .5%

homework Pg. 246 # 4, 16, 26, 29, 31, 33, 37, 39, 41, 45, 49