Given that f(−0.5) = 2 f�Œ(−0.5) = 4 , using a tangent line approximation you would estimate f(0) to be?
To estimate the value of f(0) using a tangent line approximation, we first need to find the equation of the tangent line at x = -0.5.
First, we can find the slope of the tangent line using the derivative of the function f(x). Since we are given the values of f(-0.5) and f'(-0.5), we can use these values to find the slope.
The equation of a tangent line is given by the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Given that f(-0.5) = 2 and f'(-0.5) = 4, we have:
y - 2 = 4(x - (-0.5)).
Simplifying the equation, we get:
y - 2 = 4(x + 0.5).
Next, we substitute x = 0 into the equation of the tangent line to approximate the value of f(0):
f(0) ≈ y = 4(0 + 0.5) + 2.
Calculating the equation, we get:
f(0) ≈ 4(0.5) + 2 = 2 + 2 = 4.
Therefore, using the tangent line approximation, we estimate f(0) to be 4.