The linear approximation at x = 0 to f(x) = \sin (5 x) is y =
at x=0, the slope is 5
SO, near x=0, y=5x is the linear approximation.
To find the linear approximation at x = 0 to the function f(x) = sin(5x), we can use the concept of linearization.
The linear approximation or linearization of a function at a particular point is given by the equation of a line that best approximates the behavior of the function near that point. The equation of this line is in the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.
To find the linear approximation at x = 0, we need to calculate the slope and the y-intercept.
The slope of the linear approximation can be found using the derivative of the function. In this case, f(x) = sin(5x), so we need to find the derivative of sin(5x) with respect to x.
Applying the chain rule, the derivative of sin(5x) with respect to x is:
f'(x) = 5 cos(5x)
Now that we have the derivative, we can find the slope of the linear approximation by evaluating f'(x) at x = 0:
m = f'(0) = 5 cos(5 * 0) = 5 cos(0) = 5
The y-intercept b of the linear approximation can be found by substituting x = 0 and y = f(0) into the equation y = mx + b:
f(0) = sin(5*0) = sin(0) = 0
Therefore, the linear approximation at x = 0 to f(x) = sin(5x) is y = 5x + 0, which simplifies to y = 5x.