In this activity, you will compute the limit of a difference quotient to determine the rate of change of a function at different points along its curve.

Part A

Consider the function f(x) =x³ 4x²+ 2. Calculate the limit of the difference quotient at x₀= 3 for f(x).

Part B

Find the equation of a line passing through the same point that f(x)passes through atx= 3with a slope equal to the limit you found in part A.

Part C

Graph the function f(x)and the line you identified in part B. How do the graphs relate to each other? What does the slope of the line represent about the function at the given point?