Which of the following is the best linear approximation for
f(x)=cos(x) near x= π/2 ?
a) y= x - π/2
b) y= -x + π/2
c) y= -x + π/2 + 1
d) y= x - π/2 +1
y(π/2) = 0
y' = -sinx
y'(π/2) = -1
so, using the point-slope form of the line,
y = -1(x-π/2)
To determine the best linear approximation for f(x) = cos(x) near x = π/2, we need to find the equation of a line that closely approximates the curve of the function at that point.
The general equation of a line is y = mx + b, where m is the slope and b is the y-intercept. In this case, since we are looking for a linear approximation near x = π/2, we want the line to pass through that point.
To find the slope of the line, we differentiate f(x) = cos(x) with respect to x. The derivative of cos(x) is -sin(x). Evaluating the derivative at x = π/2, we have sin(π/2) = 1. Therefore, the slope of the line is 1.
Now we have the slope, 1, and we need to find the y-intercept. Plugging in the coordinates of the point x = π/2, y = cos(π/2) = 0 into the equation y = mx + b, we can solve for b:
0 = 1 * (π/2) + b
0 = π/2 + b
b = -π/2
So the equation of the line that best approximates f(x) = cos(x) near x = π/2 is y = x - π/2.
Comparing the options given, we see that the correct answer is a) y = x - π/2.