Use a linear approximation for sin to find an approximation (this will not be linear) to the function sin(𝑥3) near 𝑥=0 .

Question

Use a linear approximation for sin to find an approximation (this will not be linear) to the function sin(𝑥3) near 𝑥=0 .

Answer 1

To use a linear approximation for the function sin(x) to approximate the function sin(x^3) near x = 0, we will need to find the first-degree Taylor polynomial of sin(x) and then substitute x^3 into it.

First, let's find the first-degree Taylor polynomial of sin(x) centered at x = 0. The Taylor polynomial of sin(x) is given by:

P₁(x) = sin(0) + cos(0)(x - 0)

Since sin(0) is equal to 0 and cos(0) is equal to 1, the Taylor polynomial simplifies to:

P₁(x) = x

Now, substitute x^3 into the Taylor polynomial to approximate sin(x^3):

P₁(x^3) = x^3

Therefore, an approximation to the function sin(x^3) near x = 0 is x^3.