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

f'(x) = 3x² - 8x

f'(3) = 3(3²) - 8(3) = 3

3

real question:

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

part b: 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?

The limit of the difference quotient is simply the derivative, so we can express this as follows:

So our function is:

Taking the derivative, we have:

So the correct option is:

The equation of the line that passes through the same point can be found as:

Where , so we need to find . Plug in that x-value in the function we have:

And the slope is:

Then, the equation of the line is:

Shown below

As you can see below, the graph of the function of is continuous. This is so because we have plotted a polynomial function whose domain is the set of all real numbers. So the function is defined at the point , so the derivative exists at this point, hence we can calculate a tangent line there. In conclusion, we get the graph shown below. The blue line is the tangent line while the red curve is the graph of

question: how many people can she serve?

answer: she can serve 16.66666667 people. so she can only give 16 people a complete serving.

question: does she have enough ice cream for her guests?

answer: no. explanation: she only has enough for 16 people to have a full serving. however, she can give a 17th person a partial serving.