How do you find the linearization function of # f(x) = sin(4x) + cos(x)#?

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Answer 1 · 2017-07-26T10:37:48

We have:

# f(x) = sin4x+cosx #

Here we will obtain an linear approximation (about #x=0#) using Maclaurin Series from First Principles.

The Maclaurin Series is define by the the infinite Power Series in ascending powers of #x#:

# f(x) = f(0) + f'(0)x/(1!) + f''(0)x^2/(2!) + f^((3))(0)x^3/(3!) + ... #

Differentiating wrt #x#' we get:

# f'(x) = 4cos4x-sinx #

Thus putting #x=0# we get:

# \ f(0) = 0+1 = 1 #
# f'(0) = 4-0 = 4#

So the linear terms of the Maclaurin Series are:

# f(x) = 1 + 4x#

We can see this approximation compared with the function on this graph near #x=0#:
graph{(y-sin(4x)-cosx)(y-1-4x)=0 [-1, 1, -1, 2.5]}