Consider the function $f(x) = 100 - 13x - x^2$. The difference quotient of this function is equal to 37 at $x=1$. Determine $\Delta x$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$\Delta x = -25\tfrac{1}{2}$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$\Delta x = -50$.
Antwoord 3 correct
Fout
Antwoord 4 optie
$\Delta x = 22$.
Antwoord 4 correct
Fout
Antwoord 1 optie
$\Delta x = -52$.
Antwoord 1 feedback
Correct: For the calculation of the difference quotient we need $f(1+\Delta x)$ and $f(1)$:
$$
\begin{align*}
f(1+\Delta x) &= 100 - 13(1+\Delta x) - (1+\Delta x)^2 \\
&= 100 -13(1+\Delta x) - (1^2 - 2\Delta x - (\Delta x)^2) \\
&= 100 - 13 - 13\Delta x - 1 - 2\Delta x - (\Delta x)^2 \\
&= 86 - 13\Delta x - (\Delta x)^2,\\[1mm]
f(1) &= 100 - 13\cdot 1 - 1^2 = 86.
\end{align*}
$$
The the difference quotient is
$$\dfrac{\Delta f}{\Delta x} = \dfrac{f(1+\Delta x)-f(1)}{\Delta x} = \dfrac{(86-15\Delta x - (\Delta x)^2) - 86}{\Delta x} = \dfrac{-15\Delta x - (\Delta x)^2}{\Delta x} = -15 - \Delta x = 37.$$
Solving this gives $\Delta x = -52$.
Go on.
$$
\begin{align*}
f(1+\Delta x) &= 100 - 13(1+\Delta x) - (1+\Delta x)^2 \\
&= 100 -13(1+\Delta x) - (1^2 - 2\Delta x - (\Delta x)^2) \\
&= 100 - 13 - 13\Delta x - 1 - 2\Delta x - (\Delta x)^2 \\
&= 86 - 13\Delta x - (\Delta x)^2,\\[1mm]
f(1) &= 100 - 13\cdot 1 - 1^2 = 86.
\end{align*}
$$
The the difference quotient is
$$\dfrac{\Delta f}{\Delta x} = \dfrac{f(1+\Delta x)-f(1)}{\Delta x} = \dfrac{(86-15\Delta x - (\Delta x)^2) - 86}{\Delta x} = \dfrac{-15\Delta x - (\Delta x)^2}{\Delta x} = -15 - \Delta x = 37.$$
Solving this gives $\Delta x = -52$.
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Wrong: Pay attention when working out the brackets. $(1+\Delta x)^2 \neq 1 + (\Delta x)^2$.
See also Example 2.
See also Example 2.
Antwoord 4 feedback