Suppose the linear approximation for a function f(x) at a = 2 is given by the tangent line y = −2x + 10.What are f(2) and f '(2)?

The linear approximation of a function at a certain point is given by the equation of the tangent line at that point. The equation of the tangent line is given in the slope-intercept form y = mx + b, where m is the slope of the line (which is also the derivative of the function at that point) and b is the y-intercept.

In this case, the equation of the tangent line is y = -2x + 10.

1. To find f(2), we substitute x = 2 into the equation of the tangent line:

   y = -2(2) + 10 y = -4 + 10 y = 6

   So, f(2) = 6.

2. To find f '(2), we look at the coefficient of x in the equation of the tangent line. This is the slope of the tangent line, which is the derivative of the function at x = 2.

   So, f '(2) = -2.