Consider the function $y(x) = x^4$. Use the property of the derivative to determine the function value approximately if $x$ increases from $x=4$ to $x=4.5$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$y\approx 410.0625$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$y\approx 288$.
Antwoord 3 correct
Fout
Antwoord 4 optie
$y \approx 227.8125$.
Antwoord 4 correct
Fout
Antwoord 1 optie
$y\approx384$.
Antwoord 1 feedback
Correct: We use the property $\Delta y \approx y'(x) \Delta x$. Moreover, we know that
$$\begin{align*}
\Delta x &= 4.5 - 4 = 0.5\\
y(4) &= 256\\
y'(x) &= 4 \cdot x^{4-1} = 4x^3\\
y'(4) &= 256\\
\Delta y &\approx y'(4)\Delta x = 256 \cdot 0.5 = 128.
\end{align*}$$
The new function value is then approximately $y \approx 256 + 128 = 384$.
Go on.
$$\begin{align*}
\Delta x &= 4.5 - 4 = 0.5\\
y(4) &= 256\\
y'(x) &= 4 \cdot x^{4-1} = 4x^3\\
y'(4) &= 256\\
\Delta y &\approx y'(4)\Delta x = 256 \cdot 0.5 = 128.
\end{align*}$$
The new function value is then approximately $y \approx 256 + 128 = 384$.
Go on.
Antwoord 2 feedback
Wrong: You are asked to approximate the new function value, not to give it exactly.
See Property derivative and the corresponding Example.
See Property derivative and the corresponding Example.
Antwoord 3 feedback
Wrong: Did you determine the derivative of $y(x)$ correctly?
See Derivatives of elementary function.
See Derivatives of elementary function.
Antwoord 4 feedback
Wrong: Note that $x$ increases from $x=4$ to $x=4.5$; not the other way around.
See Property derivative and the corresponding Example.
See Property derivative and the corresponding Example.