Use a linear approximation (or differentials) to estimate the given number. (Do
ID: 2860294 • Letter: U
Question
Use a linear approximation (or differentials) to estimate the given number. (Do not round your answer). (32.05)^4/5 Find the linearization L(x) of the function at a. f(x) = cos x, a = pi/2. Suppose that we don't have a formula for g(x) but we know that g{ 1) = -3 and g(x) = SQUAREROOTx^2 + 15 for all x. Use a linear approximation to estimate g(0.9) and g(1.1). Find the linear approximation of the function f(x) = SQUAREROOT4 - x at a = 0. Use L(x) to approximate the numbers SQUAREROOT3.9 and SQUAREROOT4.02. (Round your answers to four decimal places.)Explanation / Answer
solution- 4
Put f(x) = x^(4/5) and do a linearization about the point x_0 = 32. Note that 32.05 = x_0 + x where x = 0.05.
f(x_0 + x) f(x_0) + f '(x_0)x
f ' (x) = (4/5) x^(-1/5) ==> f ' (x_0) = (4/5)(32)^(-1/5) = (4/5)/2 = 2/5.
(32.05)^(4/5) (32)^(4/5) + (2/5)(0.05) = 16 + 1/50 = 16.02.
A calculator generated decimal approximation to the value is 16.019997
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