Bài tập 3 trang 192 SGK Toán 11 NC

Bài tập 3 trang 192 SGK Toán 11 NC

a) f (x) = ax + 3, cho x₀ một số gia Δx, ta có:

\(\begin{array}{l}
\Delta y = f({x_0} + \Delta x) - f({x_0})\\
 = a({x_0} + \Delta x) + 3 - (a{x_0} + 3) = a\Delta x\\
 \Rightarrow \frac{{\Delta y}}{{\Delta x}} = a \Rightarrow f'({x_0}) = \mathop {\lim}\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = a
\end{array}\)

b)

\(\begin{array}{l}
f(x) = \frac{1}{2}a{x^2},\Delta y = f({x_0} + \Delta x) - f({x_0})\\
 = \frac{1}{2}a{({x_0} + \Delta x)^2} - \frac{1}{2}ax_0^2\\
 = \frac{1}{2}a\Delta x(2{x_0} + \Delta x)\\
 \Rightarrow f\prime ({x_0}) = \mathop {\lim}\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\\
 = \mathop {\lim}\limits_{\Delta x \to 0} \frac{1}{2}a(2{x_0} + \Delta x) = a{x_0}
\end{array}\)

-- Mod Toán 11