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

Answer 1

Isaiah Allen

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]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.