Question

Question Attached in Screenshot Below: Only Question 1: This is NOT a test.

Answer 1

Recall that, if the limit exists:

`f^(\prime)(x)=\lim _(h\rightarrow0)(f(x+h)-f(x))/(h).`

Now, we compute f(x+h):

`\begin{gathered} f(x+h)=4.5(x+h)^2-3(x+h)+2 \\ =4.5(x^2+2xh+h^2)-3x-3h+2 \\ =4.5x^2+9xh+4.5h^2-3x-3h+2 \\ =4.5x^2-3x+2+9xh+4.5h^2-3h \\ =f(x)+9xh+4.5h^2-3h\text{.} \end{gathered}`

Therefore:

`\begin{gathered} \lim _(h\rightarrow0)(f(x+h)-f(x))/(h)=\lim _(h\rightarrow0)(f(x)+9xh+4.5h^2-3h-f(x))/(h) \\ =\lim _(h\rightarrow0)(9xh+4.5h^2-3h)/(h)=\lim _(h\rightarrow0)(9x+4.5h^{}-3) \\ =9x-3. \end{gathered}`

Therefore:

`f^(\prime)(x)=9x-3.`