A) Show that the linearization of f(x) = (1 + x)^k at x=0...

a) Show that the linearization of f(x) = (1 + x)^k at x=0 is L(x) = 1 + kx b) Use the linear approximation (1 + x)^k ? 1 + kx to find an approximation of the function f(x) = ?(1 - 2x^4) For values of x near 0 c) Use the linear approximation (1 + x)^k ? 1 + kx to estimate (1.018)^(1/3)

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Transcript

00:01 This problem, we first want to find a linearization.

00:06 So part 8, let f of x be equal to 1 plus x into the power k.

00:15 Find the linearization of our function f of x at x equals 0.

00:29 First we find the derivative of our function, and that's k times 1 plus x to the power of k.

00:35 Minus 1.

00:36 We're also told that k is some constant, well, it's not much more than the problem, but this assume k is some constant.

00:43 And the derivative of our function at 0, that's going to be k, is 1 plus 0 to the power of k minus 1.

00:51 So that's just simply k.

00:53 The linearization is l of x, which is the value of the function at 0 in this case, also derivative of our function at 0 times x minus 0.

01:04 So l of x the linearization would be f of 0 and from our function f of 0 is going to be 1 plus 0 to the power of k, which is 1 to the k power which is 1.

01:17 So this would be 1 plus the derivative of our function 0 which is k times x minus 0, which is simply k.

01:25 So that's the first part of the problem to show that the linearization is 1 plus k x.

01:30 For part b, now we are asked to use the linear approximation, use the fact that 1 plus k, sorry, 1 plus x to the power of k, that's better, is approximately equal to 1 plus kx.

01:55 And that's pretty much what part a was all about...

Question

a) Show that the linearization of f(x) = (1 + x)^k at x=0 is L(x) = 1 + kx b) Use the linear approximation (1 + x)^k ? 1 + kx to find an approximation of the function f(x) = ?(1 - 2x^4) For values of x near 0 c) Use the linear approximation (1 + x)^k ? 1 + kx to estimate (1.018)^(1/3)

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