The linear approximation at x = 0 to sin(8x) is
L(x) = A+Bx where
A =
B=
To find the linear approximation, we need to use the concept of the tangent line at a specific point on the curve. The slope of the tangent line at a point is the derivative of the function evaluated at that point.
To find A, the y-intercept of the linear approximation, we evaluate the original function sin(8x) at the given point x = 0. Since sin(0) = 0, the y-intercept A is also 0.
To find B, the slope of the tangent line, we use the derivative of the function sin(8x). The derivative of sin(8x) can be found using the chain rule. The derivative of sin(u) with respect to u is cos(u), and the derivative of 8x with respect to x is 8. Therefore, the derivative of sin(8x) with respect to x is 8cos(8x).
Now, we evaluate the derivative at x = 0 to get B. Substituting x = 0 into the derivative 8cos(8x), we get B = 8cos(0) = 8.
Therefore, the linear approximation L(x) = A + Bx is L(x) = 0 + 8x = 8x. So the value of A is 0 and B is 8.