If f(2) = 2.5 and f'(2) = -2.5, then f(2.5) is approximately:
A. 2.5
B. -2.5
C. -2
D. 2
E. 1.25
f(2.5)=f(2)+f'(2)*.5
To approximate f(2.5), we can use the concept of linear approximation. The linear approximation is given by the equation:
L(x) = f(a) + f'(a)(x - a),
where a is the known point and x is the point we want to approximate.
Given that f(2) = 2.5 and f'(2) = -2.5, we can use the linear approximation at x = 2.5, using a = 2:
L(2.5) = f(2) + f'(2)(2.5 - 2).
Substituting the known values:
L(2.5) = 2.5 + (-2.5)(2.5 - 2).
Simplifying:
L(2.5) = 2.5 - 2.5(0.5).
L(2.5) = 2.5 - 1.25.
L(2.5) = 1.25.
Therefore, the approximate value of f(2.5) is 1.25.
Hence, the correct answer is E. 1.25.