Determine the derivative of $y(x) = -2x^2-5x - 1$ in $x=-1$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$y'(-1) = -5$
Antwoord 2 correct
Fout
Antwoord 3 optie
$y'(-1) = 1$.
Antwoord 3 correct
Fout
Antwoord 4 optie
$y'(-1)$ does not exist, because the difference quotient goes to infinity if $\Delta x$ goes to 0.
Antwoord 4 correct
Fout
Antwoord 1 optie
$y'(-1)=-1$.
Antwoord 1 feedback
Correct: For the difference quotient with start value $x=-1$ and $\Delta x$ we need $y(-1)$ and $y(-1+\Delta x)$:
$$
\begin{align*}
y(2) &= -2\cdot(-1)^2 - 5\cdot(-1) - 1 = 2,\\
y(-1+\Delta x) &= -2(-1+\Delta x)^2 - 5(-1+\Delta x) - 1 = -2(1 - 2\Delta x + (\Delta x)^2) -5(-1+\Delta x) - 1 \\
&= -2 +4\Delta x -2(\Delta x)^2 +5 -5\Delta x -1 = -2(\Delta x)^2 -\Delta x + 2.
\end{align*}
$$
Plugging these values into the difference quotient gives
$$
\dfrac{\Delta y}{\Delta x} = \dfrac{y(-1+\Delta x)-y(-1)}{\Delta x} = \dfrac{-2(\Delta x)^2 -\Delta x + 2-2}{\Delta x} = \dfrac{-2(\Delta x)^2 -\Delta x }{\Delta x} = -2\Delta x - 1.
$$
If $\Delta x \rightarrow 0$, then $\tfrac{\Delta y}{\Delta x} \rightarrow -1$, hence $y'(-1)=-1$.
Go on.
$$
\begin{align*}
y(2) &= -2\cdot(-1)^2 - 5\cdot(-1) - 1 = 2,\\
y(-1+\Delta x) &= -2(-1+\Delta x)^2 - 5(-1+\Delta x) - 1 = -2(1 - 2\Delta x + (\Delta x)^2) -5(-1+\Delta x) - 1 \\
&= -2 +4\Delta x -2(\Delta x)^2 +5 -5\Delta x -1 = -2(\Delta x)^2 -\Delta x + 2.
\end{align*}
$$
Plugging these values into the difference quotient gives
$$
\dfrac{\Delta y}{\Delta x} = \dfrac{y(-1+\Delta x)-y(-1)}{\Delta x} = \dfrac{-2(\Delta x)^2 -\Delta x + 2-2}{\Delta x} = \dfrac{-2(\Delta x)^2 -\Delta x }{\Delta x} = -2\Delta x - 1.
$$
If $\Delta x \rightarrow 0$, then $\tfrac{\Delta y}{\Delta x} \rightarrow -1$, hence $y'(-1)=-1$.
Go on.
Antwoord 2 feedback
Wrong: Be careful when working out brackets. $(-1 + \Delta x)^2 \neq 1 + (\Delta x)^2$.
See also Example.
See also Example.
Antwoord 3 feedback
Wrong: Pay attention to the order of $y(-1)$ and $y(-1+\Delta x)$ in the nominator of the difference quotient.
See also Difference quotient and Example.
See also Difference quotient and Example.
Antwoord 4 feedback
Wrong: Be careful when working out brackets and when dealing with minus-signs. You probably made a mistake rewriting.
See also Example.
See also Example.