Return this question to its initial state

NUTB Incorporated is an internet$-$based company that makes money from reviewing and advertising new technological gadgets. Their profit $($in tens of dollars$)$ from reviewing and advertising $g$ new gadgets is given by the model $P(g).$ This model is graphed in black in the applet. The green dotted curve shows their marginal profit. The tangent line to point $A$ is also provided. Note that point $B$ is one horizontal unit to the right of $A($e$.$g$.,$ if point A has $g=1,$ then point $B$ has $g=2).$

Experiment with the applet and use it to answer the questions below. Respond to each question using COMPLETE SENTENCES.

$($part a$)$: Interpret $P(3)$ and $P^{\text{'}}(3).$ Include proper units and the answers to these questions: What is NUTB's total profit when $3$ gadgets are reviewed and advertised? At what rate is NUTB's profit changing when they review and advertise $3$ gadgets?

$($part b$)$: Following the directions from part $($a$),$ interpret $P(11)$ and $P^{\text{'}}(11).$

$($part c$)$: Enter $g=6$ into the applet. Explain how you can use the results for $P(6)$ and $P^{\text{'}}(6)$ to ESTIMATE $P(7).$ Make sure you use complete sentences and use only information from $g=6$ in your answer.

$($part d$)$: Explain how the tangent lines to the profit function $P(g)$ are related to the graph of the marginal profit function, $P^{\text{'}}(g)$