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
To determine the best linear approximation for f(x) = cos(x) near x = π/2, we need to find the equation of the tangent line to the graph of f(x) at x = π/2.
The equation of a straight line in slope-intercept form is given by y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the slope of the tangent line, we need to find the derivative of f(x). The derivative of f(x) = cos(x) is f'(x) = -sin(x).
At x = π/2, the slope of the tangent line is f'(π/2) = -sin(π/2) = -1.
Next, we need to find the y-coordinate of the point on the graph of f(x) at x = π/2. Plugging x = π/2 into f(x) = cos(x), we get f(π/2) = cos(π/2) = 0.
So, the point on the graph is (π/2, 0).
Now we have the slope (-1) and a point (π/2, 0), which allows us to determine the equation of the tangent line using the point-slope form:
y - y1 = m(x - x1)
y - 0 = -1(x - π/2)
y = -x + π/2
Comparing the given options, we find that Option b) y = -x + π/2 is the best linear approximation for f(x) = cos(x) near x = π/2.