Question

y = 2x − x2, x = 2, Δx = −0.5.
Δy= ?
dy= ?

Answer 1

\[
y = 2x - x^2 \\
f(x) = 2x - x^2 \\
f(2) = 2*2 - 2^2 = 4 - 4 = 0 \\
f(x+\Delta x) = f(2 - 0.5) = f(1.5) = 2*1.5 - 1.5^2 = 3.0 - 2.25 = 0.75 \\
\Delta y = f(x+\Delta x) - f(x) = 0.75 - 0.0 = 0.75 \\
\]\[
y = 2x - x^2 \\
\frac{dy}{dx} = (2 - 2x) \\
\text{At x = 2: } \\
\frac{dy}{dx} = (2 - 2*2) = -2 \\
dy = -2dx \\
dx \approx \Delta x = -0.5 \\
dy = -2 * (-0.5) = 1
\]