Determine the derivative of $y(x) = x^2 + 3x - 5$ in $x=2$.
$y'(2)=7$.
$y'(2)=3$.
$y'(2)=-7$.
$y'(2)$ cannot be determined with the given information.
Determine the derivative of $y(x) = x^2 + 3x - 5$ in $x=2$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$y'(2)=3$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$y'(2)=-7$.
Antwoord 3 correct
Fout
Antwoord 4 optie
$y'(2)$ cannot be determined with the given information.
Antwoord 4 correct
Fout
Antwoord 1 optie
$y'(2)=7$.
Antwoord 1 feedback
Correct: For the difference quotient with start value 2 and $\Delta x$ we need $y(2)$ and $y(2+\Delta x)$:
$$
\begin{align*}
y(2) &= 2^2 + 3\cdot2 - 5 = 5,\\
y(2+\Delta x) &= (2+\Delta x)^2 + 3(2+\Delta x) - 5 = 4 + 4\Delta x + (\Delta x)^2 + 6 + 3\Delta x - 5 = (\Delta x)^2 + 7\Delta x + 5.
\end{align*}
$$
Plugging this values into the difference quotient gives
$$
\dfrac{\Delta y}{\Delta x}=\dfrac{y(2+\Delta x)-y(x)}{\Delta x} = \dfrac{(\Delta x)^2 + 7\Delta x + 5 - 5}{\Delta x} = \dfrac{(\Delta x)^2 + 7\Delta x}{\Delta x}=\Delta x + 7.$$
If $\Delta x \rightarrow 0$, then $\tfrac{\Delta y}{\Delta x}\rightarrow 7$, hence $y'(2)=7$.

Go on.
Antwoord 2 feedback
Wrong: Pay attention when working out brackets. $(2+\Delta x)^2\neq 4 + (\Delta x)^2$.

See also Example.
Antwoord 3 feedback
Fout: Pay attention to the order of $y(2)$ and $y(2+\Delta x)$ in the nominator of the difference quotient.

See also Difference quotient and Example.
Antwoord 4 feedback
Wrong: $y'(2)$ can be determined.

See Example.