Question

suppose that g is the linear function g(x)=3x-1.
* if x increases by ∆x, by how much does the value of g increases?
* Evaluate g'(x).

can you please explain to me this question and how to answer it?

Answer 1

\[g(x)=3x-1\]
"If \(x\) increases by \(\Delta x\), by how much the value of \(g\) increase?"
- This question asks you to find the difference between \(g(x+\Delta x)\) and \(g(x)\). "... \(x\) increasing by \(\Delta x\)" the same as \(x+\Delta x\).
\[\begin{align*}\color{red}{g(x+\Delta x)}-\color{blue}{g(x)}&=\color{red}{3(x+\Delta x)-1}-\color{blue}{\left(3x-1\right)}\\
&=3x+3\Delta x-1-3x+1\\
&=3\Delta x
\end{align*}\]
So the answer to this question would be something along the lines of, "If \(x\) increases by \(\Delta x\), then \(g(x)\) increases by \(3\Delta x\)."

"Evaluate \(g'(x)\)."
- Here you simply find the derivative of \(g\), either by using the limit definition of the derivative or any applicable rules.

Answer 2

Thank you.